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MICHIGAN LEAGUES OF ACADEMIC GAMES

Official 2017-2018 Tournament Rules

Basic EquationsBasic On-Sets

On-WordsAdventurous EquationsAdventurous On-Sets

Wff ‘n ProofLinguiSHTIKPropagandaPresidents

Current EventsTheme

MICHIGAN LEAGUES OF ACADEMIC GAMES2017-2018 OFFICIAL TOURNAMENT RULES

TABLE OF CONTENTS

BASIC EQUATIONS BE:1 -10

GUIDE TO BASIC EQUATIONSBE:11 - 18

BASIC ON-SETS BOS:19-28

ON-WORDSOW:29-36

ON-WORDS PHONETICS GUIDE AND 2-LETTER WORD LISTOW:37-38

ADVENTUROUS EQUATIONSAE:39-64

ADVENTUROUS EQUATIONS APPENDIX AND VARIATION SHEETS AE:65-70

ADVENTUROUS ON-SETS AOS:71-90

ADVENTUROUS ON-SETS APPENDIX AND VARIATION SHEETS AOS:91-96

WFF ‘N PROOF W:97-107

WFF ‘N PROOF APPENDIX W:108

LINGUISHTIK L:109-124

DICTIONARY OF TERMS AND ORDER OF PLAY SHEETSL:125-137

PROPAGANDA PG:138-139

PRESIDENTS PZ:140-141

CURRENT EVENTS CE:142-143

THEME TH:144-146

MLAG BASIC EQUATIONS® Tournament Rules 2017-18I. Starting a Match (Round)

A. Two- or three-player matches will be played. A match is composed of one or more shakes. A shake consists of a roll of the cubes and the play of the game ending with at least one player attempting to write an Equation that contains a mathematical expression that equals the Goal and correctly uses the cubes on the playing mat.

B. The following equipment is needed to play the game:1. 24 cubes: there are six of each color (red, blue, green, and black). Every face of each

cube contains either a digit (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) or an operation sign (+, –, x, ÷, * or ^, √).

2. A playing mat: this contains four sections:a. Goal: cubes played here form the Goal.b. Required: all cubes played here must be used in any Solution.c. Permitted: any or all cubes played here may be used in any Solution.d. Forbidden: no cube played here may be used in any Solution.

Comment Many game boards have a section labeled “Resources.” However, any reference in these rules to the “playing mat” or “mat” does not include the Resources section.

3. A one-minute sand timer: this is used to enforce time limits.4. A challenge block: this is a cube or similar object, not a flat object such as a coin. It

should not be so large that two players can grab it at the same time.C. Players may use only pencils or pens and blank paper. No prepared notes, books, tables,

calculators, cell phones, or other electronic devices may be used.In Minor, Elementary and Middle Divisions, players may use a preprinted chart for recording the Resources, Goal and Solutions.

D. The Goal-setter for the first shake is determined by lot. On each subsequent shake, the Goal-setter is the player immediately to the left of the previous Goal-setter.To determine the first Goal-setter, each player rolls a red cube. The player rolling the highest digit sets the first Goal. A player who rolls an operation sign is eliminated unless all players roll an operation sign. Players tied for high digit roll again until the tie is broken.

II. Starting a ShakeTo begin a shake, the Goal-setter rolls all 24 cubes. The symbols on the top faces of the rolled cubes form the Resources for the shake.

(a) A shake begins as soon as the timing for rolling the cubes is started or the cubes are rolled.(b) During a shake, no player may turn over a cube or obstruct the other players’ view of any cube. (See

Section IX-C.)

III. Legal Mathematical ExpressionsA. A legal mathematical expression is one that names a real number and does not contain

any symbol or group of symbols that is undefined in Equations.Example a ÷ 0 for any value of a does not name a real number. (See Section C below for additional

examples.)Comment An expression written on paper may contain pairs of grouping symbols such as parentheses,

brackets, or braces even though these do not appear on the cubes. These symbols indicate how the Equation-writer would physically group the cubes if the Solution were actually built with the cubes.

B. The symbols on the cubes have their usual mathematical meanings with the following exceptions:

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1. The + and – cubes may be used only for the operations of addition and subtraction; they may not be used as positive or negative signs.Examples +7, –8, 6x+4, and 17÷ (–8) are illegal expressions.

2. If the radical sign (√) is used, it must always be preceded by an expression to denote its index unless the index equals 2. If no index is shown, it is understood to be 2.Examples 2√9 or just √9 is legal and means “the square root of 9”; 1√2 means “the first root of 2,” which

is 2. (2+1)√8 means “the cube root of 8,” which is 2. 4x√9 means “4 times the square root of 9,” which is 4 x 3 or 12. 3√√9 means “the cube root of the square root of 9,” which is the sixth root of 9. (This expression is illegal in Elementary Division – see the “General Rule” in Section III-C-3.)

3. * (or ^) means exponentiation (raising to a power). The ^ cube must be written with the point up.Example 4*2 (or 4^2) means 42, which is 4 x 4 =16.

C. Expressions involving powers and roots must satisfy these requirements:1. Even-indexed radical expressions indicate only non-negative roots.

Examples √9 equals 3, not –3; 4√16 = 2 (not –2).

2. The following expressions are undefined. Note In all cases, * may be replaced by ^.

a. 0√a where a is any numberb. 0 * a where a ≤ 0c. a √ b where a is an even integer and b is negatived. (a ÷ b) √ c where c is negative and, when a ÷ b is reduced to lowest terms, a is an

even integer and b is an odd integere. a * (b ÷ c) where a is negative and, when b ÷ c is reduced to lowest terms, b is an

odd integer and c is an even integerExamples(a) (–8)4/6 is defined, as shown by the following steps. First reduce the fractional exponent to lowest

terms: (–8)4/6 = (–8)2/3. (–8)2/3 is of the form a * (b ÷ c) where a is negative. Since b is even and c is odd, (–8)2/3 is defined. (–8)2/3 = 3√(–8)2 = 3√64 = 4.

(b) (–4)2/4 is not defined because (–4)2/4 = (–4)1/2, which is of the form a * (b ÷ c) with a negative, b odd, and c even.Note The following reasoning is not allowed since the exponent is not reduced first: (–4)2/4 = 4√(–4)2 = 4√16 = 2.

(c) 3/6√(–9) is defined because 3/6√(–9) = 1/2√(–9), which is of the form (a ÷ b) √ c, with c negative. However, a is odd and b is even. So 3/6√(–9) = (–9)6/3 = (–9)2 = 81.

(d) 8/2√(–5) is not defined because 8/2√(–5) = 4√(–5), which is of the form a √ b where a is even and b is negative.

3. In the Elementary Division, if * (or ^) is used for raising to a power, both base and exponent must be whole numbers. If √ is used for the root operation, the index must be a counting number, and the base and total value must be whole numbers.Examples (in each case, * may be replaced by ^.)(a) 3 * 2 is acceptable and equals 9. 0 * 9 equals 0 and 7 * 0 equals 1. However, 2*(1–3), 4*(1÷2), (2–5)*4, and (2÷3)*3 are not legal in Elementary.(b) 2√9 or just √9 is acceptable and equals 3. 9√0 equals 0. However, √5 and 3√9 are not legal since neither is a whole number. Also 2√(1÷3), (1÷2)√5, and 3√(1–9) are illegal in Elementary.(c) The legality of √3*4 depends on its grouping. √(3*4) is legal; (√3)*4 is not.

Note counting numbers = natural numbers = positive integers = 1, 2, 3, 4, ...;whole numbers = 0, 1, 2, 3, 4, ....

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IV. Setting the Goal

A. The player who rolls the cubes must set a Goal by transferring the cube(s) of the Goal from Resources to the Goal section of the playing mat.

B. A Goal consists of at least one and at most six cubes that form a legal expression.1. Numerals used in the Goal are restricted to one, two or three digits. The use of

operation signs is optional. All cubes should be right-side up (not sideways or upside down).Examples of legal Goals: 6, 23, 8–9, 17x8, 19+8–5, 87÷13, 3√64, √49, 125Examples of illegal Goals: 23+18+7 (too many cubes), 45x

(does not name a number), +8 (does not name a number since + means addition).

2. The order of operations of mathematics applies to the Goal. The Goal-setter may physically group the cubes in the Goal to indicate how it is to be interpreted. Examples(a) 2x 3+5 (with space between x and 3) means 2 x (3 + 5).(b) 2x3 +5 (with space between 3 and +) means (2 x 3) + 5.

Comment The Goal-setter may not be able to remove all ambiguities from the Goal. Example √ 5+4 x9 where the Goal-setter wants to apply the √ to the entire expression (5+4)x9.

Declaring orally that the √ applies to everything that follows or extending the root over the entire expression in writing is not binding. For the purposes of the game, a root in this type of expression will refer only to the first term (in parentheses) in the expression.

3. Once a cube touches the Goal section, it must be used in the Goal.a. The Goal-setter indicates the Goal has been set by saying “Goal.”b. The Goal-setter may rearrange or regroup the cubes in the Goal section until he

says “Goal.”c. If the time runs out to set the Goal or the setter turns the timer, it has been set.d. The Goal may not be changed once it has been set.

C. Before moving the first cube to the Goal section of the mat, the Goal-setter may make a bonus move. To make a bonus move, the Goal-setter must say “Bonus,” then move one cube from Resources to Forbidden, and then move one or more cubes to the Goal.

D. If the Goal-setter believes no Goal can be set that has at least one correct Solution (see Section VII), he may declare “No Goal.” Opponents have one minute to agree or disagree with this declaration.1. If all players agree, that shake is void and the same player repeats as Goal-setter for a new shake.

Comments(a) The Goal-setter would declare “No Goal” only in those rare instances when an unusual set of

Resources was rolled. For example, there are fewer than three digit cubes or only one or two operation cubes.

(b) Players receive no points for the void shake.

2. An opponent who does not agree with the “No Goal” declaration indicates disagreement by picking up the challenge block (see Section VI-B) and challenging the “No Goal” declaration. She then has two minutes to write a correct Equation. If there is a third player, he also can choose to write an Equation.Note The Challenger (and third party, if he agrees) can use any cubes from Resources as needed. See

Section X for scoring.

V. Moving CubesA. After the Goal has been set, play goes in a clockwise direction (to the left).

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B. When it is your turn to play, you must either move a cube from Resources to one of the three sections of the playing mat (Required, Permitted, Forbidden) or challenge the last Mover.The move of a cube is completed when it touches the mat. Once a cube is legally moved to the mat, it stays in the section where it was played for the duration of the shake.

C. Any player may make a bonus move before making a regular move. To make a bonus move, the Mover must say “Bonus,” then move one cube from Resources to Forbidden, and then move another cube to Forbidden, Permitted or Required.Comments(a) If you do not say ‘Bonus’ before moving the first cube to Forbidden, the move does not count as a bonus

move but as a regular move to Forbidden. You are not entitled to play a second cube.(b) When making a bonus move, the first cube must go to Forbidden. The second (bonus) cube may be

moved to Required, Permitted, or Forbidden.

VI. ChallengingA. Whether or not it is your turn, you may challenge another player who has just completed

a move or set the Goal. The two main challenges are Now and Impossible. Note Players may also challenge a “No Goal” call, see Section IV-D-2.

1. By challenging Impossible, a player claims that no correct Equation can be written regardless of how the cubes remaining in Resources may be played.

Comments(a) If the Goal is not a legal mathematical expression (see Section III), an opponent should challenge

Impossible. Examples of such Goals are +8, 65+87–3, 122, and so on.(b) A Player who challenges “Never” will be considered to have challenged “Impossible”. There will be no

penalty for saying “Never” instead of “Impossible”.

2. By challenging Now, a player claims that a correct Equation can be written using the cubes on the mat and, if needed, one cube from Resources. A player may challenge Now if all of the cubes of her Solution are in Required or Permitted.a. A player may challenge Now only if there are at least two cubes in Resources.

If a player challenges Now with fewer than two cubes in Resources, the challenge is invalid and is set aside. (See Sections VIII and IX below.)Comment If only one cube remains in Resources and no one challenges Never, then a Solution is

possible using that one cube. Since the latest Mover had no choice but to play the second-to-last Resource cube to the mat, it is not fair that he be subject to a Now challenge. (However, an Impossible challenge could be made.) See Section VIII for the procedure to be followed when one cube remains in Resources.

b. Since a correct Solution must contain at least two cubes, it is illegal to challenge Now after the Goal has been set but before a cube has been played to Required or Permitted.If a player does so, the challenge is set aside and play continues.

B. A challenge block is placed equidistant from all players. To challenge, a player must pick up the block and say “Now” or “Impossible.”A player who picks up the block and makes a challenge against himself is not penalized and the challenge is set aside.Comments(a) The purpose of the block is to determine who the Challenger is in a shake.(b) Touching the challenge block has no significance. However, players may not keep a hand, finger, or

pencil on, over, or near the block for an extended period of time. (See Section IX-C.)(c) A player must not pick up the challenge block for any reason except to challenge. For example, don’t pick

it up to say “Goal” or to charge illegal procedure or when fewer than two cubes remain in Resources.

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VII. Writing and Checking EquationsA. After a valid challenge, at least one player must write an Equation.

1. After a Now challenge, the Challenger must write an Equation. (The Mover may not present an Equation.)

2. After an Impossible challenge, the Mover must write an Equation. (The Challenger may not present an Equation.)

3. After a challenge in a three-player match (and before any Equation is presented), the Third Party must indicate by the end of the two minutes for writing Equations whether she is presenting an Equation. The Third Party may not retract her decision once she has indicated whether or not she will present an Equation.Comment To indicate his intention on the challenge, the Third Party may:(a) state whether or not he will present an Equation;(b) indicate which party, Mover or Challenger, the Third Party is “joining” (agreeing with) on the

challenge. This can be done verbally or by pointing to the party;(c) present or not present an Equation when the time limit for writing Equations expires. If the Third

Party does not present an Equation, she is assumed to be joining the player who is not writing an Equation (Challenger on a Impossible or Mover on a Now).

B. To be correct, a Solution must be a legal mathematical expression (see Section III) that satisfies the following criteria:1. The Solution must be part of a complete Equation in this form:

Solution = GoalComment While Solution = Goal is the recommended form for writing the Equation,

Goal = Solution is acceptable. (See Appendix A for all matters involving how Equations are written.)

2. The Solution must equal the interpretation of the Goal that the Equation-writer presents with the Solution.

Examples (in each case * may be replaced by ^.)Goal Sample Equation Goal Sample Equation37 (6x6) + 1 = 37 11+5 (3x2)+(5x2) = 11+5

2x7+2 (5*2) – (4 + 5) = 2x7+2 3x 5+2 (5x4)+1 = 3x(5+2)

Comments(a) An Equation-writer who does not write a complete Equation (Solution = Goal), even when there is

only one interpretation of the Goal, is automatically incorrect.(b) The Equation-writer does not write the value of the Goal except in those cases where writing the

Goal is the same as writing its value.Examples(a) The Goal is 37 or 143.(b) For a Goal like 3x5+2, the writer must write 3x5+2 and not 17.(c) If the Goal is grouped, as in 3x 5+2 , an Equation-writer must write 3x(5+2) and not 3x 5+2 (with

space between x and 5 but no parentheses).

3. The Solution uses the cubes correctly.a. The Solution contains at least two cubes.b. The Solution uses all the cubes in Required.c. The Solution may use one or more cubes in Permitted.d. The Solution uses no cube in Forbidden.Comment Since several Resource cubes may show the same symbol, it is possible to have a 2 in

Forbidden that must not be used in the Solution at the same time that there is a 2 in Required that must be used.

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e. After a Now challenge, the Solution must contain at most one cube from Resources. This means a Solution must need one more (or zero more) cube from Resources.

f. After an Impossible challenge, any cubes in Resources are considered to be in Permitted and therefore may be used in the Solution.

4. The Solution contains only one-digit numerals.5. In Basic Equations, the general order of operations of mathematics (parentheses first,

roots/exponents next, then multiplication/division, finally addition/subtraction, going from left to right) applies to Goals and Solutions. If parentheses do not remove all ambiguity, use order of operations to interpret the value.

C. After the time for writing Equations has expired (or when all Equation-writers are ready), each Equation that is presented must be checked for correctness.1. All Equations must be presented before any is checked.

a. Once a player presents an Equation to the opponent(s), she may make no further corrections or additions even if the time for writing Equations has not expired.

b. Each Equation-writer must indicate the Equation to be checked, including the interpretation of the Goal. A writer who forgets to indicate the Equation must do so when asked .

2. Opponents have two minutes to check each Equation. When more than one Equation must be checked, they may be checked in any order. In a three-player match, both opponents must check a player’s Equation during the same two minutes. No other Equation should be checked during this time.Comments(a) When both players in a two-way match present Equations after the last cube has been moved (see

Section VIII below), only one Equation should be checked at a time.(b) Players must not physically move the cubes in Required, Permitted, and Resources to form the

Solution being checked. This causes arguments over where each cube was played.

3. Within the time for checking an Equation, opponents must accept or reject the Equation. A player who claims an opponent’s Equation is not correct must give at least one of the following reasons (or cite one of the reasons in Section VII-B). An Equation is correct if no opponent shows that it is incorrect.a. The Solution is NOT in the form Solution = Goal (or Goal = Solution)b. The Solution DOES NOT equal the interpretation of the Goal that the Equation-

writer presents with the Solution. c. The Solution DOES NOT use the cubes correctly.

(i) The Solution contains only one cube.(ii) The Solution DOES NOT use all the cubes in Required.(iii) The Solution uses a cube in Forbidden.(iv) After a Now challenge, the Solution needs more than one cube from

Resources.d. The Solution contains numerals with multiple digits.e. The Goal has no legal interpretation.

Examples(a) 7 ÷ 0 (b) A Goal containing more than six cubes or a four-digit number(c) A Goal with sideways or upside-down symbols on the cubes(d) Elem: 3 √ 9

f. The Equation-writer’s interpretation of the Goal is not a legal interpretation.BE- 6

Examples(a) The writer groups the Goal in an illegal manner:

e.g.: The Goal is grouped on the mat as 5x 3+4 and the writer interprets it as (5x3)+4. g. The Solution does not equal the Equation-writer’s interpretation of the Goal.

(i) Checkers must make an effort to determine whether the Solution equals the writer’s interpretation of the Goal before rejecting the Equation.

(ii) The checker can give a general argument that the Solution does not equal the Goal.Examples(a) The Goal is a fraction or an irrational number, but the solution equals an integer (or vice-

versa). (b) The Solution equals a value greater than 1000 when the Goal is 50x10. That is, the Solution

is clearly too big (or too small) even without calculating its exact value.

(iii) One or both of the checkers may ask a judge to determine whether the Solution equals the Goal. However, the checkers will be restricted in that No further objections to the Equation will be allowed even if the time limit for checking has not expired. If there are two checkers, both must agree that there are no other questions (cubes on the mat, parentheses, procedures, etc.), as this is the final question that a judge will answer.

VIII. Last Cube ProcedureA. If one cube remains in Resources, the next Mover must either play that cube to Required

or Permitted or challenge Impossible. When the cube has been moved, each player has two minutes to write an Equation.The last cube in Resources may not be moved to Forbidden. If a player does so, any challenge that is made is set aside and the cube is returned to Resources. There is no penalty unless the player’s time to move expires. (See Section XI.)

B. An opponent may challenge Impossible against the player who moved the last cube from Resources to Required or Permitted, provided the challenge is made by the end of the first minute for writing Equations. If the challenge is made, the Mover (and the Third Party if siding with the Mover) has the rest of the original two minutes to write an Equation.Comment Any Now challenge with one cube or zero cubes left in Resources is invalid, as is any

Impossible challenge made after the first minute for writing Equations. In both cases, the challenge is set aside and there is no point penalty.

IX. Illegal ProceduresA. Any action that violates a procedural rule is an illegal procedure. A player charging illegal

procedure must specify clearly (within 15 seconds) the exact nature of the illegal procedure.1. If a move is an illegal procedure, the Mover must return any illegally moved cube(s) to

their previous position(s) (usually Resources) and, if necessary, make another move.The Mover must be given at least 10 seconds to make this correction, unless the original move was made after the ten-second countdown (see Section XI-A-3 below), in which case the time limit rule (Section XI-A) is enforced. In general, there is no direct penalty except that the Mover may lose a point if he does not legally complete his turn during the time limit.Examples of illegal proceduresMoving out of turn, moving two cubes without calling “Bonus” before the first cube touches the mat in Forbidden, moving the last cube in Resources to Forbidden.

2. If the move is not an illegal procedure, the cube stands as played.Comment There is no penalty for erroneously charging illegal procedure. However, see Section IX-C

below if a player does so frequently.

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B. An illegal procedure is insulated by a legal action (for example, a move or challenge) by another player so that, if the illegal procedure is not corrected before another player takes a legitimate action, it stands as completed.

Example Suppose a player makes an invalid bonus move (such as moving two cubes to Required). Before anyone notices the illegal procedure, the next mover moves (or a valid challenge is issued). In this case, the illegal bonus move stays without penalty.

C. Certain forms of behavior interfere with play and annoy or intimidate opponents. If a player is guilty of such conduct, a judge will warn the player to discontinue the offensive behavior. Thereafter during that round or subsequent rounds, if the player again behaves in an offensive manner, the head judge may penalize the player one point for each violation after the warning. Flagrant misconduct or continued misbehavior may cause the player’s disqualification for that round or all subsequent rounds. The head judge may even decide to have the other two opponents replay one or more shakes or the entire round because play was so disrupted by the third party. In some cases, the head judge may order the shake replayed by all three players.Examples This rule applies to use of a cell phone, constant talking, tapping on the table, humming or

singing, loud or rude language, keeping a hand or finger over or next to the challenge block, making numerous false accusations of illegal procedure, and so on. It also includes not playing to win but rather trying only to ruin the perfect scores of one or both opponents (for example, by erroneously challenging Now or Impossible at or near the beginning of each shake so that both opponents will score 5 for the round), constantly charging illegal procedure erroneously, counting down the 10-second warning in an obnoxious manner, etc.

X. Scoring a ShakeA. After a challenge, a player is correct according to the following criteria:

1. That player had to write an Equation and did so correctly.If the Third Party agrees with the person who must write an Equation, the Third Party must write a correct Equation also.

2. That player did not have to write an Equation (someone else did), and no opponent wrote a correct Equation.Exception: After an Impossible Challenge in a three-player match, a player who does not present an Equation for a shake scores 2 if he accepts another player’s Equation as correct even if that Equation is subsequently proven wrong by the other checker.

B. After a challenge, points are awarded as follows:1. Any player who is not correct scores 2.2. Any player who is correct scores 6, unless that player is the Third Party agreeing with

the Challenger, in which case the score is 4.C. After the last cube from Resources is moved to the playing mat and no one challenges

Impossible, points are awarded as follows:1. Any player who writes a correct Equation scores 4.2. Any player who does not write a correct Equation scores 2.

D. A player who is absent for a shake scores 0 for that shake.

XI. Time LimitsA. Each task a player must complete has a specific time limit (listed below). The one- and two-

minute time limits are enforced with the timer. If a player fails to meet a deadline, he loses one point and has one more minute to complete the task. If he is not finished at the end of this additional minute, he loses his turn or is not allowed to complete the task.Note In Minor, Elementary and Middle Divisions, each one-point penalty must be approved by a judge

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Two-Player Matches PointsFirst place 6Tie for first 5

Second place 4

initialing the scoresheet.

1. The time limits are as follows:a. rolling the cubes and setting the Goal 2 minutesb. first turn of the player to the left of the Goal-setter 2 minutesc. all other regular turns (including any bonus moves) 1 minuted. stating a valid challenge after picking up the challenge block 15 secondse. deciding whether to challenge Impossible when no more cubes

remain in ResourcesIf an Impossible challenge is made, any time (up to a minute) that the Challenger took deciding to challenge counts as part of the two minutes for writing an Equation.

1 minute

f. writing an EquationDuring this time, the Third Party (if there is one) must decide whether to present an Equation after a Now or an Impossible challenge. At the end of these two minutes she must present her solution.

2 minutes

g. deciding whether an opponent’s Equation is correct 2 minutes2. Often a player completes a task before the time limit expires. When sand remains in

the timer from the previous time limit, the next player will receive additional time. An opponent timing the next player may either flip or not flip the timer so as to give the opponent the lesser amount of time before the remaining sand runs out and the next time limit can be started.

3. A player who does not complete a task before all the sand runs out for the time limit must be warned that time is up. An opponent must then count down 10 seconds loud enough for the opponent to hear. The one-point penalty for exceeding a time limit can be imposed only if the player does not complete the required task by the end of the countdown.The countdown must be done at a reasonable pace; for example, “1,010; 1,009; 1,008…”

B. A round lasts a specified amount of time (usually 30 minutes). When that time is up, players are told not to start any more shakes. Players have five minutes to finish the last shake. After these five minutes, players still involved in a shake in which no challenge has been made and one or more cubes remain in Resources will be told: “Stop, don’t move another cube – this is the end of the round. Each player has two minutes to write a correct Equation that may use any of the cubes remaining in Resources. Any player who presents a correct Equation scores 4 points for the shake; an incorrect Equation scores 2.”

XII. Scoring a MatchA. Each player is awarded points for the match based on the sum of his scores for the

shakes played during that match according to the following tables:

B. When a round ends, each player must sign (or initial) the scoresheet and the winner (or one of

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Three-Player Matches PointsFirst place 6

Two-way tie for first 5Three-way tie for first 4

Second place 4Tie for second 3

Third place 2

those tied for first) turns it in. If a player signs or initials a scoresheet on which his score is listed incorrectly and the error was a simple oversight, then, with the agreement of all players, correct the scores. However, if there is evidence that there was intent to deceive and the error was not a simple oversight, then do the following:1. If the error gives the player a lower score, she receives the lower score.2. If the error gives the player a higher score, she receives 0 for that round.

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RESOURCES

FORBIDDEN PERMITTED REQUIRED

GUIDE TO BASIC EQUATIONS

MATERIALSPlaying mat (see diagram), one-minute timer, 24 cubes (6 of each color)Red 0, 1, 2, 3, +, - Blue 0, 1, 2, 3, x, ÷Green 4, 5, 6, *, x, - Black 7, 8, 9, √, +, ÷

______________ GOAL

EXPONENTS AND ROOTSIn the game of Equations, the * or ^ cube is used to represent the exponent operation. (Note: older games used the * symbol, newer versions use the ^ symbol.) 52 is represented as 5 * 2 or 5 ^ 2 and has a value of 25. If the ^ symbol is used, the only acceptable orientation of the ^ is upward, not sideways or pointing down.

The √ symbol is used for the root operation. √ 81 represents the square root of 81 and has a value of 9. The expression 4 √ 81 represents the fourth root of 81 and has a value of 3.

NOTE: In the Minor and Elementary divisions (grade 6 and below) version of Equations, the following rules apply:If * or ^ is used for raising to a power, both base and power must be whole numbers (0, 1, 2, 3, 4, …). If √ is used for the root operation, the index must be a counting number (1, 2, 3, 4, …) and the base and total value must be whole numbers.

ROLLING FOR FIRSTTwo or three player games will be played. All players roll a red cube to determine who goes first. If a player rolls an operation, that player is out and does not reroll. The player with the highest number is the Goal Setter. If there is a tie for first place only, those players will roll again. After first is determined, the play continues clockwise. The player to the left of the Goal Setter is next.

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BEGINNING A SHAKEA shake consists of rolling the cubes, setting the Goal, and moving cubes until a Challenge is made and settled or the last cube is moved to the board (see Last Game Procedure). A shake begins as soon as the stall for rolling the cubes (turning the timer over and announcing “stall”) or a player has picked up the cubes to roll them. Once the cubes are rolled to form the Resources for the shake, no player may alter the face of the cubes nor obstruct the other players’ view of any cubes remaining in Resources. If cubes fall outside the Resources area of the playing mat when rolled, only the Goal Setter can carefully move them into the Resources section, taking care not to turn over any cubes.

SETTING THE GOALThe Goal Setter will roll all 24 cubes and set the goal.The goal may be: a single digit number - ex 8 a two digit number – ex 23 a three digit number – ex 345 an expression - ex 3 x 4 ex (4 x 5) + 2 to show the use of parentheses on the Goal line, leave a space between the groupings

– ex 4 x 5 + 2 = (4 x 5) + 2 The goal may not contain more than 6 cubes. The Goal setter has 2 minutes to set the Goal. Once a cube touches the Goal line, it cannot be moved back to Resources. The Goal

Setter may rearrange the cubes. When finished, the Goal Setter will say “goal set”. If the two minute time limit expires, the Goal is considered set.

ORDER OF OPERATIONSIn Basic Equations, order of operations is used to interpret a Goal or Equation. The order is parentheses first, roots and exponents (moving left to right), multiplication and division (moving left to right), and addition and subtraction (moving left to right). It is the responsibility of the Goal Setter and Equation Writer to ensure parentheses are added as needed.MAKING A MOVEThe player to the left of the Goal Setter is the first mover. That player has 2 minutes to make a move or challenge. This will give the mover time to study the Resources and write possible solutions for the Goal. The mover will move a cube to one of the three areas of the playing mat.

FORBIDDEN PERMITTED REQUIRED

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Forbidden – Cubes in this area may not be used in a Solution.Permitted – Cubes in this area may be used in a Solution.Required - Cubes in this area must be used in a Solution.

BONUS MOVEA player may choose to make a bonus move and move 2 cubes on his turn. The player should say “Bonus” before moving. The first cube moved must go to Forbidden. The second cube can be moved anywhere on the board (Forbidden, Permitted, Required, or the Goal).

TIME LIMITS FOR MOVESThe first player after the Goal is set has 2 minutes to make a move or challenge. All players after the first have a 1 minute time limit.

ILLEGAL PROCEDUREA move that violates a procedure is labeled an illegal procedure. Examples include: moving out of turn, moving two cubes without calling Bonus.The player who charges Illegal Procedure must specify what caused the Illegal Procedure within 15 seconds. The Mover must correct the error within 10 seconds (ex. return the cube(s) to Resources and, if necessary, make another move). There is no penalty assessed.

If another legal move has been made before the Illegal Procedure is noticed, that move insulates the Illegal Procedure and play continues.

CALLING A CHALLENGEThe game stops when a Challenge has been made.

You do not have to wait for your turn to challenge. The only player who cannot challenge is the player who made the last move. If a player

challenges himself, the incorrect challenge is set aside and the shake continues. The third player must declare which player he agrees with before the two minute time allowed

to write Equations expires. If the third player agrees with a player who has to write an Equation, he must also present his Equation at the end of the second minute. This player may indicate his intention by stating if he is or is not presenting an Equation, by stating whether he agrees with the Mover or the Challenger, or presenting or not presenting an Equation.

A flub ball (challenge block) will be place equidistant from all players. To challenge, a player must pick up the ball. If he has not picked up the ball, he has not challenged.

The Challenger must pick up the “challenge block” or “flub ball” and state his challenge. If both players challenge at the same time, the player with the “flub ball” in his hand is the Challenger.

A player must not pick up the “flub ball” unless he intends to challenge. If a player does not specify a legitimate challenge, a one point penalty is enforced and play continues.

The Challenge block or flub ball should not be picked up for any reason but to challenge. If it is necessary to move the flub ball closer to all players, they should move it with a pencil and/or announce their intention.

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CHALLENGESIMPOSSIBLE- Call this challenge when you believe a Solution cannot be made.

The Mover has the burden of proof and must write a Solution. If the Joiner agreed with the Mover, he also has to write a Solution. Solution writers have two minutes to write a present an Equation. Equations must be written in the form Solution = Goal or Goal = Solution. The writer must

include all grouping symbols (parentheses, brackets, etc.) in his Equation.

NOW – Call this challenge when you believe a Solution can be written with the cubes available on the board and at most, one more cube from Resources.

The Challenger has the burden of proof and must write an Equation. The third player may choose to write an Equation within the two minute writing time. Equation writers have two minutes to write an Equation.

Note - Do not call Now if there are one or fewer cubes left in Resources. If a player calls Now with one or fewer cubes in Resources, it is set aside , and play continues. (See Last Cube Played)

WRITING AND CHECKING AN EQUATIONAfter a valid Challenge, at least one player must write an Equation. After a Now Challenge, the Challenger must write an Equation. The Mover may not present an

Equation. After an Impossible Challenge, the Mover must write an Equation. The Challenge may not

present an Equation. In a three player match, the Third Party must decide whether to agree with the Challenger or

the Mover. If the player with who the Third Party agrees must write an Equation, then the Third party must also write an Equation.

The Third Party must indicate by the end of the two minutes for writing Equations whether she is presenting an Equation. The Third Party may not retract her declaration once she has indicated whether or not she will present an Equation.

The Third Party may indicate by stating whether or not he will present an Equation, by indicating which party (Mover or Challenger) he agrees with, either verbally or by pointing to the party, or by presenting or not presenting an Equation when the time for writing Equations expires.

To be correct, a Solution must be a legal expression that satisfies the following criteria.

The Solution must be part of a complete Equation in this form:

Solution = Goal

Note: This is the recommended form but the form Goal = Solution will be accepted.

The Solution must equal the interpretation of the Goal, using Order of Operations.

The Solution uses the cubes correctlyBE- 14

o The solution contains at least two cubes.

o The solution uses all the cubes in Required.

o The Solution uses no cubes in Forbidden.

o The Solution may use one or more cubes from Permitted.

o After a Now challenge, the Solution must contain at most one cube from Resources.

o After an Impossible challenge, any cube in Resources may used in the Solution.

The Solution contains only one digit numerals.

All Equations must be presented before any is checked.

o Once a player presents an Equation to the opponents, she may make no further corrections or additions, even if the time for writing Equations has not expired.

o Each Equation writer must circle the Equation to be checked. A writer who forgets to circle the Equation must do so immediately when asked by an opponent.

Within the time for checking an Equation, opponents must accept or reject the Equation. If the Equation is rejected, an opponent must show that it violates at least one of the criteria listed below. An Equation is correct if not opponent shows that it is incorrect.

In a three player match, a player who does not present an Equation for a shake scores 2 if he accepts another player’s Equation as correct and that Equation is proven wrong by the other player.

All Equations are in writing. Players must not physically move the cubes from Required, Permitted, Forbidden and Resources to form the Equation to be checked. This causes arguments over where each cube was played.

A player who claims an opponent’s Equation is not correct must give at least one of the following reasons.

The Goal has no legal interpretation

o The Goal contains more than six cubes

o The Goal contains a four or more digit number.

o The Goal is mathematically impossible. ex 7 ÷ 0

o The Goal is not allowed. ex Elem/ Minor 3 √9

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The Solution does not equal the Goal.

The Equation writer did not use appropriate cubes based on their placement on the playing mat. (ex did not use a 5 in Required, used a x when all x cubes were placed in Forbidden)

The solution uses less than 2 cubes.

The Equation writer used cubes that were not available.

One or both of the checkers may ask a judge to determine whether the Solution equals the Goal. However, no further objections to the Equation will be allowed even if the time limit for checking has not expired.

LAST CUBE PROCEDURE

If one cube remains in Resources, the next Mover must either play that cube to Required or Permitted or challenge Impossible.

When the cube has been moved, each player has two minutes to write an Equation.

The last cube in Resources may not be moved to Forbidden. If a player does so, any challenge that is made is set aside and the cube is returned to Resources. There is no penalty.

An opponent may challenge Impossible against the player who moved the last cube from Resources to Required or Permitted provided the challenge is made by the end of the first minute for writing Equations. If the challenge is made, the Mover (and the Third Party if siding with the Mover) has the rest of the original two minutes to write an Equation.

An Impossible challenge made after the first minute is an illegal procedure and is set aside.

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SCORING

Challenger correct

6 pts Mover correct 6 pts

Challenger wrong

2 pts Mover wrong 2 pts

Joiner correct (with Challenger)

4 pts Joiner wrong

2 pts

Joiner correct (with Mover)

6 pts

Last Cube Procedure

Equation correct

4 pts Last Cube Procedure

Equation incorrect

2 pts

TIME LIMITSRolling cubes and Setting the Goal 2 minutesFirst move after Goal is set 2 minutesAll other moves 1 minuteStating a valid challenge after

picking up flub ball 15 secondsDeciding whether to challenge Impossible when no

more cubes remain in Resources 1 minuteWriting a Equation 2 minutesChecking each Equation 2 minutes

TOURNAMENT SCORING

At the end of a 30 minute round, the tournament official will call a 5 minute warning. At that time, finish the shake you are playing but do not start another shake. A shake is considered to be started if the cubes have been rolled.

After 5 minutes, the tournament director will announce that the round is now over. At that time, if a game has not ended in a Challenge, each player has two minutes to write a Equation for the Goal using all Required cubes and any other available cubes. A correct Equation will score 4 points; an incorrect Equation will score 2 points.

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When the last challenge or last cube has been played and scored, the players will total their scores. The player with the highest score receives 6 team points, the second place will receive 4 points, and the low score receives 2 points. This method will equalize the different raw scores at each table. The raw score varies depending on the number of shakes played.

The three players are splitting a total of 12 points. A two player game will split 10 points.

SCORING CHART

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HIGH RAW SCORE

MIDDLE RAW SCORE

LOW RAW SCORE

ALL DIFFERENT SCORES

6 4 2

TIE FOR FIRST PLACE

5 5 2

TIE FOR SECOND PLACE

6 3 3

THREE WAY TIE 4 4 4

2 PLAYER GAME 6 4

2 PLAYER TIE 5 5

MLAG BASIC ON-SETS® Tournament Rules 2017-18 I. Starting a Match (Round)

A. Two- or three-player matches will be played. A match is composed of one or more shakes. A shake begins with the rolling of the cubes, the dealing of some cards, and the setting of a number as the Goal for that shake. A shake ends with at least one player attempting to write a Solution consisting of a Set-Name that correctly uses the cubes on the playing mat and describes a set containing the number of cards specified by the Goal.

B. The following equipment is needed to play the game:1. 16 cards: each card contains a unique combination of zero to four dots colored blue

(B), red (R), green (G), or yellow (Y). No card contains more than one dot of any color. At the start of a shake, some of these cards are dealt face up to form the Universe for that shake.Comment Players should make sure all 16 cards are in the game, with no duplicates. One of the cards

is blank.

2. 18 cubes: these consist of the following:a. 3 digit cubes: each face has one of the digits 1 through 5.

Comment The digit cubes are used only in setting the Goal.

b. 8 color cubes: each face has a dot colored B, R, G, or Y. Each dot names the set of all cards in the Universe that contain a dot of that same color.

c. 4 operation cubes: each face has one of the symbols U, ∩, –, or '.(i) U means the union of two sets.

Example B U G is the set of cards in the Universe that are either B or G.

(ii) ∩ means the intersection of two sets.Example R ∩ Y is the set of cards that are both R and Y.

(iii) – means set subtraction.Example B – Y is the set of cards that are B but not Y.

(iv) ' means the complement of a set.Example G' (often read “green prime”) is the set of cards that are not G.

d. 3 restriction cubes: each face has one of the symbols V, Ʌ, =, or C.(i) V names the set of all the cards in the Universe for the shake.(ii) Ʌ names the set of no cards (the null or empty set).(iii) = and C are wild cubes. = can be used in a Solution to represent any of the sets

(B, R, G, Y, V or Ʌ). C can be used to represent any of the operations (U, ∩, –, or '). If multiple = or C cubes are rolled, each wild symbol must stand for the same set or operation (as applicable) each time it is used in the Solution.

3. A playing mat: this contains four sections.a. Goal: digit cubes played here form the Goal.b. Required: all cubes played here must be used in any Solution.c. Permitted: any or all cubes played here may be used in any Solution.d. Forbidden: no cube played here may be used in any Solution.

Comment Many game boards have a section labeled “Resources.” However, any reference in these rules to the “playing mat” or the “mat” does not include the Resources section.



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calculators, cell phones or other electronic devices may be used, except that players’ paper may contain preprinted Universe charts on which the cards that are dealt may be marked.Comment The chart a player uses may not have sets pre-shaded or pre-marked in any way. (See Appendix

B for samples.)

D. The Goal-setter for the first shake is determined by lot. On each subsequent shake, the Goal-setter is the player immediately to the left of the previous Goal-setter.To determine the first Goal-setter, each player rolls a digit cube. The player rolling the highest digit sets the first Goal. Players tied for high digit roll again until the tie is broken.

II. Starting a ShakeA. To begin a shake, the Goal-setter rolls all 18 cubes. The symbols on the top faces of the

rolled cubes form the Resources for the shake.1. A shake begins as soon as the timing for rolling the cubes and dealing the cards is

started or the cubes are rolled or the first card is dealt.2. During a shake, no player may turn over a cube or obstruct the other players’ view of

any cube. (See Section IX-C.)B. While the Goal-setter rolls the cubes, the player to the right of the Goal-setter shuffles

and deals the cards.1. At least six but no more than 12 cards must be dealt. 2. The dealer may not take back a card that has been dealt unless the number of

cards exceeds the maximum allowed. In that case, the extra card(s) must be removed from the Universe.

III. Setting the GoalA. The player who rolls the cubes must set a Goal by transferring the cube(s) of the Goal from

Resources to the Goal section of the playing mat.B. A Goal consists of at least one and at most three digit cubes that form an expression that

names a whole number.1. If more than one cube is used to set the Goal, the way the cubes are placed in the

Goal determines the Goal’s value.a. The sum of two numbers is indicated by placing the cubes in a horizontal line (side

by side).b. The product of two numbers is indicated by placing the cubes in a vertical line.c. The negative of a number is indicated by placing the cube so that its numeral is

upside-down.

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The following are the only legal configurations of the cubes for the Goal. Any other configuration is incorrect and a player should challenge Impossible.

Goal Meaning Goal Meaning

A

A + B A x B x C

A + B + C (A x B) + C

A x B A x (B + C)

Comment Any digit cubes not used in the Goal must be placed in Forbidden, since they are not used in Solutions.

2. Once a digit cube touches the Goal section of the mat, it must be used in the Goal.a. The Goal-setter indicates the Goal has been set by saying “Goal.”b. The Goal-setter may rearrange or regroup the cubes in the Goal section until she


C. Before moving the first digit cube to the Goal section of the mat, the Goal-setter may make a bonus move.1. To make a bonus move, the Goal-setter must say “Bonus,” then move one non-digit

cube from Resources to Forbidden, and then set the Goal.D. If the Goal-setter believes no Goal can be set that has at least one correct Solution (see

Section VII), he may declare “No Goal.” Opponents have one minute to agree or disagree with this declaration.1. If all players agree, that shake is void and the same player repeats as Goal-

setter for a new shake.Comments(a) The Goal-setter would declare “No Goal” only in those rare instances when an unusual set of

Resources was rolled. For example, there are three 1’s, the operations are all U signs, and each color appears on at least four cards.

(b) Players receive no points for the void shake.2. An opponent who does not agree with the “No Goal” declaration indicates

disagreement by picking up the challenge block (see Section V-B) and challenging the “No Goal” declaration. She then has two minutes to write a legal Goal and a correct Solution. If there is a third player, he also can choose to write a legal Goal and a correct Solution.Note The Challenger (and third party, if he agrees) can use as many cubes from Resources as needed.

See Section X for scoring.

IV. Moving CubesA. After the Goal has been set, play goes in a clockwise direction (to the left).B. When it is your turn to play, you must either move a cube from Resources to one of the

three sections of the playing mat (Required, Permitted, Forbidden) or challenge the last

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Mover.The move of a cube is completed when it touches the mat. Once a cube is legally moved to the mat, it stays in the section where it was played for the duration of the shake.




V. ChallengingA. Whether or not it is your turn, you may challenge another player who has just completed a

move or set the Goal. The two main challenges are Now and Impossible.Note Players may also challenge a “No Goal” call, see Section III-D-2.

1. By challenging Impossible, a player claims that no correct Solution can be written regardless of how the cubes remaining in Resources may be played.

Comments(a) If the Goal is not in a legal configuration (see Section III-B-1) or the Goal equals a negative number, an

opponent should challenge Impossible.(b) A Player who challenges “Never” will be considered to have challenged “Impossible”. There will be no

penalty for saying “Never” instead of “Impossible”. 2. By challenging Now, a player claims that a correct Solution can be written using the

cubes on the mat and, if needed, one cube from Resources. A player may challenge Now if all of the cubes of her Solution are in Required or Permitted.a. A player may challenge Now only if there are at least two cubes in Resources.

If a player challenges Now with fewer than two cubes in Resources, the challenge is invalid and is set aside. (See Sections VIII and IX below.)Comment If only one cube remains in Resources and no one challenges Impossible, then a Solution

is possible using that one cube. Since the latest Mover had no choice but to play the second-to-last Resource cube to the mat, it is not fair that he be subject to a Now challenge. (However, an Impossible challenge could be made.) See Section VIII for the procedure to be followed when one cube remains in Resources.

b. Since a correct Solution must contain at least two cubes, it is illegal to challenge Now after the Goal has been set but before a cube has been played to Required or Permitted.If a player does so, the challenge is set aside and play continues.




VI. The SolutionA Solution consists of a legal Set-Name that specifies a set of cards in the Universe and does not contain any symbol or group of symbols that is undefined in On-Sets.

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Examples of Solutions R', G U Y, (R ∩ B) – Ʌ, (V – G)' U RComments(a) A Solution written on paper may contain pairs of grouping symbols such as parentheses, brackets or

braces even though these do not appear on the cubes. These symbols indicate how the Solution-writer would physically group the cubes if the Solution were built with the cubes.

(b) The Solution-writer must not write “= Goal” after the Solution. Doing so makes the Solution incorrect.

(c) If the Solution uses an = or C cube as a wild cube, the writer must indicate what the wild cube is used for in writing.

VII. Writing and Checking SolutionsA. After a valid challenge, at least one player must write a Solution.

1. After a Now challenge, the Challenger must write a Solution. (The Mover may not present a Solution.)

2. After an Impossible challenge, the Mover must write a Solution. (The Challenger may not present a Solution.)

3. After a challenge in a three-player match (and before any Solution is presented), the Third Party must indicate by the end of the two minutes for writing Solutions whether she is presenting a Solution. The Third Party may not retract her decision once she has indicated whether or not she will present a Solution.Comment To indicate his intention on the challenge, the Third Party may:(a) state whether or not he will present a Solution;(b) indicate which party, Mover or Challenger, the Third Party is “joining” (agreeing with) on the

challenge. This can be done verbally or by pointing to the party.(c) present or not present a Solution when the time limit for writing Solutions expires. If the Third Party

does not present a Solution, she is assumed to be joining the player who is not writing a Solution (Challenger on an Impossible or Mover on a Now).

B. To be correct, a Solution must satisfy the following criteria:1. The Solution equals the Goal. That is, the number of cards selected from the Universe

equals the Goal.CommentUnlike Equations, the Solution-writer must not write “= Goal” after the Solution.

2. The Solution uses the cubes correctly.a. The Solution contains at least two cubes.b. The Solution uses all the cubes in Required.c. The Solution may use one or more cubes in Permitted.d. The Solution uses no cube in Forbidden.

Comment Since several Resource cubes may show the same symbol, it is possible to have a U in Forbidden that must not be used in the Solution at the same time that there is a U in Required that must be used.



3. Every legal interpretation of the Solution equals the Goal.a. An ambiguous Solution is one that has more than one legal interpretation. Such a

Solution is incorrect if an opponent shows that one of the interpretations does not equal the Goal.

b. The only defined order of operations in On-Sets is that the ' operation takes priority over all other operations (U, ∩, and –). Consequently, a Solution may be ambiguous

BOS- 23

if the writer does not use parentheses (or other symbols of grouping such as brackets or braces) to indicate the order of operations. ‘ is a unary operation; therefore, a player may not insert a parenthesis to split a ‘ from a Set-Name.

C. After the time for writing Solutions has expired (or when all Solution-writers are ready), each Solution that is presented must be checked for correctness.1. All Solutions must be presented before any is checked.

a. Once a player presents a Solution to the opponent(s), she may make no further corrections or additions even if the time for writing Solutions has not expired.

b. Each Solution-writer should indicate the Solution to be checked. A writer who forgets to indicate the Solution must do so when asked.

2. Opponents have two minutes to check each Solution. When more than one Solution must be checked, they may be checked in any order. In a three-player match, both opponents must check a player’s Solution during the same two minutes. No other Solution should be checked during this time.Comments(a) When both players in a two-way match present Solutions after the last cube has been moved (see Section

VIII below), only one Solution should be checked at a time.(b) Players must not physically move the cubes in Required, Permitted, and Resources to form the Solution

being checked. This causes arguments over where each cube was played.

3. Within the time for checking a Solution, opponents must accept or reject the Solution. A player who claims an opponent’s Solution is not correct must show that it violates at least one of the criteria in Section VII-B or cite one of the reasons below. A Solution is correct if no opponent shows that it is incorrect.a. The Goal has no legal interpretation.

Examples(a) The Goal is in the shape of a backwards L, which is not a legal configuration.(b) The Goal equals a negative number.

b. The Solution equals a value that is not the value of the Goal. (i) Checkers must make an effort to determine whether the Solution equals the Goal before

rejecting the Solution. This can usually be done by applying the Solution to the Universe and turning over cards and/or selecting out the cards that are included in the Solution.

(ii) The checker can give a general argument that the Solution does not equal the Goal.Example: The Goal is 0, and the Solution clearly does not give the null set.

(iii) One or both of the checkers may ask a judge to determine whether the Solution equals the Goal. However, the checkers will be restricted in that No further objections to the Solution will be allowed even if the time limit for checking has not expired. If there are two checkers, both must agree that there are no other questions (cubes on the mat, parentheses, procedures, etc.), as this is the final question that a judge will answer.

c. The Solution may be grouped so that it does not equal the value of the Goal. If an opponent believes there is an interpretation of a Solution that does not equal the Goal, that opponent must copy the Solution onto his own paper and add symbols of grouping to create a wrong interpretation. If this revised Solution does not equal the Goal, the Solution is incorrect. However, each checker has only one opportunity to prove ambiguity.

Examples(a) The Solution B U G – R is ambiguous and may be interpreted by an opponent as

(B U G) – R or as B U (G – R). If the interpretation the opponent selects does not equal the Goal, the Solution is incorrect.

(b) R U G' is not ambiguous. It must be interpreted as R U (G') since ' takes priority over U.

BOS- 24

Comment The = and C cubes are wild and may stand for any set or operation, respectively; but each must stand for the same thing wherever it is used. A Solution-writer must indicate clearly and unambiguously in writing what each wild cube represents.

d. A symbol or group of symbols in the Solution has no defined meaning.Examples: R U 'B or R Ʌ G.


or Permitted or challenge Impossible. When the cube has been moved, each player has two minutes to write a Solution.The last cube in Resources may not be moved to Forbidden. If a player does so, any challenge that is made is set aside and the cube is returned to Resources. There is no penalty unless the player’s time to move expires. (See Section XI.)

B. An opponent may challenge Impossible against the player who moved the last cube from Resources to Required or Permitted, provided the challenge is made by the end of the first minute for writing Solutions. If the challenge is made, the Mover (and the Third Party if siding with the Mover) has the rest of the original two minutes to write a Solution.Comment Any Now challenge with one cube or zero cubes left in Resources is invalid, as is any

Impossible challenge made after the first minute for writing Solutions. In both cases, the challenge is set aside and there is no point penalty.

IX. Illegal ProceduresA. Any action that violates a procedural rule is an illegal procedure. A player charging illegal

procedure must specify clearly (within 15 seconds) the exact nature of the illegal procedure.1. If a move is an illegal procedure, the Mover must return any illegally moved cube(s) to

their previous position(s) (usually Resources) and, if necessary, make another move.The Mover must be given at least 10 seconds to make this correction, unless the original move was made after the 10-second countdown (see Section XI-A-3 below), in which case the time-limit rule (Section XI-A) is enforced. In general, there is no direct penalty except that the Mover may lose a point if she does not legally complete her turn during the time limit.Examples of illegal proceduresMoving out of turn, moving two cubes without calling “Bonus” before the first cube touches the mat in Forbidden, or moving the last cube in Resources to Forbidden.

2. If the move is not an illegal procedure, the cube stands as played.Comment There is no penalty for erroneously charging illegal procedure. However, see Section IX-C below if a player does so frequently.

B. An illegal procedure is insulated by a legal action (for example, a move or challenge) by another player so that, if the illegal procedure is not corrected before another player takes a legitimate action, it stands as completed.Example Suppose a player makes an invalid bonus move (such as moving two cubes to Required). Before

anyone notices the illegal procedure, the next mover moves (or a valid challenge is issued). Then the illegal bonus move stays without penalty.

C. Certain forms of behavior interfere with play and annoy or intimidate opponents. If a player is guilty of such conduct, a judge will warn the player to discontinue the offensive behavior. Thereafter during that round or subsequent rounds, if the player again behaves in an offensive manner, the head judge may penalize the player one point for each violation after the warning. Flagrant misconduct or continued misbehavior may cause the player’s disqualification for that round or all subsequent rounds. The head judge may even decide to have the other two opponents replay one or more shakes or the entire round because play was so disrupted by the third party. In some cases, the head judge may order the shake replayed by all three players.

BOS- 25

Examples This rule applies to use of a cell phone, constant talking, tapping on the table, humming or singing, loud or rude language, keeping a hand or finger over or next to the challenge block, making numerous false accusations of illegal procedure, and so on. It also includes not playing to win but rather trying only to ruin the perfect scores of one or both opponents (for example, by erroneously challenging Now or Impossible at or near the beginning of each shake so that both opponents will score 5 for the round), counting down the 10-second warning in an obnoxious manner, etc.


1. That player had to write a Solution and did so correctly.If the Third Party agrees with the person who must write a Solution, the Third Party must write a correct Solution also.

2. That player did not have to write a Solution (someone else did), and no opponent wrote a correct Solution.Exception: After an Impossible Challenge in a three-player match, a player who does not present a Solution for a shake scores 2 if he accepts another player’s Solution as correct even if that Solution is subsequently proven wrong by the other checker.



Impossible, points are awarded as follows:1. Any player who writes a correct Solution scores 4.2. Any player who does not write a correct Solution scores 2.



minute time limits are enforced with the timer. If a player fails to meet a deadline, he loses one point and has one more minute to complete the task. If he is not finished at the end of this additional minute, he loses his turn or is not allowed to complete the task.Note In Minor, Elementary and Middle Divisions, each one-point penalty must be approved by a judge initialing the

scoresheet.

1. The time limits are as follows:a. setting the Universe 1 minuteb. rolling the cubes and setting the Goal 2 minutesc. first turn of the player to the left of the Goal-setter 2 minutesd. all other regular turns (including any bonus moves) 1 minutee. stating a valid challenge after picking up the challenge

block15 seconds

f. deciding whether to challenge Impossible when no more cubes remain in ResourcesIf an Impossible challenge is made, any time (up to a minute) that the Challenger took deciding to challenge counts as part of the two minutes for writing a Solution.

1 minute

g. writing a SolutionDuring this time, the Third Party (if there is one) must decide whether to present a Solution after a Now or Impossible challenge. At the end of these two minutes she must present her

2 minutes

BOS- 26

solution.

h. deciding whether an opponent’s Solution is correct 2 minutes

2. Often a player completes a task before the time limit expires. When sand remains in the timer from the previous time limit, the next player will receive additional time. An opponent timing the next player may either flip or not flip the timer so as to give the opponent the lesser amount of time before the remaining sand runs out and the next time limit can be started.

3. A player who does not complete a task before all the sand runs out for the time limit must be warned that time is up. An opponent must then count down 10 seconds loud enough for the opponent to hear. The one-point penalty for exceeding a time limit can be imposed only if the player does not complete the required task by the end of the countdown.The countdown must be done at a reasonable pace; for example, “1,010; 1,009; 1,008…”

B. A round lasts a specified amount of time (usually 30 minutes). When that time is up, players are told not to start any more shakes. Players have five minutes to finish the last shake. After these five minutes, players still involved in a shake in which no challenge has been made and one or more cubes remain in Resources will be told: “Stop, don’t move another cube — this is the end of the round. Each player has two minutes to write a correct Solution that may use any of the cubes remaining in Resources. Any player who presents a correct Solution scores 4 points for the shake; an incorrect Solution scores 2.”



Two-Player Matches Pointsfirst place 6

two-way tie for first 5second place 4

B. When a round ends, each player must sign (or initial) the scoresheet and the winner (or one of those tied for first) turns it in. If a player signs or initials a scoresheet on which his score is listed incorrectly and the error was a simple oversight, then, with the agreement of all players, correct the scores. However, if there is evidence that there was intent to deceive and the error was not a simple oversight, then do the following:1. If the error gives the player a lower score, he receives the lower score.2. If the error gives the player a higher score, he receives 0 for that round.

BOS- 27

MLAG ON-WORDS® Tournament Rules 2017-18I. Starting a Match (Round)

A. Two- or three-player matches will be played. A match is composed of one or more shakes. A shake consists of a roll of the cubes and the play of the game ending with at least one player attempting to write a Solution consisting of a word or network of words that correctly uses the cubes on the playing mat and has a value that is equal to the Goal.

B. The following equipment is needed to play the game:1. 31 cubes (with only the letters listed on each color cube):

6 black (E, I, N, O, R, T)4 blue (A, C, D, E, L, S)4 red (A, F, H, M, P, U or C)3 green (B, E, G, T, V, W)2 pink (I, N, O, R, S, Y)2 yellow (J, K, Q, X, Y, Z)3 digit cubes


7 orange phonetics cubes (including h, ô, t, s, ou, n, d)


Comment Many game boards have a section labeled “Resources.” However, any reference in these rules to the “playing mat” or the “mat” does not include the Resources section.



cell phones, or other electronic devices may be used.In Minor, Elementary and Middle Divisions, players may use a preprinted chart for recording the Resources, Goal and Solutions.



rolled cubes form the Resources for the shake.1. A shake begins as soon as the timing for rolling the cubes is started or the cubes are

rolled.2. During a shake, no player may turn over a cube or obstruct the other players’ view

of any cube. (See Section VIII-C.)

III. Setting the Goal

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A. The player who rolls the cubes must set a Goal by transferring the cube(s) of the Goal from Resources to the Goal section of the playing mat.

B. A Goal consists of at least one and at most three digit cubes that form an expression that names a whole number. 1. On each shake, the goal must equal 3 or greater. A goal less than 3 is impossible and

should be challenged Impossible.2. If more than one cube is used to set the Goal, the way the cubes are placed in the Goal

determines the Goal’s value.a. The sum of two numbers is indicated by placing the cubes in a horizontal line (side by

side).b. The product of two numbers is indicated by placing the cubes in a vertical line.c. The negative of a number is indicated by placing the cube so that its numeral is

upside-down.3. The following are the only legal configurations of the cubes for the Goal. Any other

configuration is incorrect and a player should challenge Impossible.Comment Any digit cubes not used in the Goal must be placed in Forbidden, since they are not used in

Solutions.

4. Once a digit cube touches the Goal section of the mat, it must be used in the Goal.a. The Goal-setter indicates the Goal has been set by saying “Goal.”b. The Goal-setter may rearrange or regroup the cubes in the Goal section until he

says “Goal.” c. If the time runs out for setting the Goal or the setter turns the timer, it has been set.d. The Goal may not be changed once it has been set.


cube from Resources to Forbidden, and then set the Goal.D. If the Goal-setter believes no Goal can be set that has at least one correct Solution (see

Section VI), he may declare “No Goal.” Opponents have one minute to agree or disagree with this declaration.1. If all players agree, that shake is void and the same player repeats as Goal-setter

for a new shake.Comments(a) The Goal-setter would declare “No Goal” only in those rare instances when an unusual set of

Resources was rolled where no word could be formed. OW - 29

(b) Players receive no points for the void shake.

2. An opponent who does not agree with the “No Goal” declaration indicates disagreement by picking up the challenge block (see Section V-B) and challenging the “No Goal” declaration. She then has two minutes to write a legal Goal and a correct Solution. If there is a third player, he also can choose to write a legal Goal and a correct Solution.Note: The Challenger (and third party, if he agrees) can use as many cubes from Resources as needed.

See Section XI for scoring.


three sections of the playing mat (Required, Permitted, Forbidden) or challenge the last Mover.The move of a cube is completed when it touches the mat. Once a cube is legally moved to the mat, it stays in the section where it was played for the duration of the shake.




V. ChallengingA. Whether or not it is your turn, you may challenge another player who has just completed

a move or set the Goal. The two main challenges are Now and Impossible. Note: Players may also challenge a “No Goal” call, see Section III-D-2.


Comments(a) If the Goal is not in a legal configuration (see Section III-B-3) or the Goal equals a number less than

three, an opponent should challenge Impossible.(b) A Player who challenges “Never” will be considered to have challenged “Impossible”. There will be no

penalty for saying “Never” instead of “Impossible”

2. By challenging Now, a player claims that a correct Solution can be written using the cubes on the mat and, if needed, one cube from Resources. A player may challenge Now if all of the cubes of her Solution are in Required or Permitted.a. A player may challenge Now only if there are at least two cubes in Resources.

If a player challenges Now with fewer than two cubes in Resources, the challenge is invalid and is set aside. (See Sections VII and VIII below.)Comment If only one cube remains in Resources and no one challenges Impossible, then a

Solution is possible using that one cube. Since the latest Mover had no choice but to play the second-to-last Resource cube to the mat, it is not fair that he be subject to a Now challenge. (However, an Impossible challenge could be made.) See Section VII for the procedure to be followed when one cube remains in Resources.

b. Since a correct Solution must contain at least three cubes, it is illegal to challenge Now after the Goal has been set but before two total cubes have been played to Required and/or Permitted.If a player does so, the challenge is set aside and play continues.

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pencil on, over, or near the block for an extended period of time. (See Section VIII-C.)(c) A player must not pick up the challenge block for any reason except to challenge. For example, don’t pick


VI. Writing and Checking SolutionsA. After a valid challenge, at least one player must write a Solution.



3. After any challenge in a three-player match (and before any Solution is presented), the Third Party must indicate by the end of the two minutes for writing Solutions whether she is presenting a Solution. The Third Party may not retract her decision once she has indicated whether or not she will present a Solution.Comment To indicate his intention on the challenge, the Third Party may:(a) state whether or not he will present a Solution;(b) indicate which party, Mover or Challenger, the Third Party is “joining” (agreeing with) on the


does not present a Solution, she is assumed to be joining the player who is not writing a Solution (Challenger on an Impossible or Mover on a Now).

B. To be correct, a Solution must be a word or network of words that satisfies the following criteria:

1. The Solution equals the Goal. That is, value of the Solution equals the Goal.CommentUnlike Equations, the Solution-writer must not write “= Goal” after the Solution.

2. The Solution uses the cubes correctly.a. The Solution contains at least three cubes.b. The Solution uses all the cubes in Required.c. The Solution may use one or more cubes in Permitted.d. The Solution uses no cube in Forbidden. Comment Since several Resource cubes may show the same symbol, it is possible to have an E in

Forbidden that must not be used in the Solution at the same time that there is an E in Required that must be used.



3. The Solution may consist of either a single word or a network of words that have one or more letters in common.a. The official dictionary of ON-WORDS is Merriam Webster’s online Unabridged

Dictionary (dictionary.eb.com). Only words in this dictionary and those on the MLAG

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Official Two-Letter Word List are legal words. For two-letter words, only those on the Official MLAG Two-Letter Word List are permitted. The Official Scrabble Players Dictionary will be used to check for foreign words.

b. All words must be 2 or more letters in length. c. Contractions, abbreviations, proper names, words marked as foreign, obsolete, slang

or archaic or hyphenated words are not allowed unless they are on the MLAG Official Two-Letter Word List.

d. Phonetics cubes represent the phonetics usage in the manual accompanying the game. They are to be used as constraints on the sounds contained in the words in the Solutions. The phonetic cubes may not be used as letters to spell words in the Solutions.

e. If the solution-checker is in doubt regarding the spelling or pronunciation of a word, s/he may call a judge to verify the word. Each player may only ask the judge to verify one word per shake.

f. Required phonetic cubes need only be used once in the solution. Forbidden phonetics cubes are forbidden from being used in any word in the solution.

g. The only allowed pronunciations of a word are those that are listed first for each entry in Merriam Webster’s online Unabridged Dictionary (dictionary.eb.com) or are listed on the MLAG Official Two-Letter Word List.

h. If there is more than one correct pronunciation of a word, the pronunciation(s) that is/are assumed is/are the one(s) that make(s) the solution correct.

C. After the time for writing Solutions has expired (or when all Solution-writers are ready), each Solution that is presented must be checked for correctness.1. All Solutions must be presented before any is checked.

a. Once a player presents a Solution to the opponent(s), she may make no further corrections or additions even if the time for writing Solutions has not expired.

b. Each Solution-writer must indicate the Solution to be checked. A writer who forgets to indicate the Solution must do so when asked.

2. Opponents have two minutes to check each Solution. When more than one Solution must be checked, they may be checked in any order. In a three-player match, both opponents must check a player’s Solution during the same two minutes. No other Solution should be checked during this time.Comments(a) When both players in a two-way match present Solutions after the last cube has been moved (see

Section VII below), only one Solution should be checked at a time.(b) Players must not physically move the cubes in Required, Permitted, and Resources to form the


3. Within the time for checking a Solution, opponents must accept or reject the Solution. A player who claims an opponent’s Solution is not correct must show that it violates at least one of the criteria in Section VI-B or cite one of the reasons below. A Solution is correct if no opponent shows that it is incorrect.a. The Goal has no legal interpretation.

Examples(a) The Goal is in the shape of a backwards L, which is not a legal configuration.(b) The Goal equals a number less than 3.

b. The Solution equals a value that is not a legal value of the Goal.

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(i) Checkers must make an effort to determine whether the Solution equals the Goal before rejecting the Solution. This can be done by counting the number of letters in the word or, for networks, the sum of the number of letters in each word in the network.

(ii) The checker can give a general argument that the Solution does not equal the Goal.

(iii) One or both of the checkers may ask a judge to determine whether the Solution equals the Goal. However, the checkers will be restricted in that No further objections to the Solution will be allowed even if the time limit for checking has not expired. If there are two checkers, both must agree that there are no other questions (cubes on the mat, procedures, etc.), as this is the final question that a judge will answer.

c. The Solution is not in a legal format (e.g., two separate networks...)d. One or more of the words in the Solution is misspelled.e. One of the words in the Solution contains a sound that is forbidden.f. None of the words in the Solution contains a sound that is required.

VII. Last Cube ProcedureA. If one cube remains in Resources, the next Mover must either play that cube to Required

or Permitted or challenge Impossible. When the cube has been moved, each player has two minutes to write a Solution.The last cube in Resources may not be moved to Forbidden. If a player does so, any challenge that is made is set aside and the cube is returned to Resources. There is no penalty unless the player’s time to move expires. (See Section X.)

B. An opponent may challenge Impossible against the player who moved the last cube from Resources to Required or Permitted, provided the challenge is made by the end of the first minute for writing Solutions. If the challenge is made, the Mover (and the Third Party if siding with the Mover) has the rest of the original two minutes to write a Solution.Comment Any Now challenge with one cube or zero cubes left in Resources is invalid, as is any

Impossible challenge made after the first minute for writing Solutions. In both cases, the challenge is set aside and there is no point penalty.

VIII. Illegal ProceduresA. Any action that violates a procedural rule is an illegal procedure. A player charging illegal

procedure must specify (within 15 seconds) the exact nature of the illegal procedure.1. If a move is an illegal procedure, the Mover must return any illegally moved cube(s) to

their previous position(s) (usually Resources) and, if necessary, make another move.The Mover must be given at least 10 seconds to make this correction, unless the original move was made after the ten-second countdown (see Section X-A-3 below), in which case the time-limit rule (Section X-A) is enforced. In general, there is no direct penalty except that the Mover may lose a point if he does not legally complete his turn during the time limit.Examples of illegal proceduresMoving out of turn, moving two cubes without calling “Bonus” before the first cube touches the mat in Forbidden, moving the last cube in Resources to Forbidden.

2. If the move is not an illegal procedure, the cube stands as played.Comment There is no penalty for erroneously charging illegal procedure. However, see Section VIII-C

below if a player does so frequently.


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Example Suppose a player makes an invalid bonus move (such as moving two cubes to Required). Before anyone notices the illegal procedure, the next mover moves (or a valid challenge is issued). Then the illegal bonus move stays without penalty.

C. Certain forms of behavior interfere with play and annoy or intimidate opponents. If a player is guilty of such conduct, a judge will warn the player to discontinue the offensive behavior. Thereafter during that round or subsequent rounds, if the player again behaves in an offensive manner, the head judge may penalize the player one point for each violation after the warning. Flagrant misconduct or continued misbehavior may cause the player’s disqualification for that round or all subsequent rounds. The head judge may even decide to have the other two opponents replay one or more shakes or the entire round because play was so disrupted by the third party. In some cases, the head judge may order the shake replayed by all three players.Examples This rule applies to constant talking, tapping on the table, humming or singing, loud or rude

language, keeping a hand or finger over or next to the challenge block, making numerous false accusations of illegal procedure, and so on. It also includes not playing to win but rather trying only to ruin the perfect scores of one or both opponents (for example, by erroneously challenging Now or Impossible at or near the beginning of each shake so that both opponents will score 5 for the round), counting down the 10-second warning in an obnoxious manner, etc.

IX. Scoring a ShakeA. After a challenge, a player is correct according to the following criteria:


2. That player did not have to write a Solution (someone else did), and no opponent wrote a correct Solution.Exception: After a Challenge in a three-player match, a player who does not present a Solution for a shake scores 2 if he accepts another player’s Solution as correct even if that Solution is subsequently proven wrong by the other checker.





X. Time LimitsA. Each task a player must complete has a specific time limit (listed below). The one- and two-

minute time limits are enforced with the timer. If a player fails to meet a deadline, he loses one point and has one more minute to complete the task. If he is not finished at the end of this additional minute, he loses his turn or is not allowed to complete the task.Note: In Minor, Elementary and Middle Divisions, each one-point penalty must be approved by a judge initialing the scoresheet.

1. The time limits are as follows:a. rolling the cubes and setting the Goal 2 minutesb. first turn of the player to the left of the Goal-setter 2 minutes

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Second place 4

c. all other regular turns (including any bonus moves) 1 minuted. stating a valid challenge after picking up the challenge block 15 secondse. deciding whether to challenge Impossible when no more 1 minute cubes

remain in ResourcesIf an Impossible challenge is made, any time (up to a minute) that the Challenger took deciding to challenge counts as part of the two minutes for writing a Solution.

f. writing a Solution 2 minutesDuring this time, the Third Party (if there is one) must decide whether to present a Solution after a Now or Impossible challenge. At the end of these two minutes she must present her solution.

g. deciding whether an opponent’s Solution is correct 2 minutes2. Often a player completes a task before the time limit expires. When sand remains in the

timer from the previous time limit, the next player will receive additional time. An opponent timing the next player may either flip or not flip the timer so as to give the opponent the lesser amount of time before the remaining sand runs out and the next time limit can be started.

3. A player who does not complete a task before all the sand runs out for the time limit must be warned that time is up. An opponent must then count down 10 seconds loud enough for the opponent to hear. The one-point penalty for exceeding a time limit can be imposed only if the player does not complete the required task by the end of the countdown.The countdown must be done at a reasonable pace; for example, “10-10, 10-09, ..., zero.”

B. A round lasts a specified amount of time (usually 30 minutes). When that time is up, players are told not to start any more shakes. Players have five minutes to finish the last shake. After these five minutes, players still involved in a shake in which no challenge has been made and one or more cubes remain in Resources will be told: “Stop, don’t move another cube – this is the end of the round. Each player has two minutes to write a correct Solution that may use any of the cubes remaining in Resources. Any player who presents a correct Solution scores 4 points for that shake; an incorrect Solution scores 2.”

XI. Scoring a MatchA. Each player is awarded points for the match based on the sum of his scores for the


B. When a round ends, each player must sign (or initial) the scoresheet and the winner (or one of those tied for first) turns it in. If a player signs or initials a scoresheet on which his score is listed incorrectly and the error was a simple oversight, then, with the agreement of all players, correct the scores.

However, if there is evidence that there was intent to deceive and the error was not a simple oversight, then do the following:

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Third place 2

1. If the error gives the player a lower score, she receives the lower score.2. If the error gives the player a higher score, he receives 0 for that round.

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Phonetics Symbols for On-Words

symbol sounds like the underlined part in:

ä. ah, cot, not, hoopla, hurrah, alms,

calm

ă. attic, hat, laugh

ā. aim, bait, neigh, say, date

b. bubble

d. dog, rudder

f. fit, phone, taffy, enough, graph

ĕ. elm, met, dread, friend, genuine

ē. eat, easy, feet, thief, deceive, me

ə. above, opinion, opposition, circus

g. get, wiggle, bog, rogue

h. heat, hen, who

k. keep, can, cake, quack, take, book

ĭ. if, bit, myth

ī. iris, find, sigh, sign, buy, by, pie

p. pet, popped

r. rat, rhythm, writer, merry, star, dare

t. top, butter

y. young, opinion

ô. aw, caught, ought, haul, long, fall

ō. oh, owe, loan, roe, sew

ŏŏ. look, push, would, sugar

ōō. ooze, dude, do, you, blue

th. thin, lengthy, breath (italics, not

underlined, for clarity)

th. the, father, breathe

l. let, lily, wallet

m. moo, drummer, tomb

n. need, gnome, knock, dinner

symbol sounds like the underlined part in:

ng. song, link, thinking

ŭ. other, luck, above, blood, duh

ū. use, you, few

ch. achoo, chair, choose, witch

j. = j. adjure, jail, jaw, gem, badger, fudge

zh. azure, measure, rouge, azure,

prestige

sh. assure, shape, sugar, special,

patience

hw. wheat, when

s. sewer, sit, cent, scene, hustle

oi. oyster, soil

ər. measure, assure, sugar, dirty, cellar

ou. ouch, owl

v. vat, haven

w. win, one

z. zip, fuzz, rose

Notes:

ə (by itself) is the schwa sound. It stands for

the vowel sound in unaccented syllables.

ər can occur in accented or unaccented

syllables

ŭ is the short u sound. It can only occur in an

accented syllable (if unaccented, it’s ə).

The ŏŏ and ōō symbols on the actual phonetics cubes look slightly different, as the symbols here are constrained by the available fonts.

OW - 37

OW - 38

Michigan League of Academic Games Official On-Words Two-Letter Word List

AA /ää/- rough, cindery lavaAB /ăb/- abdominal muscleAD /ăd/- advertisementAE /ā/- oneAG /ăg/- pertaining to

agricultureAH /ä/- expresses delightAI /ī/- three-toed slothAL /äl/- an East Indian treeAM /ăm/- form of “to be”AN /ǝn/, /ăn/- indefinite

articleAR /är/- the letter “R”AS /ăz/ - to the same degreeAT /ăt/- in the position ofAW /ô/- expresses protestAX /ăks/- cutting toolAY /ī/- affirmative voiceBA /bä/- eternal soulBE /bē/- to have actualityBI /bī/- twoBO /bō/- a palBY /bī/- a side issueDE /dē/, /dǝ/- of; fromDO /dō/, /dōō/- a tone of the

scale; carry outED /ĕd/- educationEF /ĕf/- the letter “F”EH /ā/, /ĕ/- expresses doubtEL /ĕl/- elevated railroadEM /ĕm/- the letter “M”EN /ĕn/- the letter “N”ER /ǝ/, /ŭ/- expresses

hesitationES /ĕs/- the letter “S”ET /ĕt/- a past tense of eatEX /ĕks/- the letter “X”FA /fä/– a tone of the scaleFE /fā/– a Hebrew letterGO /gō/- to move alongHA /hä/– sound of surpriseHE /hē/- male personHI /hī/– a greetingHM /hm/– expresses

considerationHO /hō/– expresses surpriseID /ĭd/- part of the psycheIF /ĭf/- a possibility

IN /ĭn/- to harvestIS /ĭz/- form of “to be”IT /ĭt/- neuter pronounJO /jō/– sweetheartKA /kä/– spiritual selfKI /kē/– inner strength, life

energyLA /lä/– a tone of the scaleLI /lē/– Chinese unit of

distanceLO /lō/– expresses surpriseMA /mä/– motherME /mē/– personal pronounMI /mē/– a tone of the scaleMM /m/– expresses assentMO /mō/– a momentMU /mū/– a Greek letterMY /mī/– possessive

pronounNA /nǝ/ or /nŭ/– no; notNE /nā/– born with the name

ofNO /nō/– a negative replyNU /nū /– a Greek letterOD /äd/– a hypothetical forceOE /ō/– Faeroe Isl. whirlwindOF /ǝv/, /ŭv/– fromOH /ō/– to exclaim in

surpriseOI /oi/– expresses dismayOM /ōm/, /ôm/– a mantraON /ôn/, /än/– batsman’s

side of a wicketOP /äp/- a style of abstract

artOR /ôr/, /ôǝr/, /ǝr/– the

heraldic color gold; conjunction between alternatives

OS /äs/– a boneOW /ou/– expresses painOX /äks/– a clumsy personOY /oi/– expresses dismayPA /pä/– fatherPE /pā/– a Hebrew letterPI /pī/– a Greek letterQI /chē/– circulating life

energy

RE /rā/– a tone of the scaleSH /sh/– urges silenceSI /sē/– a tone of the scaleSO /sō/– a tone of the scaleTA /tä/– expression of

gratitudeTI /tē/– a tone of the scaleTO /tōō/– in the direction ofUH /ŭ/, /ǝ/– expresses

hesitationUM /ŭm/, /mmm…/–

indicates hesitationUN /ŭn/– oneUP /ŭp/– to raiseUS /ŭs/– personal pronounUT /ŭt/– a tone of the scaleWE /wē/– pronounWO /wō/– woeXI /zī/– a Greek letterXU /sōō/– Vietnamese

moneyYA /yŭ/, /yǝ/– youYE /yē/- youYO /yō/– used to call

attentionZA /zä/- a pizza

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MLAG ADVENTUROUS EQUATIONS® Tournament Rules 2017-18I. Starting a Match (Round)

A. Two- or three-player matches will be played. A match is composed of one or more shakes. A shake consists of a roll of the cubes and the play of the game ending with at least one player attempting to write an Equation, which contains a mathematical expression that equals the Goal and correctly uses the cubes on the playing mat.

B. The following equipment is needed to play the game:1. 24 cubes: there are six of each color (red, blue, green, and black). Every face of each

cube contains either a digit (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) or an operation sign (+, –, x, ÷, * or ^, √).

2. A playing mat: this contains four sections.a. Goal: cubes played here form the Goal.b. Required: all cubes played here must be used in any Solution.c. Permitted: any or all cubes played here may be used in any Solution.d. Forbidden: no cube played here may be used in any Solution.

Comment Many game boards have a section labeled “Resources.” However, any reference in these rules to the “playing mat” or “mat” does not include the Resources section.


should not be so large that two players can grab it at the same time.C. Players may use only pencils or pens, blank paper, and (for Adventurous Equations)

variation sheets. No prepared notes, books, tables, calculators, cell phones, or other electronic devices may be used.In Elementary and Middle Divisions, players may use a preprinted chart for recording the Resources, variations, Goal and Solutions.

D. The Goal-setter for the first shake is determined by lot. On each subsequent shake, the Goal-setter is the player immediately to the left of the previous Goal-setter.To determine the first Goal-setter, each player rolls a red cube. The player rolling the highest digit sets the first Goal. A player who rolls an operation sign is eliminated unless all players roll an operation sign. Players tied for high digit roll again until the tie is broken.


rolled cubes form the Resources for the shake.1. A shake begins as soon as the timing for rolling the cubes is started or the cubes are

rolled.2. During a shake, no player may turn over a cube or obstruct the other players’ view of

any cube. (See Section IX-C.)B. In Adventurous Equations, after the cubes are rolled but before the Goal is set, each player

must select a variation from the appropriate list in Section XIII of these rules. A variation is a special rule which, if it conflicts with any of the regular tournament rules, supersedes those rules.1. The Goal-setter makes the first selection, then the player to the left of the Goal-setter,

then the third player if there is one.a. Each player has 15 seconds to make a variation selection.b. To begin a shake, the Goal-setter has one minute to roll the cubes. At the end of this

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minute, he has 15 seconds to select a variation. However, if the Goal-setter selects a variation before the minute for rolling the cubes expires, the next player has the rest of that minute plus 15 seconds to select a variation. If the second player also selects a variation before that minute expires, the third player (if there is one) has the rest of that minute plus 15 seconds to select.

c. A player selects a variation by circling its name in the list for that shake. This list is located on the reverse side of the scoresheet or on a separate sheet. For certain variations (e.g., Base or Multiple of k), the player must also fill in a blank to indicate which base or value of k is chosen, and so on.

2. If a player selects a variation that has no effect on the shake, a variation that conflicts with one already chosen for the shake, or a variation that has already been chosen for the shake, the player loses one point and must pick another variation. If, on the second try, the player still does not select an appropriate variation, he loses another point and may not pick a variation for that shake.If a player’s illegal variation selection is not pointed out before the next player selects a legal variation or a legal Goal is set (whichever comes first), the player making the illegal selection is not penalized. However, the illegal variation is ignored for the shake.Examples It is illegal to choose 0 Wild when no 0 cube is in Resources or +=Average when no + was

rolled.

3. In two-player matches in Elementary, Middle, and Junior Divisions, the player who is not the Goal-setter must select two variations for the shake. In Senior Division, any player may pick two variations for any shake in either a two- or three-player match.A player picking two variations must select both within the 15-second time limit. (See Section XI- A-1-b.)

III. Legal Mathematical ExpressionsA. A legal mathematical expression is one that names a real number and does not contain

any symbol or group of symbols that is undefined in Equations.Example a ÷ 0 for any value of a does not name a real number. (See Section C below for additional

examples.)Comment An expression written on paper may contain pairs of grouping symbols such as parentheses,

brackets, or braces even though these do not appear on the cubes. These symbols indicate how the Equation-writer would physically group the cubes if the Equation were actually built with the cubes.

B. The symbols on the cubes have their usual mathematical meanings with the following exceptions:1. The + and – cubes may be used only for the operations of addition and subtraction;

they may not be used as positive or negative signs.Examples +7, –8, 6x+4, and 17÷ (–8) are illegal expressions.

2. If the radical sign (√) is used, it must always be preceded by an expression to denote its index unless the index equals 2. If no index is shown, it is understood to be 2.Examples 2√9 or just √9 is legal and means “the square root of 9”; 1√2 means “the first root of 2,” which

is 2. (2+1)√8 means “the cube root of 8,” which is 2. 4x√9 means “4 times the square root of 9,” which is 4 x 3 or 12. 3√√9 means “the cube root of the square root of 9,” which is the sixth root of 9. (This expression is illegal in Elementary Division – see the “General Rule” in Section XIII-A.)

3. * (or ^) means exponentiation (raising to a power). The ^ cube must be written with the point up.Example 4*2 (or 4^2) means 42, which is 4 x 4 =16.

C. Expressions involving powers and roots must satisfy these requirements:1. Even-indexed radical expressions indicate only non-negative roots.

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Examples √9 equals 3, not –3; 4√16 = 2 (not –2).

2. The following expressions are undefined. Note: In all cases, * may be replaced by ^.a. 0√a where a is any numberb. 0 * a where a ≤ 0c. a √ b where a is an even integer and b is negatived. (a ÷ b) √ c where c is negative and, when a ÷ b is reduced to lowest terms, a is an

even integer and b is an odd integere. a * (b ÷ c) where a is negative and, when b ÷ c is reduced to lowest terms, b is an

odd integer and c is an even integerExamples(a) (–8)4/6 is defined, as shown by the following steps. First reduce the fractional exponent to lowest

terms: (–8)4/6 = (–8)2/3. (–8)2/3 is of the form a * (b ÷ c) where a is negative. Since b is even and c is odd, (–8)2/3 is defined. (–8)2/3 = 3√(–8)2 = 3√64 = 4.

(b) (–4)2/4 is not defined because (–4)2/4 = (–4)1/2, which is of the form a * (b ÷ c) with a negative, b odd, and c even.Note The following reasoning is not allowed since the exponent is not reduced first: (–4)2/4 = 4√(–4)2 = 4√16 = 2.

(c) 3/6√(–9) is defined because 3/6√(–9) = 1/2√(–9), which is of the form (a ÷ b) √ c, with c negative. However, a is odd and b is even. So 3/6√(–9) = (–9)6/3 = (–9)2 = 81.

(d) 8/2√(–5) is not defined because 8/2√(–5) = 4√(–5), which is of the form a √ b where a is even and b is negative.

IV. Setting the GoalA. The player who rolls the cubes must set a Goal by transferring the cube(s) of the Goal

from Resources to the Goal section of the playing mat.B. A Goal consists of at least one and at most six cubes that form a legal expression.

1. Numerals used in the Goal are restricted to one or two digits. The use of operation signs is optional.Examples of legal Goals 6, 23, 8–9, 17x8, 19+8–5, 87÷13, 3√64, √49Examples of illegal Goals 125 (three-digit numerals not allowed), 23+18+7 (too many cubes), 45x

(does not name a number), +8 (does not name a number since + means addition).

2. The order of operations of mathematics does not apply to the Goal. The Goal-setter may physically group the cubes to indicate how it is to be interpreted. If the Goal-setter does not group the cubes, the Goal may be interpreted in any valid way.Examples(a) 2x 3+5 (with space between x and 3) means 2 x (3 + 5).(b) 2x3 +5 (with space between 3 and +) means (2 x 3) + 5.(c) The Goal 2x3+5 (with no spaces) may be interpreted as either 2 x (3 + 5) or (2 x 3) + 5.

Comment The Goal-setter may not be able to remove all ambiguities from the Goal. Example √ 5+4 x9 where the Goal-setter wants to apply the √ to the entire expression (5+4)x9.

Declaring orally that the √ applies to everything that follows or extending the root over the entire expression in writing is not binding.

Players may interpret this Goal as [√(5+4)]x9 or as √[(5+4)x9].

3. Once a cube touches the Goal section, it must be used in the Goal.a. The Goal-setter indicates the Goal has been set by saying “Goal.”b. The Goal-setter may rearrange or regroup the cubes in the Goal section until he


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C. Before moving the first cube to the Goal section of the mat, the Goal-setter may make a bonus move.1. To make a bonus move, the Goal-setter must say “Bonus,” then move one cube from

Resources to Forbidden, then move one or more cubes to the Goal.2. A Goal-setter who is leading in the match may not make a bonus move.

If the Goal-setter makes a bonus move while leading in the match and an opponent points out the error before the next player moves or someone legally challenges, the cube in Forbidden is returned to Resources. The mover also receives a one-point penalty.

D. If the Goal-setter believes no Goal can be set that has at least one correct Solution (see Section VII), he may declare “No Goal.” Opponents have one minute to agree or disagree with this declaration.1. If all players agree, that shake is void and the same player repeats as Goal-setter


Resources was rolled. For example, there are less than three digit cubes or only one or two operation cubes. (Even in these cases, the Goal-setter could choose a variation like 0 Wild that might allow a Goal to be set.)

(b) Players receive no points for the void shake.(c) If the Goal-setter makes a Bonus move, he is committed to setting a goal and may not declare “No Goal”

2. An opponent who does not agree with the “No Goal” declaration indicates disagreement by picking up the challenge block (see Section VI-B) and challenging the “No Goal” declaration. She then has two minutes to write a correct Equation. If there is a third player, he also can choose to write an Equation. Scoring for a challenged “No Goal” is the same as the scoring in an Impossible Challenge.Note: The Challenger (and third party, if he agrees) can use as many cubes from Resources as needed.


V. Moving CubesA. After the Goal has been set, play progresses to the left (clockwise direction).B. When it is your turn to play, you must either move a cube from Resources to one of the

three sections of the playing mat (Required, Permitted, Forbidden) or challenge the last Mover.The move of a cube is completed when it touches the mat. Once a cube is legally moved to the mat, it stays where it was played for the duration of the shake.

C. If you are not leading in the match, then on your turn you may take a bonus move before making a regular move.1. To make a bonus move, the Mover must say “Bonus,” then move one cube from

Resources to Forbidden, then another cube to Forbidden, Permitted or Required.Comments(a) If you do not say ‘Bonus’ before moving the cube to Forbidden, the move does not count as a bonus



2. If the player in the lead makes a bonus move and an opponent points out the error before another player makes a legal move or challenge, the Mover must return the second cube played on that turn to Resources. The Mover also receives a one-point penalty.Comments(a) Players tied for the lead may make Bonus moves.(b) Players often call “Bonus” and move two cubes simultaneously to Forbidden. If the player did not

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call “Bonus”, he may return either of the two cubes to Resources.


a move or set the Goal. The two main challenges are Now and Impossible.

Note Players may also challenge a “No Goal” call, see Section IV-D-2.

1. By challenging Impossible, a player claims that no correct Equation can be written regardless of how the cubes remaining in Resources may be played.Comments(a) If the Goal is not a legal expression, an opponent should challenge Impossible. Examples of such

Goals are +8, 65+87–3, 122, and so on.(b) Occasionally it is obvious before the Goal-setter completes the Goal that no Solution is possible.

Examples: Using more than six cubes in the Goal or (in Mid/Jr/Sr) using an 8 or 9 in the Goal when Base eight was called. However, opponents must still wait until the Goal-setter indicates the Goal is finished before challenging. You may not pick up the challenge block and “reserve” the right to challenge when the Goal is completed.

(c) A Player who challenges “Never” will be considered to have challenged “Impossible”. There will be no penalty for saying “Never” instead of “Impossible”.

2. By challenging Now, a player claims that a correct Equation can be written using the cubes on the mat and, if needed, one cube from Resources.a. A player may challenge Now only if there are at least two cubes in Resources.

If a player challenges Now with fewer than two cubes in Resources, the challenge is invalid and is set aside. The player who called Now also receives a one point penalty.Comment If only one cube remains in Resources and no one challenges Impossible, then a

Solution is possible using that one cube. Since the latest Mover had no choice but to play the second-to-last Resource cube to the mat, it is not fair that he be subject to a Now challenge. (However, an Impossible challenge could be made.) See Section VIII for the procedure to be followed when one cube remains in Resources.

b. Since a correct Solution must contain at least two cubes, it is illegal to challenge Now after the Goal has been set but before a cube has been played to Required or Permitted. If a player challenges Now before any cubes have been played to Required or Permitted, the challenge is invalid and is set aside. The player who called Now also receives a one point penalty.

B. A challenge block is placed equidistant from all players. To challenge, a player must pick up the block and say “Now” or “Impossible.”If a player picks up the block, then decides not to challenge (without saying “Now” or “Impossible”), the player accepts a one-point penalty and play continues. A player who picks up the block and makes a challenge against himself is also penalized one point, and the challenge is set aside.

Comments(a) The purpose of the block is to determine who the Challenger is in a shake.(b) Touching the challenge block has no significance. However, players may not keep a hand, finger, or



VII. Writing and Checking Equations

A. After a valid challenge, at least one player must present an Equation.1. After a Now Challenge,

the Challenger must present an Equation. the Mover may not present an Equation. the Third Party in a three-player match may present an Equation.

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2. After an Impossible Challenge, the Challenger may not present an Equation. the Mover must present an Equation. the Third Party in a three-player match may present an Equation.

B. To be correct, a Solution must be a legal expression (see Section III) that satisfies the following criteria:1. The Solution must be part of a complete Equation in this form:

Solution = GoalComment While Solution = Goal is the recommended form for writing the Equation,

Goal = Solution is acceptable. (See Appendix for all matters involving how Equations are written.)

2. The Solution must equal the interpretation of the Goal that the Equation-writer presents with the Solution.

Examples (in each case * may be replaced by ^.)Goal Sample Equation Goal Sample Equation37 (6x6) + 1 = 37 11+5 (3x2)+(5x2) = 11+5

3x5+2 (5*2) – 4 + 0 = 3x(5+2) 3x 5+2 (5x4)+1 = 3x(5+2)03 (5x5) – 2 = 23 (with 0 Wild)

↑ ↑0 0

0+3 1*(7+8-4x3) = 0+3 (0 Wild, Upside-down)↑

usd 230x7 [(5*2)x8] + 6 + 4 = 30x7

(with 0 Wild, 0 defaults to 0)9÷8 5 + 4 = 9! ÷ 8!

with Factorial variation

Note: See Appendix for a complete list of ways of indicating what ambiguous cubes (such as wild cubes) represent in Equations. The Appendix also lists the default values of ambiguous symbols if an Equation-writer does not indicate the interpretation. However, there is no default order of operations in Equations (except when certain variations, such as Factorial, are played– see Section 6b below).

Comments(a) An Equation-writer who does not write a complete Equation (Solution = Goal), even when there is

only one interpretation of the Goal, is automatically incorrect.(b) The Equation-writer does not write the value of the Goal except in those cases where writing the

Goal is the same as writing its value.Examples

(i) The Goal is 37.(iii) The Goal is 40 with 0 Wild, and the writer writes 45 to indicate what 0 represents.(iii) For a Goal like 3x5+2, the writer must write either (3x5)+2 or 3x(5+2) and not 17 or 21.

(c) For a Goal like 3x5+2, the writer must write either (3x5)+2 or 3x(5+2) and not 17 or 30.(d) If the Goal is grouped, as in 3x 5+2 , an Equation-writer must write 3x(5+2) and not 3x 5+2 (with

space between x and 5 but no parentheses). In the latter case, the Goal is ambiguous and a checker may group it in such a way as to make the Equation wrong.

3. The Solution uses the cubes correctly.a. The Solution contains at least two cubes.b. The Solution uses all the cubes in Required.c. The Solution uses no cube in Forbidden.Comment “Since several Resource cubes may show the same symbol, it is possible to have a 2 in

Forbidden which must not be used in the Solution at the same time that there is a 2 in Required which must be used.”

d. The Solution may use one or more cubes in Permitted.e. After a Now challenge, the Solution must contain at most one cube from

Resources.f. After an Impossible challenge, any cubes in Resources are considered to be in

Permitted and therefore may be used in the Solution.4. The Solution contains only one-digit numerals.

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Comment Certain variations (see Section XIII) allow exceptions to this rule; for example, Two-digit Numerals in Elementary Division and Base m in Middle/Junior/Senior.

5. In Adventurous Equations, the Equation satisfies all conditions imposed by the variations selected for that shake. (See Section XIII for a list of the variations.)Examples(a) If the Elementary variation Three-operation Solution has been chosen, any Solution that contains

fewer than three operations is incorrect.(b) In Middle, Junior, and Senior, the Multiple of k variation requires that any Solution not equal the

Goal. Instead the Solution must differ from the Goal by a multiple of k.

6. The Solution is not ambiguous. An ambiguous Solution is one that has more than one legal interpretation. Such a Solution is incorrect if an opponent shows that one of its values does not equal the interpretation of the Goal provided with the Solution.Comment For the procedure to be followed when an Equation-checker thinks a Solution is ambiguous,

see Section C-5-c below.

a. In Adventurous Equations, the general order of operations of mathematics (roots/exponents first, then multiplication/division, finally addition/subtraction) does not apply to Solutions. Consequently, a Solution may be ambiguous if the writer does not use parentheses (or other symbols of grouping such as brackets or braces) to indicate the order of operations.

b. Certain symbols have a default interpretation as regards grouping, as follows. (i) The radical sign (√) applies to just the numeral immediately behind it unless grouping symbols are used. Examples √4+5 means (√4)+5. An opponent may not interpret √4+5 as √(4+5). √(4+5) + 7 means (√9)

+ 7. An opponent may not interpret √(4+5) +7 as √[(4+5) + 7]. (ii)For the Factorial variation (see Section XIII below), ! applies to just the numeral in front of it unless the Equation-writer uses grouping symbols to indicate otherwise.Examples 4 + 7! means 4 + (7!). An opponent may not interpret it as (4 + 7)!. Suppose the Goal is 4 +

7. If an Equation-writer wants 4+(7!), just write 4+7! If the writer wants 11!, write (4+7)!Note: There are some variations (Exponent, Number of Factors, Smallest Prime, Imaginary Numbers) that have default interpretations in the AGLOA rules, but DO NOT have a default interpretation in the MLAG rules.

Note: None of the default interpretations of symbols restricts a player’s right to interpret an ungrouped Goal in any acceptable way

c. When the default interpretation for two symbols conflict, the expression is ambiguous, and the Equation-writer must use grouping symbols to remove the ambiguity.Example The expression √9! Is ambiguous because the default interpretation for Factorial, which

says the expression means √(9!), conflicts with the default interpretation of √, which specifies the interpretation as (√ 9)!.

C. After the time for writing Equations has expired (or when all Equation-writers are ready), each Equation that is presented must be checked for correctness.1. After a challenge in a three-player match, the Third Party has two minutes to decide

whether or not to present an Equation. If the Third-Party indicates her decision before the two minutes expire, she may not retract the decision.Comment The Third Party is not obligated to indicate whether he is presenting an Equation before the

time limit expires. However, if the Third Party decides to indicate his decision before time expires, he may:

(a) State whether or not he will present an Equation;(b) indicate which party, Mover or Challenger, the Third Party is “joining” (agreeing with) on the

challenge. This can be done verbally or by pointing to the party when asked.(c) present or not present and Equation when the time limit for writing Equations expires. If the Third

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Party does not present an Equation, she is assumed to be joining the player who is not writing an Equation..

2. All Equations must be presented before any is checked.a. Once a player presents an Equation to the opponent(s), she may make no further

corrections or additions even if the time for writing Equations has not expired.b. An Equation is considered presented when the paper is out of the hand of the

writer.c. Each Equation-writer must indicate the Equation to be checked, including the

interpretation of the Goal. A writer who forgets to indicate the Equation must do so when asked..

3. Opponents have two minutes to check each Equation. When more than one Equation must be checked, they may be checked in any order. In a three-player match, both opponents must check a player’s Equation during the same two minutes. No other Equation should be checked during this time.Comments(a) When both players in a two-way match present Equations after the last cube has been moved (see

Section VIII below), only one Equation should be checked at a time.(b) Players must not physically move the cubes in Required, Permitted, and Resources to form the


4. Within the time for checking an Equation, opponents must accept or reject the Equation. A player who claims an opponent’s Equation is not correct must give at least one of the following reasons (or cite one of the reasons in Section VII-B).a. The Equation is NOT in the form Solution = Goal (or Goal = Solution)b. The Solution DOES NOT equal the interpretation of the Goal that the Equation-

writer presents with the Solution. c. The Solution DOES NOT use the cubes correctly.

(i) The Solution contains only cube.(ii) The Solution DOES NOT use all the cubes in Required.(iii) The Solution uses a cube in Forbidden.(iv) After a Now challenge, the Solution needs more than one cube from

Resources.d. The Goal has no legal interpretation.

Examples(a) 7 ÷ 0 when 0 is not wild(b) A Goal containing more than six cubes or a three-digit number(c) Mid/Jr/Sr: 39 in Base eight(d) Elem: 3 √ 9(e) Elem/Mid: 8 x when Sideways was not chosen

e. The Solution contains numerals with multiple digits.Comment Certain variations (see Section XIII) allow exceptions to this rule: for example, Two-digit

Numerals in Elementary Division and Base m in Middle/Junior/Senior.

f. The Equation DOES NOT satisfy all conditions imposed by the variations selected for that shake.

g. The Equation-writer’s interpretation of the Goal is not a legal interpretation.Examples(a) The writer makes 0 in the Goal equal another numeral when 0 Wild was not chosen.(b) The writer groups the Goal in an illegal manner:

e.g., the Goal is grouped on the mat as 5x 3+4 and the writer interprets it as (5x3)+4. (c) With Multiple Operations, the Equation-writer uses an operation cube in the Goal multiple times.(d) With Red Exponent, the writer interprets the Goal 312 (red 2) as a three-digit numeral.

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h. One or both sides of the Equation is ambiguous and may be grouped so that the Solution does not equal the Goal. If an opponent believes there is a legal interpretation of a Solution or Goal that makes the Equation wrong, that opponent should copy the Solution and/or Goal to his own paper and add symbols of grouping to create a wrong interpretation. If the checker is able to do so, the Equation is incorrect. However, each checker has only one opportunity to prove ambiguity. If there is a second checker, the checkers may either work together to prove ambiguity or work separately. If working separately, the second checker may simultaneously and independently try to prove ambiguity. When both checkers are ready (or one is ready and the other has nothing to show), follow the same procedure used for checking the original Equation. That is, each attempt at proving ambiguity is checked. If either shows a legal interpretation of the Solution and/or Goal such that the Solution does not equal the Goal, then the original Equation is incorrect. If each attempt at proving ambiguity fails, the Equation must be accepted as correct. That is, once the Equation-writer starts checking the attempt(s) at proving ambiguity, no further objections to the Equation are allowed.

Comments(a) In the case where the checkers work separately to prove ambiguity, if the time for checking the

Equation runs out, either or both checkers may take an additional minute (paying the one-point penalty to do so). If only one checker wishes to take the additional minute, the other checker may make no further changes to his revision of the Equation. If he tries to do so, then he incurs the one-point penalty also.

(b) Just as two players writing Equations after a challenge or forceout may not communicate with each other, so two checkers attempting to prove ambiguity separately may not communicate while doing so.

(c) Each checker working separately has only one opportunity to prove ambiguity. Similarly, checkers working together have just one joint chance to prove ambiguity.

(d) While only one checker is attempting to prove ambiguity, the other checker may continue to check other aspects of the Equation.

(e) If each checker separately trying to prove ambiguity is ready with his revision of the Equation before the time for checking expires, no -1 penalty is enforced during the time the original Equation-writer checks any attempts at proving ambiguity.

Examples (in each case, * may be replaced by ^.)(a) The Solution in 5*2–4+0 = 3x(5+2) is ambiguous. An opponent may rewrite the Solution as

5*(2–4)+0 so that it does not equal 3x(5+2).(b) The Solution in 2x4–(3+1) = 4 is ambiguous. An opponent may rewrite it as

2x[4–(3+1)] so that it does not equal 4. However, an opponent may not rewrite it as 2x[4–(3]+1) since the brackets interfere with a grouping already in the Solution.

(c) The Goal in (6x4)–2 = 7+5x3 is ambiguous. A checker who rewrites the Goal as (7+5)x3 has shown that the Equation is incorrect.

(d) In Elementary Division, an expression like 9√5*9 must be grouped 9√(5*9) because 9√5 is illegal (undefined) in Elementary. So if an Equation-writer does not group 9√5*9, an opponent may group it (9√5)*9 to make the Equation wrong.

Comment Certain variations (such as 0 Wild) allow cubes to be used for other symbols. If a cube stands for anything other than what is on the cube, the Equation-writer must indicate clearly and unambiguously in writing what each such cube represents. See Appendix for a list of suggested ways of doing this. Appendix also lists any default interpretations when players do not write what symbols represent.

i. The Solution does not equal the Equation-writer’s interpretation of the Goal.(i) Checkers must make an effort to determine whether the Solution equals the

writer’s interpretation of the Goal before rejecting the Equation.(ii) The checker can give a general argument that the Solution does not

equal the Goal.Examples(a) The Goal is a fraction or an irrational number, but the Solution equals an integer (or vice-

versa). AE-48

(b) The Solution equals a value greater than 1000 when the Goal is 50x10. That is, the Solution is clearly too big (or too small) even without calculating its exact value.

(iii) One or both of the checkers may ask a judge to determine whether the Solution equals the Goal. However, the checkers will be restricted in two ways:• No further objections to the Equation will be allowed even if the time limit for

checking has not expired. Both checkers must agree that there are no other questions (cubes on the mat, parentheses, procedures, etc.), as this is the final question that a judge will answer.

• If the Solution and/or the Goal in the Equation is ambiguous, the judge will answer “Yes, the Solution equals the Goal” when one legal value of the Solution equals a legal value of the Goal since the checkers did not raise the issue of ambiguity. Furthermore, checkers may not make the catch-all objection, “The Solution does not unambiguously equal the Goal.” General claims of ambiguity are not allowed. The checker must provide a specific grouping of either or both sides of the Equation that makes it incorrect. (See III-A on page E3 for the definition of legal mathematical expression.)Example The Equation is: (5+1)! = √10! ÷ 7

The writer has not removed the ambiguity for √10! However, she clearly wants √(10!÷7). Since the Solution equals that interpretation of the Goal, the judge will rule the Equation correct if no opponent raised the ambiguity issue. Even if a checker claims ambiguity, he must group the Goal as (√10)!÷7 or [√(10!)]÷7 in order to prove the Equation incorrect.

j. A symbol or group of symbols is used ambiguously in the Solution or Goal, and one interpretation of the symbol(s) gives a value that makes the Solution not equal the writer’s Goal.ExampleJr/Sr: With Base Twelve, the Solution in 6 + √4 = 5 + 3 is ambiguous because √ can mean root or the digit eleven.Note: See Appendix for the default meaning of symbols that may be ambiguous. For example, if 0 is wild and the Equation-writer does not indicate what 0 means, 0 equals 0.

k. A variation is applied wrongly or not at all.Examples of incorrect Equations(a) With 0 Wild, the Equation uses a 0 for one symbol and another 0 for a different symbol.(b) Elem/Mid: With Average, the Solution equals the Goal if + is interpreted as addition but not as

average.(c) Mid/Jr/Sr: With Multiple of k, the Solution equals the Goal rather than differing from it by a

multiple of k.


or Permitted or challenge Impossible. When the cube has been moved, each player has two minutes to write an Equation.The last cube in Resources may not be moved to Forbidden. If a player does so, any challenge that is made is set aside and the cube is returned to Resources. There is no penalty unless the player’s time to move expires. (See Section XI.)

B. An opponent may challenge Impossible against the player who moved the last cube from Resources to Required or Permitted provided the challenge is made by the end of the first minute for writing Equations. If the challenge is made, the Mover (and the Third Party, if siding with the Mover) has the rest of the original two minutes to write an Equation.Comment Any Now challenge with one cube or zero cubes left is invalid, as is any Impossible challenge

made after the first minute for writing Equations. In both cases, the challenge is set aside. A player challenging Now in this case receives a one point penalty.

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IX. Illegal ProceduresA. Any action that violates a procedural rule is illegal procedure. A player charging illegal


their previous position (usually Resources) and, if necessary, make another move.The Mover must be given at least 10 seconds to make this correction, unless the original move was made after the 10-second countdown (see Section XI-A-3 below), in which case the time limit rule (Section XI-A) is enforced. In general, there is no direct penalty except that the Mover may lose a point if he does not legally complete his turn during the time limit.Examples of illegal proceduresMoving out of turn, moving two cubes without calling “Bonus” before the first cube touches the mat in Forbidden, moving the last cube in Resources to Forbidden.

2. If the move is not an illegal procedure, the cube stands as played.Comment There is no penalty for erroneously charging illegal procedure. However, see Section IX-C if a player does so frequently.


Example Suppose a player makes an illegal bonus move (two cubes to Required). Before anyone notices the illegal procedure, the next mover moves (or a valid challenge is issued). In this case, the illegal bonus move stays without penalty.


language, keeping a hand or finger over or next to the challenge block, making numerous false accusations of illegal procedure, and so on. It also includes not playing to win but rather trying only to ruin the perfect scores of one or both opponents (for example, by erroneously challenging Now or Impossible at or near the beginning of each shake so that both opponents will score 5 for the round), saying one variation but circling another, constantly charging illegal procedure erroneously, counting down the 10-second warning in an obnoxious manner, etc.


1. That player had to write an Equation and did so correctly.If the Third Party agrees with the person who must write an Equation, the Third Party must write a correct Equation also.

2. That player did not have to write an Equation (someone else did), and no opponent wrote a correct Equation.Exception: After a challenge in a three-player match, a player who does not present an Equation for a shake scores 2 if he accepts another player’s Equation as correct even if that Equation is subsequently proven wrong by the other checker.

B. After a challenge, points are awarded as follows.1. Any player who is not correct scores 2.

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2. Any player who is correct scores 6, unless that player is the Third Party agreeing with the Challenger, in which case the score is 4.

C. After the last cube from Resources is moved to the playing mat and no one challenges Impossible, points are awarded as follows:1. Any player who writes a correct Equation scores 4.2. Any player who does not write a correct Equation scores 2.



minute time limits are enforced with the timer. If a player fails to meet a deadline, he loses one point and has one more minute to complete the task. If he is not finished at the end of this additional minute, he loses his turn or is not allowed to complete the task.Note: In Elementary and Middle Divisions, each one-point penalty must be approved by a judge initialing the scoresheet.1. The time limits are as follows:

a. rolling the cubes 1 minuteb. making a variation selection

This time limit does not begin until after the one minute for rolling the cubes.15 seconds

c. setting the Goal 2 minutesd. first turn of the player to the left of the Goal-setter 2 minutese. all other regular turns (including any bonus moves) 1 minutef. stating a valid challenge after picking up the challenge block 15 secondsg. deciding whether to challenge Impossible when no more cubes

remain in ResourcesIf the Impossible challenge is made, any time (up to a minute) the Challenger takes deciding to challenge counts as part of the two minutes for writing an Equation.

1 minute

h. writing an Equation 2 minutesDuring this time, the Third Party (if there is one) must decide whether to present an Equation after a Now or an Impossible challenge. At the end of these two minutes they must present their Equation.

i. deciding whether an opponent’s Equation is correct 2 minutes2. Often a player completes a task before the time limit expires. When sand remains in

the timer from the previous time limit, the next player will receive additional time. An opponent timing the next player may either flip or not flip the timer so as to give the opponent the lesser amount of time before the remaining sand runs out and the next time limit can be started.

3. A player who does not complete a task before all the sand runs out for the time limit must be warned that time is up. An opponent must then count down 10 seconds loud enough for the opponent to hear. The one-point penalty for exceeding a time limit may be imposed only if the player does not complete the required task by the end of the countdown.The countdown must be done at a reasonable pace; for example, “1,010; 1,009; 1,008…,”An exception to this rule occurs when a player picks up the Challenge Block but does not state a valid challenge within the 15-second time limit. If the player does not wish to challenge, he loses one point and play continues.

B. A round lasts a specified amount of time (usually 30 minutes). After 30 minutes, players

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Second place 4

are told not to start any more shakes. Players have five minutes to finish the last shake. After these five minutes, players still involved in a shake in which no challenge has been made and one or more cubes remain in Resources will be told: “Stop, don’t move another cube – this is the end of the round. Each player has two minutes to write a correct Equation that may use any of the cubes remaining in Resources. Any player who presents a correct Equation scores 4 points for that shake; an incorrect Equation scores 2.”

XII. Scoring a MatchA. Each player is awarded points for the match based on the sum of his scores for the shakes

played during that match according to the following tables:


However, if there is evidence of intent to deceive and the error was not a simple oversight, then do the following:1. If the error gives the player a lower score, she receives the lower score.2. If the error gives the player a higher score, she receives 0 for that round.

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Third place 2

XIII. Adventurous VariationsComment See Section II-B for the procedure to be followed when selecting variations.

A . Elementary Variations (grade 6 and below) Note counting numbers = natural numbers = positive integers = 1, 2, 3, 4, ...

whole numbers = 0, 1, 2, 3, 4, ...GENERAL RULE: If * (or ^) is used for raising to a power, both base and exponent must be whole numbers. If √ is used for the root operation, the index must be a counting number, and the base and total value must be whole numbers.Examples (in each case, * may be replaced by ^.)(a) 3 * 2 is acceptable and equals 9. 0 * 9 equals 0 and 7 * 0 equals 1. However, 2*(1–3), 4*(1÷2), (2–

5)*4, and (2÷3)*3 are not legal in Elementary.(b) 2√9 or just √9 is acceptable and equals 3. 9√0 equals 0. However, √5 and 3√9 are not legal since neither is

a whole number. Also 2√(1÷3), (1÷2)√5, and 3√(1–9) are illegal in Elementary.(c) The legality of √3*4 depends on its grouping. √(3*4) is legal; (√3)*4 is not.

1. Sid e ways: A cube representing a non-zero number may be used sideways in the Goal or Solution to equal the reciprocal of that number.Examples 1 + 2 + = 1 + 2 + 0.5 = 3.5;

1 ÷ = 1 ÷ (1/3) = 1 x 3 = 3Comment See Appendix for ways to indicate a sideways cube in an Equation.

2. Upsi d e- d o wn: A cube representing a number may be used upside-down in the Goal or Solution to equal the additive inverse of that number.Examples 6 x = 6 x (–2) = –12. However, 6 is not legal for 6 – 2 or 60 + (–2).Comment See Appendix for ways to indicate an upside-down cube in an Equation.

Note: The Sideways rule and the Upside-down rule may be used on a single-digit numeral at the same time if both variations have been selected, but only in a Solution.Examples 3 = -⅓; 8 = -⅛

sw sw

ud ud

3. 0 Wild: The 0 cube may represent any numeral on the cubes, but it must represent the same numeral everywhere it occurs (Goal and Solution). Each Equation-writer must specify in writing the interpretation of the 0 cube if it stands for anything other than 0 in the Equation.Examples(a) (0 x 6) – 0 = 15, where both 0’s stand for 3, is allowed but (0 x 6) – 0 = 14, where the first 0 stands for 3

and the second for 4, is not allowed.(b) (0 x 6) – 0 = 12, where the first 0 stands for 2 and the second for 0, is not allowed.(c) A 0 in the Goal and any 0 in the Solution must equal the same number. So (8 x 5) + 0 equals the Goal 40 if

each 0 equals 2. However, (9 x 5) – 0, where this 0 stands for 5, does not equal the Goal 40.

4. Factorial: There are two occurrences of the factorial operator (!) available to be used in the Solution and/or the Goal as the Equation-writer chooses to use them. All uses of ! in the Equation must be in writing.Comments(a) 5! (“5 factorial”) means 5 x 4 x 3 x 2 x 1, which equals 120. n! is defined only for whole number values of n.

0! is defined as 1.(b) In the absence of grouping symbols, ! applies to just the numeral in front of it. See Section VII-B-6 for

examples.Examples(a) For the Goal 4 x 30, a Solution of 5! is not correct since it contains only one cube.

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(b) If the Goal is 9 ÷ 8, an Equation-writer may interpret it as 9! ÷ 8!, which is 9. However, the Solution may not contain an ! since both allotted factorial signs have been placed in the Goal (unless Multiple Operations has been called – see below).

(c) The Equation 5 x 4 ÷ 0! = 20 is correct since 0! equals 1 as defined in mathematics texts.(d) The Equation (8 – 5)! + 2 = 4! ÷ 3 is correct since the Goal is (4 x 3 x 2) ÷ 3 = 4 x 2 = 8.(e) 3!! = (3!)! = (3 x 2)! = 6! = 6 x 5 x 4 x 3 x 2 = 720

5. M ultiple O perations: Every operation sign in Required or Permitted may be used many times in any Solution. If the Factorial variation is also chosen for the shake, an unlimited number of factorial operators may be used in each Solution. At most two factorials may be used in the Goal.Comments(a) After an Impossible challenge, any operation sign in Resources may be used many times in a Solution. After

a Now challenge, if the one cube allowed from Resources is an operation sign, it may be used multiple times.

(b) Players may simply write an operation sign multiple times in Solutions without any additional comment since an operation cube is not being used to represent another symbol.

6. Three-o p eration Solution: Any Solution must contain at least three operation symbols. The operation symbols are +, –, x, ÷, * (or ^), √ (and ! if Factorial is chosen).Comment x used for number of factors is an operation symbol, as is an upside-down radical used for percent.

However, a * (or ^) used as a decimal point does not count as an operation.

7. Remainder: A B ( is a sideways ÷) equals the remainder when A is divided by B. A and B are positive integers, and A is less than or equal to 1000.Examples(a) 15 2 = 1 since 15 divided by 2 gives a quotient of 7 with remainder 1.(b) 89 15 = 14, the remainder when 89 is divided by 15.(c) 45 70 = 45 [In general, A B = A when A < B.](d) 87 87 = 0(e) 54 10 = 4 [In general, A 10 = the last digit of A].

The following even-year variations will be played in Elementary in 2017-18.8. Next Prime Number: xA means “the next prime number bigger than A,” where A is a rational

number less than or equal to 200.Comment This variation does not rule out using x for multiplication. In the Goal or Solution, the meaning of an x

cube will usually be clear from the context since next prime is a unary operation and multiplication is a binary operation. For example, the Goal 4xx6 has only one interpretation: 4 x (x6), which is 4 x 7 = 28.

(a) x7 = 11, the next prime number bigger than 7.(b) x(9 ÷ 2) = 5, the next prime bigger than 4.5.(c) x(0 – 3) = 2, the next prime number bigger than –3. (Note: 1 is not prime.)(d) xx5 = x(x5) = x7 = 11.(e) x√49 = x7 = 11. x√67 is undefined because √67 is not a rational number.(f) In the expression 2xx5, the first x means multiplication and the second means next prime number. The value

of this expression is 2 x (x5) = 2 x 7 = 14.(g) In the expression x5x7, the first x means next prime number and the second means multiplication. The value

of this expression is either x(5x7) = 37, the next prime number bigger than 35, or (x5)x7 = 7 x 7 = 49.(h) There is no limit to the number of consecutive x’s in an expression, especially with the Multiple Operations

variation also in effect. Thus xxx9 = xx(x9) = x(x11) = x13 = 17.

9. Percent: (upside-down radical) means “percent of.” That is, A B = A% of B where A and B are numbers. In the Goal or Solution, A and/or B may be a two-digit numeral.

Examples(a) 25 16 = 25% of 16 = 0.25 x 16 = 4.(b) 6 8 = 6% of 8 = 0.06 x 8 = 0.48.(c) In a Solution, (8 – 3) (4 + 2) = 5% of 6 = 0.05 x 6 = 0.3.(d) If the Decimal Point variation (see below) is also in effect, an expression like 1.5 .25 is legitimate in a

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Solution and equals 1.5% of 0.25 = 0.015 x 0.25 = 0.00375. Similarly, in a Solution, 25 (1.5 x 2) = 25% of 3 = 0.75. And 12.5 16 = 2.

10. Decimal P o i n t: * (or ^) may represent a decimal point. If so used in the Goal or Solution, an * (or ^) may be combined with at most three digits to form a numeral. When used as a decimal, * (or ^) takes precedence over all other operations. Comment This variation does not rule out using * (or ^) for exponentiation. Therefore, Equation-writers are encouraged to write a decimal point instead of * (or ^) when they want to use * (or ^) as a decimal point. Also one * (or ^) may be used as a decimal point and another for exponentiation. If * (or ^) is used as a decimal point, this must be indicated in writing in the Equation. (See Appendix.)Examples (in each case, * may be replaced by ^.)(a) 2*5 means either 2.5 or 25; 3*0 means either 3.0 or 30. The Equation-writer must indicate whether the decimal interpretation is desired. One way to do this is to write 2.5 or 3.0 rather than 2*5 or 3*0.(b) 2*4x2 = 2.4 x 2 or 28 or 24 x 2; it does not mean 2.8. That is, it may not be grouped as 2.(4x2). Similarly 4*3! may not be interpreted as 4.(3!) or 4.6.(c) 12*5 means either 12.5 or (in the Goal only unless the Two-digit Numerals variation has been chosen) 125; 1*25 means 1.25 or (in the Goal only unless the Two-digit Numerals variation has been chosen) 125.(d) 15*0 means either 15.0 or (in the Goal only unless the Two-digit Numerals variation has been chosen) 150.(e) In the Goal or Solution, 255* means 255 and *050 means 0.05.(f) 15*25 = 1525 (in the Goal only unless the Two-digit Numerals variation has been chosen) but has no legitimate interpretation as a decimal since * is used with four digits.(g) 122*5, 1*225, *1225, and 1225* have no interpretations as decimals or powers.(h) The expression *37*5 may not be interpreted as 0.37 1/2 and has no defined interpretation in Elementary Division.(i) The “digits” are the symbols 0,1,2,3,4,5,6,7,8 and 9. A digit turned sideways or upside-down is no longer a digit. Therefore, with Sideways or Upside-down in effect, you may not use an expression like * , *, * or * and interpret the * as a decimal point.

11. += Average: + shall not represent addition; instead, it shall represent the operation of averaging two numbers.Examples(a) 7 + 9 = 8, the average of 7 and 9. 7 – (0 – 9) equals 16, as usual.(b) 5 + (4 x 2) = the average of 5 and 8 = 6.5.(c) 4+6+9 has two values: (4+6)+9 = 5+9 = 7; 4+(6+9) = 4+7.5 = 5.75. Notice that neither answer equals 19/3, the usual (mathematical) average of 4, 6, and 9.

The following odd-year variations will be played again in 2018-19.12. Two-digit Numerals: Two-digit numerals are allowed in Solutions.

13. LCM: √ may represent the LCM (least common multiple) of two counting numbers.Comment This variation does not rule out using √ for root, so each Equation-writer must indicate in writing which

√ in the Equation represents LCM. (See Appendix.)Examples(a) 6 √ 8 = 24, the smallest integer divisible by both 6 and 8.(b) (2 x 3) √ (5 + 4) = 6 √ 9 = 18.(c) 2 √ 4 = 4 (or 2, the square root of 4).(d) 0 √ 5 is undefined.

(e) 6√√9 means the LCM of 6 and the square root of 9; that is, the LCM of 6 and 3, which is 6.

14. GCF: * (or ^) may represent the GCF (greatest common factor) of two whole numbers, provided at least one of them is not 0.GCF(A, B), “the greatest common factor of A and B,” is defined if A and B are counting numbers or if A is a counting number and B = 0 or if B is a counting number and A = 0. GCF(A, 0) = A and GCF(0, B) = B.

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Comment This variation does not rule out using * (or ^) for exponentiation. So each Equation-writer must indicate in writing which * (or ^) in the Equation represents GCF. (See Appendix.)

Examples (in each case, * may be replaced by ^.)(a) 8 * 6 = either 86 or 2, the largest integer that is a divisor of both 8 and 6.(b) (4 x 2) * (6 + 3) = 1 (with GCF) or 89 (with exponentiation).

15. Number of Factors: xA means “the number of counting number factors of A,” where A is a counting number less than or equal to 200.Comments(a) This variation does not rule out using x for multiplication. In the Goal or Solution, the meaning of an x cube

will usually be clear from the context since number of factors is a unary operation and multiplication is a binary operation.

Examples(a) x(6 x 2) = 6 (since 12 has six factors: 1, 2, 3, 4, 6, 12)(b) x(4 x 4) = 5 (since the factors of 16 are 1, 2, 4, 8, 16)(c) x12 = 6 (for use in any Goal or, if the Two-digit Numerals variation is chosen, in a Solution)(d) x0, x(1 ÷ 2), x(1 – 4), and x(5 * 4) [or x(5 ^ 4)] are not defined.(e) xx12 = x(x12) = x6 = 4(f) In the expression 3xx7, the first x means multiplication and the second means number of factors. 3xx7 = 3 x

(x7) = 3 x 2 = 6.(g) In the expression x4x2, the first x means number of factors and the second means multiplication. By default,

the value of this expression is (x4)x2 = 3 x 2 = 6. To obtain the number of factors of 8, the Equation-writer must write x(4x2).

B . Middle Division Variations (grades 7- 8 )

1. Sid e ways: A cube representing a non-zero number may be used sideways in the Goal or Solution to equal the reciprocal of that number. Examples 1 + 2 + = 1 + 2 + 0.5 = 3.5;


2. Upsi d e- d o wn: A cube representing a number may be used upside-down in the Goal or Solution to equal the additive inverse of that number. Examples 6 x = 6 x (–2) = –12. However, 6 is not legal for 6 – 2 or 60 + (–2).

Comment See Appendix for ways to indicate an upside-down cube in an Equation.


sw sw

ud ud

3. 0 Wild: The 0 cube may represent any symbol (numeral or operation) on the cubes, but it must represent the same symbol everywhere it occurs (Goal and Solution). Each Equation-writer must specify in writing the interpretation of the 0 cube if it stands for anything other than 0 in the Equation. Examples

(a) (0 x 6) – 0 = 15, where both 0’s stand for 3, is allowed but (0 x 6) – 0 = 14, where the first 0 stands for 3 and the second for 4, is not allowed.

(b) (0 x 6) – 0 = 12, where the first 0 stands for 2 and the second for 0, is not allowed.(c) A 0 in the Goal and any 0 in the Solution must equal the same number. So (8 x 5) + 0 equals the Goal

40 if each 0 equals 2. However, (9 x 5) – 0, where this 0 stands for 5, does not equal the Goal 40.

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Comments(a) If a player interprets 0 in the Goal as x, then any 0 in that player’s Solution must also be an x.(b) If 0 Wild and Multiple Operations are both chosen, 0 may be used multiple times in a Solution only if it

stands for an operation sign, not a numeral.(c) If Base eight is also chosen (see below), 0 may not represent the digits “8” or “9”. If Base nine is chosen,

0 may not represent “9”.

4. Factorial: There are two occurrences of the factorial operator (!) available to be used in the Solution and/or the Goal as the Equation-writer chooses to use them. All uses of ! in the Equation must be in writing. Comments

(a) 5! (“5 factorial”) means 5 x 4 x 3 x 2 x 1, which equals 120. n! is defined only for whole number values of n. 0! is defined as 1.

(b) In the absence of grouping symbols, ! applies to just the numeral in front of it. See Section VII-B-6 for examples.

Examples(a) For the Goal 4 x 30, a Solution of 5! is not correct since it contains only one cube.(b) If the Goal is 9 ÷ 8, an Equation-writer may interpret it as 9! ÷ 8!, which is 9. However, the Solution may

not contain an ! since both allotted factorial signs have been placed in the Goal (unless Multiple Operations has been called – see below).


5. M ultiple O perations: Every operation sign in Required, Permitted, or Resources may be used many times in any Solution. If the Factorial variation is also chosen for the shake, an unlimited number of factorial operators may be used in each Solution. At most two factorials may be used in the Goal. Comments

(a) After an Impossible challenge, any operation sign in Resources may be used many times in a Solution. After a Now challenge, if the one cube allowed from Resources is an operation sign, it may be used multiple times.


6. Base m : Both the Goal and the Solution must be interpreted as base m expressions, where the player choosing this variation specifies m for the shake as eight, nine, or ten. Two-digit numerals are allowed in Solutions. Examples (in each case, * may be replaced by ^.)

(a) For Base eight, 37 + 5 = 6 * 2 is a correct Equation. Any Solution or Goal containing the digit “8” or “9” is an illegal expression.

(b) For Base nine, 34 + 5 = 6 * 2 is a correct Equation. Any Solution or Goal containing the digit “9” is an illegal expression.

(c) In Base eight, a Goal like 3 + 8 or 39 should be challenged Impossible. A Goal like 39 is also illegal in Base nine.

7. Multiple of k : A Solution must not equal the Goal but must differ from the Goal by a non-zero multiple of k, where the player choosing this variation specifies k for the shake as a whole number from six to eleven, inclusive. The Goal must not be greater than 1000(in Base 10) or less than –1000 (in Base 10). Example If k = 6 and the Goal is 5, then a Solution must equal 11, 17, 23, 29, and so on,

or –1, –7, –13, –19, and so on. A Solution equal to 5 is incorrect.Comment Multiple of k does not require any special writing of the Goal by an Equation-writer. As always,

write the interpretation of the Goal, indicating wild cubes and grouping. You do not have to indicate the multiple of k difference.

The following even-year variations will be played in Middle in 2017-18:

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8. Percent: (upside-down radical) means “percent of.” That is, A B = A% of B, where A and B are numbers. In the Goal or Solution, A and/or B may be a two-digit numeral. If Base m and Percent are both chosen, the meaning of percent (“per 100”) changes with the base. “Percent”

means “per sixty-four” for Base eight and “per eighty-one” for Base nine. Examples

(a) 25 16 = 25% of 16 = 0.25 x 16 = 4. (b) 6 8 = 6% of 8 = 0.06 x 8 = 0.48.

(c) In a Solution, (8 – 3) (4 + 2) = 5% of 6 = 0.05 x 6 = 0.3.(d) If the Decimal Point variation (see below) is also in effect, an expression like 1.5 .25 is legitimate in a Solution and equals 1.5% of 0.25 = 0.015 x 0.25 = 0.00375. Similarly, in a Solution, 25 (1.5 x 2) = 25% of 3 = 0.75. And 12.5 16 = 2.

(e) In Base eight, 60 11 = (60eight ÷ 100eight) x 11eight = (48ten ÷ 64ten) x 9ten = (3ten ÷ 4ten) x 9ten = 27ten ÷ 4ten = (6 + 3/4)ten = (6 + 6/8)ten = 6.6eight

9. Decimal Po i n t: * (or ^) may represent a decimal point. If so used in the Goal or Solution, an * (or ^) may be combined with at most three digits to form a numeral. When used as a decimal, * takes precedence over all other operations.

Example A Goal of 4**5 (or 4^^5) can equal either 4.5 = 1028 or 4.5 = √4 = 2.Comment If 0 Wild and Decimal Point are both chosen, 0 may represent a decimal point. Also, one 0 in the Equation may be decimal point and another 0 may be exponentiation since 0 is the symbol * in both cases.

10.+= Average: + shall not represent addition; instead, it shall represent the operation of averaging two numbers.

Examples(a) 7 + 9 = 8, the average of 7 and 9. 7 – (0 – 9) equals 16, as usual.(b) 5 + (4 x 2) = the average of 5 and 8 = 6.5.(c) 4+6+9 has two values: (4+6)+9 = 5+9 = 7; 4+(6+9) = 4+7.5 = 5.75. Notice that neither answer equals 19/3, the usual (mathematical) average of 4, 6, and 9.Comment If 0 Wild is chosen along with Average, any 0 that represents + must mean average.

11. Next Prime Number: xA means “the next prime number bigger than A,” where A is a rational number less than or equal to 200.

Comment This variation does not rule out using x for multiplication. In the Goal or Solution, the meaning of an x cube will usually be clear from the context since next prime is a unary operation and multiplication is a binary operation. For example, the Goal 4xx6 has only one interpretation: 4 x (x6), which is 4 x 7 = 28.(a) x7 = 11, the next prime number bigger than 7.(b) x(9 ÷ 2) = 5, the next prime bigger than 4.5.(c) x(0 – 3) = 2, the next prime number bigger than –3. (Note: 1 is not prime.)(d) xx5 = x(x5) = x7 = 11.(e) x√49 = x7 = 11. x√67 is undefined because √67 is not a rational number.(f) In the expression 2xx5, the first x means multiplication and the second means next prime number. The value of this expression is 2 x (x5) = 2 x 7 = 14.(g) In the expression x5x7, the first x means next prime number and the second means multiplication. The value of this expression is either x(5x7) = 37, the next prime number bigger than 35, or (x5)x7 = 7 x 7 = 49.(h) There is no limit to the number of consecutive x’s in an expression, especially with the Multiple Operations variation also in effect. Thus xxx9 = xx(x9) = x(x11) = x13 = 17.

The following odd-year variations will be played again in 2018-19.

12. Powers of the Base: 1 (one) may represent any integral power of ten. (If 1 is used in a two-digit numeral, it stands for 1.) If Base m is also chosen, 1 represents any integral power of m.

Examples

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(a) For Base ten, 9 + 1 may be interpreted as 9 + 1 (since 100 = 1), 9 + 10, 9 + 100, 9 + 1000, and so on, or as 9 + 0.1 (since 10–1 = 0.1), 9 + 0.01, 9 + 0.001, and so on.

(b)If Base eight is chosen, then 1 may represent one, eight, sixty-four, and so on, or one-eighth, one-sixty-fourth, and so on. For Base nine, 1 represents one, nine, eighty-one, one-ninth, etc.

13. Number of Factors: xA means “the number of counting number factors of A,” where A is a counting number and A is less than or equal to 1000. Comments

(a) “1000” in the statement of the variation refers to the number one-thousand in base ten. The limit is not reinterpreted if Base Eight or Nine is chosen.

(b) This variation does not rule out using x for multiplication. In the Goal or Solution, the meaning of an x cube will usually be clear from the context since number of factors is a unary operation and multiplication is a binary operation.

(c) If 0 Wild is chosen along with Number of Factors, one 0 may represent number of factors while another 0 may be multiplication since 0 is the symbol x in both cases.

Examples(a) x(6 x 2) = 6 (since 12 has six factors: 1, 2, 3, 4, 6, 12)(b) x(4 x 4) = 5 (since the factors of 16 are 1, 2, 4, 8, 16)(c) x12 = 6 (for use in any Goal or, if the Two-digit Numerals variation is chosen, in a Solution)(d) x0, x(1 ÷ 2), x(1 – 4), and x(5 * 4) [or x(5 ^ 4)] are not defined.(e) xx12 = x(x12) = x6 = 4(f) In the expression 3xx7, the first x means multiplication and the second means number of factors. 3xx7 =

3 x (x7) = 3 x 2 = 6.(g) In the expression x4x2, the first x means number of factors and the second means multiplication. By

default, the value of this expression is (x4)x2 = 3 x 2 = 6. To obtain the number of factors of 8, the Equation-writer must write x(4x2).

14. Any Color Exponent: Any numeral on a cube may be used as an exponent without being accompanied by an * (or ^) cube. The player selecting this variation chooses a color: red, blue, green, or black and should announce “Red Exponent” or “Blue Exponent”, etc. to indicate the color. Examples

(a) If the chosen exponent color is red, the Goal 253, where the 3 is red, must mean 253 since three-digit numerals are illegal.

(b) If blue is the chosen color, a Goal or a Solution like 52, where the 2 is on a blue cube, is legal.(c) If red is the chosen color, an expression like 523 [2 and 3 red] must be interpreted as either 523, or (52)3,

which is 56. It may not be interpreted as 5*(2*3) or 58, because the 2 by itself is no longer an exponent of the 5.

Comments(a) If Factorial is also chosen, a ! may be placed behind an exponent. So with red exponent, a Goal of 23

(red 3) may be interpreted as 23! or 26.(b) If a player selects Exponent when no digits of the selected color were rolled, that player is penalized

one point and must pick another variation.

15. AB+: The Goal and/or the Solution may be or may include a three-cube expression of the form AB+, which is interpreted as a repeating decimal. It may be interpreted as .ABABAB... or as .ABBBBB... A player who presents an Equation must specify in writing which interpretation is used in the Goal and Solution. No decimal points may be used in the Equation (except when the decimal point variation is also chosen for the shake). When the form AB+ is used in an Equation, the Equation-writer should indicate in writing which fraction the AB+ is used for (.ABAB or .ABBB). Examples

(a) 33+ = 0.333333…(.ABAB interpretation, which is 33/99 and simplifies to 1/3) or 0.333333 (.ABBB interpretation, which also simplifies to 1/3).

(b) 90+ = 0.909090…(.ABAB interpretation, which is 90/99 and simplifies to 10/11) or 0.900000 (.ABBB interpretation, which is [90-9]/90, which simplifies to 9/10).

(c) 1-(55+) = (99/99) – (55/99) = 44/99 (which simplifies to 4/9 in either interpretation).Comments

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(a) If Base m is also chosen, the decimal places also change to the appropriate base m expression, meaning the fractions change to AB/77 for Base 8, AB/88 for Base 9, etc.

(b) In Base m, the numerators and denominators in fractions also must be converted to the appropriate base.

C . Junior Division Variations (grades 9- 10 ) The following two variations are in effect for all shakes in Junior and Senior Divisions.

Sideways: A cube representing a non-zero number may be used sideways in the Goal or Solution to equal the reciprocal of that number.Examples 1 + 2 + = 1 + 2 + 0.5 = 3.5;


Upsi d e- d o wn: A cube representing a number may be used upside-down in the Goal or Solution to equal the additive inverse of that number.Examples 6 x = 6 x (–2) = –12. However, 6 is not legal for 6 – 2 or 60 + (–2).Comment See Appendix for ways to indicate an upside-down cube in an Equation.


sw sw

ud ud

Note that 0 or x Wild, Base m and Multiple of k are expanded from Middle variations. 1. 0 or x Wild: The 0 or x cube may represent any symbol on the cubes, but it must represent

the same symbol everywhere it occurs (Goal and Solution). Each Equation-writer must specify in writing the interpretation of the 0 or x cube if it stands for anything other than itself in the Equation. The player selecting this variation specifies whether 0 or x (but not both) is wild for the shake.Examples for x Wild (for 0 Wild, see the examples for Elementary and Middle Divisions)(a) x – (x ÷ 3) = 4, where both x’s stand for 6, is a correct Equation.(b) (9 x 3) x 5 = 1, where both x’s stand for –, is a correct Equation.(c) x – (3 x 2) = 2, where the first x is 7 and the second x is +, is not a correct Equation.(d) An x in the Goal and any x in the Solution must represent the same symbol. For example, the Equation (4 *

2) – 2 = 2x7 is incorrect since x stands for * (or ^) on the left side and for ↑ multiplication (by default) in the Goal. x

2. Factorial: There are two occurrences of the factorial operator (!) available to be used in the Solution and/or the Goal as the Equation-writer chooses to use them. All uses of ! in the Equation must be in writing. However, if Multiple of k is also chosen for the shake, no factorial may be placed in the Goal.Comments(a) 5! (“5 factorial”) means 5 x 4 x 3 x 2 x 1, which equals 120. n! is defined only for whole number values of

n. 0! is defined as 1.(b) In the absence of grouping symbols, ! applies to just the numeral in front of it. See Section VII-B-6 for

examples.Examples(a) For the Goal 4 x 30, a Solution of 5! is not correct since it contains only one cube.(b) If the Goal is 9 ÷ 8, an Equation-writer may interpret it as 9! ÷ 8!, which is 9. However, the Solution may not

contain an ! since both allotted factorial signs have been placed in the Goal (unless Multiple Operations has been called – see below).

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3. M ultiple O perations: Every operation sign in Required or Permitted may be used many times in any Solution. If the Factorial variation is also chosen for the shake, an unlimited number of factorial operators may be used in each Solution. At most two factorials may be used in the Goal.Comments(a) After an Impossible challenge, any operation sign in Resources may be used many times in a Solution.

After a Now challenge, if the one cube allowed from Resources is an operation sign, it may be used multiple times.


4. Any Color Exponent: Any numeral on a cube may be used as an exponent without being accompanied by an * (or ^) cube. The player selecting this variation fills the blank in the previous sentence with one of the colors red, blue, green, or black. The player should announce “Red Exponent” or “Blue Exponent”, etc. to indicate the color.

Examples(a) If the chosen exponent color is red, the Goal 253, where the 3 is red, must mean 253 since three-digit

numerals are illegal.(b) If blue is the chosen color, a Goal or a Solution like 52, where the 2 is on a blue cube, is legal.(c) If red is the chosen color, an expression like 523 [2 and 3 red] must be interpreted as either 523, or (52)3,

which is 56. It may not be interpreted as 5*(2*3) or 58, because the 2 by itself is no longer an exponent of the 5.

Comments(a) If Factorial is also chosen, a ! may be placed behind an exponent. So with red exponent, a Goal of 23

(red 3) may be interpreted as 23! or 26.(b) If a player selects Any Color Exponent when no digits of the selected color were rolled, that player is

penalized one point and must pick another variation.

5. Base m : Both the Goal and the Solution must be interpreted as base m expressions, where the player choosing this variation specifies m for the shake as eight, nine, ten, eleven, or twelve. Two-digit numerals are allowed in Solutions. For bases eleven and twelve, * (or ^) may be used for the digit ten; in base twelve, √ may be used for the digit eleven.If Sideways and Base Eleven (or Twelve) are both chosen, an * (or ^) may be used sideways to represent one-tenth. If the * (or ^) is part of a two-digit numeral, it may not be interpreted as sideways. If an * (or ^) is a one-digit numeral in the Goal, the Equation-writer may interpret the * (or ^) as right-side up or sideways regardless of the way the * (or ^) is physically placed in the Goal. In a Solution, the writer must clearly indicate if an * (or ^) is sideways.If Upside-down and Base Eleven (or Twelve) are both chosen, an * (or ^) may be used upside-down to represent -10. If the * (or ^) is part of a two-digit numeral, it may not be interpreted as upside-down. If an * (or ^) is a one-digit numeral in the Goal, the Equation-writer may interpret the * (or ^) as right-side up or upside-down regardless of the way the * (or ^) is physically placed in the Goal. In a Solution, the writer must clearly indicate if an * (or ^) is upside-down.Note: For newer games with ^ on the cubes instead of *, a ^ must never be placed sideways or upside-down in the Goal or written sideways or upside-down in the Equation. This is consistent with the principle that ^ inherits all the properties of *, one of which is that * is ambiguous with regard to sideways and upside-down.Examples (in each case, * may be replaced by ^.)(a) For Base eight, 37 + 5 = 6 * 2 is a correct Equation. Any Solution or Goal containing the digit “8” or “9” is an

illegal expression.(b) For Base nine, 34 + 5 = 6 * 2 is a correct Equation. Any Solution or Goal containing the digit “9” is an illegal

expression.(c) In Base eight, a Goal like 3 + 8 or 39 should be challenged Impossible. A Goal like 39 is also illegal in Base

nine.

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Comments(a) In bases eleven and twelve, * (or ^) may still represent exponentiation; in base twelve, √ may still represent

root. If the interpretation of an * (or ^) or √ is not clear from the context of the Solution, the Equation-writer must indicate which meaning is desired so as to eliminate any ambiguity in the Solution or the Goal. (See Appendix.)

(b) If Powers of the Base is chosen with Base Eleven, 1 may mean one, eleven, one-hundred twenty-one, and so on, or one-eleventh, one one-hundred twenty-first, and so on. If Powers of the Base is chosen along with Base Twelve, 1 may mean one, twelve, one-hundred forty-four, and so on, or one-twelfth, one one-hundred-forty-fourth, and so on.

(c) If 0 (or x) wild is chosen along with Base Eleven or Twelve, a wild cube may represent * (or ^) for ten or √ for eleven (or exponentiation or root as long as each wild cube represents the same symbol).

6. Powers of the Base: 1 (one) may represent any integral power of ten. (If 1 is used in a two-digit numeral, it stands for 1.) If Base m is also chosen, 1 represents any integral power of m.Examples(a) For Base ten, 9 + 1 may be interpreted as 9 + 1 (since 100 = 1), 9 + 10, 9 + 100, 9 + 1000, and so on, or

as 9 + 0.1 (since 10–1 = 0.1), 9 + 0.01, 9 + 0.001, and so on.(b) If Base eight is chosen, then 1 may represent one, eight, sixty-four, and so on, or one-eighth, one-sixty-

fourth, and so on. For Base nine, 1 represents one, nine, eighty-one, one-ninth, etc.

7. Multiple of k : A Solution must not equal the Goal but must differ from the Goal by a non-zero multiple of k, where the player choosing this variation specifies k for the shake as a whole number from six to twelve, inclusive. Example If k = 6 and the Goal is 5, then a Solution must equal 11, 17, 23, 29, and so on,

or –1, –7, –13, –19, and so on. A Solution equal to 5 is incorrect.Comment Multiple of k does not require any special writing of the Goal by an Equation-writer. As always, write

the interpretation of the Goal, indicating wild cubes and grouping. You do not have to indicate the multiple of k difference.

8. Number of Factors: xA means “the number of counting number factors of A,” where A is a counting number.Comment Since there is no limit to the size of A, it is possible to present an Equation that is uncheckable. For

example, with Multiple of k =11 and Factorial:x(8!! + 1) = 5

Any such Equation that cannot be verified (even with a calculator) by opponents and judges as correct or incorrect will be ruled incorrect.Comments(a) This variation does not rule out using x for multiplication. In the Goal or Solution, the meaning of an x cube

will usually be clear from the context since number of factors is a unary operation and multiplication is a binary operation.

(b) If 0 Wild is chosen along with Number of Factors, one 0 may represent number of factors while another 0 may be multiplication since 0 is the symbol x in both cases.

Examples(a) x(6 x 2) = 6 (since 12 has six factors: 1, 2, 3, 4, 6, 12)(b) x(4 x 4) = 5 (since the factors of 16 are 1, 2, 4, 8, 16)(c) x0, x(1 ÷ 2), x(1 – 4) are not defined.(d) xx12 = x(x12) = x6 = 4(e) In the expression 3xx7, the first x means multiplication and the second means number of factors. 3xx7 = 3

x (x7) = 3 x 2 = 6.(f) In the expression x4x2, the first x means number of factors and the second means multiplication. By

default, the value of this expression is (x4)x2 = 3 x 2 = 6. To obtain the number of factors of 8, the Equation-writer must write x(4x2).

The following even-year variations will be played in Junior/Senior in 2017-18:

9. += Average: + shall not represent addition; instead, it shall represent the operation of averaging two numbers.

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Examples(a) 7 + 9 = 8, the average of 7 and 9. 7 – (0 – 9) equals 16, as usual.(b) 5 + (4 x 2) = the average of 5 and 8 = 6.5.(c) 4+6+9 has two values: (4+6)+9 = 5+9 = 7; 4+(6+9) = 4+7.5 = 5.75. Notice that neither answer equals 19/3, the usual (mathematical) average of 4, 6, and 9.Comment If 0 Wild is chosen along with Average, any 0 that represents + must mean average.

10. Next Prime Number: xA means “the next prime number bigger than A,” where A is a rational number less than or equal to 200.Comment This variation does not rule out using x for multiplication. In the Goal or Solution, the meaning of an x cube will usually be clear from the context since next prime is a unary operation and multiplication is a binary operation. For example, the Goal 4xx6 has only one interpretation: 4 x (x6), which is 4 x 7 = 28.(a) x7 = 11, the next prime number bigger than 7.(b) x(9 ÷ 2) = 5, the next prime bigger than 4.5.(c) x(0 – 3) = 2, the next prime number bigger than –3. (Note: 1 is not prime.)(d) xx5 = x(x5) = x7 = 11.(e) x√49 = x7 = 11; x√67 is undefined because √67 is not a rational number.(f) In the expression 2xx5, the first x means multiplication and the second means next prime number. The value of this expression is 2 x (x5) = 2 x 7 = 14.(g) In the expression x5x7, the first x means next prime number and the second means multiplication. The value of this expression is either x(5x7) = 37, the next prime number bigger than 35, or (x5)x7 = 7 x 7 = 49.(h) There is no limit to the number of consecutive x’s in an expression, especially with the Multiple Operations variation also in effect. Thus xxx9 = xx(x9) = x(x11) = x13 = 17

The following odd-year variations will be played again in 2018-19:

11. AB+: The Goal and/or the Solution may be or may include a three-cube expression of the form AB+, which is interpreted as a repeating decimal. It may be interpreted as .ABABAB... or as .ABBBBB... A player who presents an Equation must specify in writing which interpretation is used in the Goal and Solution. No decimal points may be used in the Equation (except when the decimal point variation is also chosen for the shake). When the form AB+ is used in an Equation, the Equation-writer should indicate in writing which fraction the AB+ is used for (.ABAB or .ABBB).Examples(a) 33+ = 0.333333…(.ABAB interpretation, which is 33/99 and simplifies to 1/3) or 0.333333 (.ABBB

interpretation, which also simplifies to 1/3).(b) 90+ = 0.909090…(.ABAB interpretation, which is 90/99 and simplifies to 10/11) or 0.900000 (.ABBB

interpretation, which is [90-9]/90, which simplifies to 9/10).(c) 1-(55+) = (99/99) – (55/99) = 44/99 (which simplifies to 4/9 in either interpretation).

Comments(a) If Base m is also chosen, the decimal places also change to the appropriate base m expression, meaning the

fractions change to AB/77 for Base 8, AB/88 for Base 9, etc.(b) In Base m, the numerators and denominators in fractions also must be converted to the appropriate base.

12. Add to Goal: On his turn, instead of a regular move, a player may physically add a cube to the Goal. The cube may be placed anywhere in the Goal. However, the limit of six cubes in the Goal, with no numeral containing more than two consecutive digits, still prevails.Comments(a) A goal must be set before a challenge is made (i.e., at least one cube must be on the goal line).(b) The original Goal-setter must put at least one cube on the Goal line. If he states that the Goal is set with no

cubes (the goal line is blank), a one-point penalty is assessed.

D. S enior Divis i on Variations (g r ades 11- 12 )

Players may choose any of the Junior variations (except for the two that are in effect for every

AE-63

shake) plus the following:

13. Imaginary : | (sideways minus) shall represent the imaginary number i (such that i 2 = –1). The | may be placed immediately before or after a numeral without the x sign. When this variation is selected, all roots of a^b, where a is a complex number and b is a rational number are available.Note: This variation may be selected even if no – signs (or wild cubes) are avail able in Resources.With this variation, the rules for legal expressions in Section III-C are amended to allow expressions like a * (b ÷ c) [or a ^ (b ÷ c)] where a is a negative real number, b is an integer, and c is an even non-zero integer (when b ÷ c is reduced to lowest terms). Furthermore, in a Goal or Solution, any expression of the form a * (b ÷ c) [or a ^ (b ÷ c)] (where c ≠ 0) may equal any one of the complex roots equal to the expression. An Equation-writer using such an expression must indicate in writing which one of the complex roots the expression equals. [See examples (f) and (g) and comment (d) below.]If | is multiplied by a number before or after it, as allowed by this variation without a x sign, the implied multiplication takes precedence over any other operations in the expression (in the absence of parentheses). See the examples below.A player may use an x before or after a |. In that case, the explicit multiplication involving the i also takes precedence over other operations.

Comments (in each case, * may be replaced by ^.)(a) “Numeral” means “any expression that names a number, real or otherwise.” i itself is a numeral, which means

that expressions like ||, |||, and so on, are legal and equal i 2, i 3, etc.(b) An expression like 4*| is not allowed because the exponent is not a real number. However, |*4 is permitted.(c) | is ambiguous as regards right-side up and upside down. The default is right-side up. So any | in the Goal

may be interpreted as i or -i. However, | My not be interpreted as sideways.(d) If 0 (or x) wild is also chosen, any wild cube may be used as | to equal i or -i. Another wild cube may also be

used for – in the same Equation since it stands for the same symbol in both cases. If Multiple Operations is also chosen, the wild cube may be used multiple times, but only once as i.

(e) With Imaginary, the Goal and the Solution may equal non-real (complex) numbers.(f) If Multiple Operations is also chosen, | may not be used multiple times because it represents a numeral and

not an operation.(g) A Goal like 2|*88 may not be interpreted as 2(i*88) since the variation allows a numeral in front of |

without x but not in front of an expression like |*88 without a x. Similarly, the expression 2|*8 in a Solution must be interpreted as (2|)*8 and not 2(|*8) whether the Equation-writer includes the parentheses around “2|” or not.

Examples (in each case, * may be replaced by ^.)(a) 2i may be represented in a Goal or Solution by either 2| or |2. | or | is ±0.25i.

| or | is ±3i.(b) 3 + 4i may be represented as either 3 + 4| or 3 + |4.(c) (3+4) | or | (3+4) equals 7i. (d) i 6 may be represented as | * 6.(e) 14i may be represented as 7| 2.(f) A Goal of 4 * (1 ÷ 2) may equal 2 or –2. A Goal of 16 * may equal 2, –2, 2i, or –2i. Each Equation-writer

must eliminate any such ambiguities in his Equation.(g) Suppose the Goal is 0 – 8 |. Then a Solution might be this: (8 x 2) * (3 ÷ 4). The Equation-writer must indicate

in a clear and unambiguous manner which root is being used. One way is this: (8 x 2) * (3 ÷ 4)

(2i)3

Examples of the default order of operations with | = i Expression Default Interpretation Expression Default Interpretation (a) 3+2| 3+(2|) (b) |2+3 (|2)+3 (c) | |*2 (| |)*2 (d) 5| |2–7 (5| |2) –7 (e) 3+2x| 3+(2x|) (f) |√4 |(√4) (g) √9+| (√9)+| (h) 5x|–|4 (5x|)–(|4) (i) The expression √9| contains a clash of defaults. The rule for √ says it must be interpreted as (√9)| but the

AE-64

default for | says it must be √(9|). So the Equation-writer must indicate which interpretation they want to use. (j) If an expression like 3+2| is in the Goal, a player may interpret it as (3+2)| but must write the parentheses on the right-side of the Equation.

14. Decimal i n Goal: Each Equation-writer may determine where decimal points occur in the Goal.Examples(a) A Goal of 20 may be interpreted as 20, 2.0, or 0.2(b) A Goal of 2 * 3 may be interpreted as 2 * 3, 0.2 * 3, 2 * 0.3, or 0.2 * 0.3. (Each * may be replaced by ^ for

newer games.)Comment A decimal point may be placed in front of only a right-side-up digit. Therefore, no decimal point may be

placed in front of a sideways or upside-down cube or in front of i (since i is not a digit).

15. ÷ as log: (sideways ÷) represents the log operation. Thus, if a and b are positive real numbers (b ≠ 1), a b equals logba. A sideways sign means that log must be used; a normal ÷ sign is ambiguous and can be log or regular division. Examples(a) [(6 x 4) + 1] 5 = log525 = 2.(b) 3 2 = log23, which is an irrational number.(c) a 1 is undefined for any value of a. 0 5, (0 –1) 2, (0 – 8) (3 – 1), and 4 (0 – 2) are all undefined.

AE-65

Equations Appendix

The following is an extensive (but not necessarily exhaustive) list of the examples of ways of indicating what cubes mean in Equations solutions. In some cases when a variation is not written the context in which a solution is written dictates which variations are used. When this is not clear the solution defaults to the situation as if a variation is not used.

Division Variations Examples DefaultEMJS Sideways Cube 1/7, 7 Cube is right side up

sw, sw

EMJS Upside Down -7, 7 Cube is right side up ud ud

EMJS Zero Wild 2X(6+5) Every zero in a solution must have in 0W writing what they are wild for. A zero

in a solution without anything written stands for a zero.

EMJS Sideways, Zero Wild 1/7, 0, 0, 0 Cube is right side up and zero = 0 sw/0W sw/0W7 1/7

EMJS Upside Down,0 Wild , -7, 7, 0, 0 Cube is right side up and zero =00 usd/0W usd/0W -7 1

EMJS Multiple Operation Write the operation sign as many times as needed in your solution. Nospecial indication of Mult. Op. isnecessary.

EMJS Next Prime Number 5X8Multiply,X67Next Prime Context dictates the interpretation of 7XX3 or7X(X3)1st X Mult,2nd N the XXX8 or X(X8)Both NP

EMJS # of Factors Same as Next Prime Number

E LCM 8√2 √ = rootLCM ( must be indicated since ambiguous)

E GCF 9*2 or 9^2 * or ^ = exponentiationGCF ( must be indicated since ambiguous )

EM Decimal Point *23Dec. Pt., 23*+5Dec. Pt. Context: If context does not (* or ^) 2*3Default to power unless determine, default to power

Player writes 2*3 or 2*3 (player must write Decimal Point) DP

EM Percent 50 34, 50% of 34, 50√34 √ is right side up = root %

M Decimal Point/AB+ 66+*5 is ambiguous so default to * or ^= Exponentiation66+to the power of 5

MJS Powers of the Base 1 100 100 1=one100 POB 1

MJS AB+ No indication for + of repeating Context dictates interpretation of +45++5 45+5Repeating Decimal Addition

JS AB+, +=Average Same as above Context dictates

JS AB+, Base 11 or12 66+*2 is ambiguous Solution writer must indicate Interpretation of * which then Determines interpretation of +

JS Base 11 or 12 7+*4= 100, 6**2 is ambiguous Context: If context cannot be So write (6*)*2 which means determined, expression is ambiguous(6*) to the 2 power or 6*(*2) In Base 12, rules for √ are the same which means 6 to the *2 power or as * or ^6* *2Power ten

JS Add to Goal Solution writer must write the goal used for the solution if one or more cubes are added to the goal on the mat. If necessary, the solution writer

AE-66

must also explain orally how the goal can be obtained by individual moves from the mat to the goal; E.G. there is no way to obtain 23X0 from 23.

S Imaginary |*4I * 4 The default for placement is right-, side up.

S X Wild See examples for Zero Wild X=multiplication. Also see Note for Zero Wild

S X Wild, Next Prime Number Context determines

S X Wild, # of Factors Context determines

S Division as Log 8÷2, 8Log2, log2 8 Division=division,·|·=Log Log

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Michigan Leagues of Academic GamesElementary Equations Variations Even Year

GENERAL RULE: If * or ^ is used for an exponent, both base and exponent must be whole numbers. If √ is used for the root operation, the index must be a counting number and the base and total value must be whole numbers.

Note: counting numbers = natural numbers = positive integers = 1,2,3,4…whole numbers = 0,1,2,3,4…

The following may be used in September and OctoberSideways Cube: A cube representing a non-zero number may be used sideways in the Goal or aSolution to equal the reciprocal of the number it represents.

Upside-Down Cube: In the Goal or a Solution, any numeral may be used upside-down to equal the additive inverse of the number represented by that numeral.

0 Wild: The 0 cube may represent any numeral on the cubes, but it must represent the same numeral everywhere it occurs (Goal and Solution). Each Equation-writer must specify in writing the interpretation of the 0 cube if it stands for anything other than 0 in the Equation.

Factorial (!): There are two occurrences of the factorial operator (!) available to be used in theSolution and/or the Goal as the Equation-writer chooses to use them. All uses of ! in theEquation must be in writing.

Three-operation Solution: Any Solution must contain at least three operation symbols. The operation symbols are +, –, x, ÷, * (or ^), √, and !.

Multiple Operations: Every operation sign in Required or Permitted may be used many times in any Solution. If the Factorial variation is also chosen for the shake, an unlimited number of factorial operators may be used in each Solution. At most two factorials may be used in the Goal.

The following may be added in November:

Remainder: A B ( is a sideways ÷) equals the remainder when A is divided by B. A and B are positive integers, and A is less than or equal to 1000.

Percent: (upside down root) means "percent of.” That is, A B = A% of B, where A and B are numbers. In the Goal or a Solution, A and/or B may be a two-digit numeral.

Decimal point: * (or ^) may be used as a decimal point. If so used in the Goal or a Solution, an * (or ^) may be combined in a numeral with at most three digits. When used as a decimal, * (or ^) takes precedence over all other operations.

Next Prime Number: xA means “the next prime number bigger than A”, where A is a rational number ≤ 200.

+ = Average: + shall not represent addition; instead, it shall represent the operation of averaging two numbers.

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Michigan Leagues of Academic GamesMiddle Equations Variations Even Year

The following may be used in September and OctoberSideways Cube: A cube representing a non-zero number may be used sideways in the Goal or aSolution to equal the reciprocal of the number it represents

Upside-Down Cube: In the Goal or a Solution, any numeral may be used upside-down to equal the additive inverse of the number represented by that numeral

0 Wild: The 0 cube may represent any symbol (numeral or operation) on the cubes, but it must represent the same symbol everywhere it occurs (Goal and Solution). Each Equation-writer must specify in writing the interpretation of the 0 cube if it stands for anything other than 0 in the Equation.

Factorial (!): There are two occurrences of the factorial operator (!) available to be used in theSolution and/or the Goal as the Equation-writer chooses to use them. All uses of ! in theEquation must be in writing.


The following may be added in NovemberBase m: Both the Goal and the Solution must be interpreted as base m expressions, where the player choosing this variation specifies m for the shake as eight, nine, or ten. Two-digit numerals are allowed in Solutions.

Multiple of k: A Solution must not equal the Goal but must differ from the Goal by a non-zero multiple of k, where the player choosing this variation specifies k for the shake as a whole number from six to eleven, inclusive. The Goal must not be greater than 1000 or less than –1000.

Percent: (upside down root) means "percent of.” That is, A B = A% of B, where A and B are numbers. In the Goal or a Solution, A and/or B may be a two-digit numeral.

Decimal point: * (or ^) may be used as a decimal point. If so used in the Goal or a Solution, an * (or ^) may be combined in a numeral with at most three digits. When used as a decimal, * (or ^) takes precedence over all other operations.



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Michigan Leagues of Academic GamesJunior Equations Variations Even Year

The following two variations will be in effect for every shake:Sideways Cube: A cube representing a non-zero number may be used sideways in the Goal or a Solution to equal the reciprocal of the number it representsUpside-Down Cube: In the Goal or a Solution, any numeral may be used upside-down to equal the additive inverse of the number represented by that numeral

The following may be used in September and October

0 or x Wild: The 0 or x cube may represent any symbol on the cubes, but it must represent the same symbol everywhere it occurs (Goal and Solution). Each Equation-writer must specify in writing the interpretation of the 0 or x cube if it stands for anything other than itself in the Equation. The player selecting this variation specifies whether 0 or x (but not both) is wild for the shake.

Factorial (!): There are two occurrences of the factorial operator (!) available to be used in theSolution and/or the Goal as the Equation-writer chooses to use them. All uses of ! in theEquation must be in writing. However, if Multiple of k is also chosen for the shake, no factorial may be placed in the Goal.


Powers of the Base: 1 (one) may represent any integral power of ten. (If 1 is used in a two-digit numeral, it stands for 1.) If Base m is also chosen, 1 represents any integral power of m.

Any Color Exponent: Any numeral on a ___ cube may be used as an exponent without being accompanied by an * (or ^) cube. The player selecting this variation chooses a color: red, blue, green, or black (for example, “Red Exponent”).

The following may be added in NovemberBase m: Both the Goal and the Solution must be interpreted as base m expressions, where the player choosing this variation specifies m for the shake as eight, nine, ten, eleven or twelve. Two-digit numerals are allowed in Solutions. For bases eleven and twelve, * (or ^) may be used for the digit ten; in base twelve, √ may be used for the digit eleven.

Multiple of k: A Solution must not equal the Goal but must differ from the Goal by a non-zero multiple of k, where the player choosing this variation specifies k for the shake as a whole number from six to twelve, inclusive.

Number of Factors: xA means “the number of counting number factors of A,” where A is a counting number.



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Michigan Leagues of Academic GamesSenior Equations Variations Even Year

The following two variations will be in effect for every shake:Sideways Cube: A cube representing a non-zero number may be used sideways in the Goal or a Solution to equal the reciprocal of the number it representsUpside-Down Cube: In the Goal or a Solution, any numeral may be used upside-down to equal the additive inverse of the number represented by that numeral

The following may be used in September and October0 or x Wild: The 0 or x cube may represent any symbol on the cubes, but it must represent the same symbol everywhere it occurs (Goal and Solution). Each Equation-writer must specify in writing the interpretation of the 0 or x cube if it stands for anything other than itself in the Equation. The player selecting this variation specifies whether 0 or x (but not both) is wild for the shake.

Factorial (!): There are two occurrences of the factorial operator (!) available to be used in the Solution and/or the Goal as the Equation-writer chooses to use them. All uses of ! in the Equation must be in writing. However, if Multiple of k is also chosen for the shake, no factorial may be placed in the Goal.


Any Color Exponent: Any numeral on a ___ cube may be used as an exponent without being accompanied by an * (or ^) cube. The player selecting this variation chooses a color: red, blue, green, or black (for example, “Red Exponent”).

Base m: Both the Goal and the Solution must be interpreted as base m expressions, where the player choosing this variation specifies m for the shake as eight, nine, ten, eleven or twelve. Two-digit numerals are allowed in Solutions. For bases eleven and twelve, * (or ^) may be used for the digit ten; in base twelve, √ may be used for the digit eleven.

Powers of the Base: 1 (one) may represent any integral power of ten. (If 1 is used in a two-digit numeral, it stands for 1.) If Base m is also chosen, 1 represents any integral power of m.

Number of Factors: xA means “the number of counting number factors of A,” where A is a counting number.


The following may be added in NovemberMultiple of k: A Solution must not equal the Goal but must differ from the Goal by a non-zero multiple of k, where the player choosing this variation specifies k for the shake as a whole number from six to twelve, inclusive.


Decimal in Goal: Each Equation-writer may determine where decimal points occur in the Goal. Any decimals must be indicated in writing when the Equation is presented.

Imaginary |(sideways minus) shall represent the imaginary number I (such that i2 = -1) | may be placed immediate before or immediately after a numeral without the x sign. When this variation is selected, all roots of a^b where a is a complex number and b is a rational number are available. Note: This variation may be chosen even if no – signs (or wild cubes) are in Resources.

÷ as log: (sideways ÷) represents the log operation. Thus, if A and B are positive real numbers (b ≠ 1), A B equals logBA. A sideways sign means that log must be used; a normal ÷ sign is ambiguous and can be log or normal division.

.

AE-72

MLAG ADVENTUROUS ON-SETS® Tournament Rules 2017-18I. Starting a Match (Round)

A. Two- or three-player matches will be played. A match is composed of one or more shakes. A shake begins with the rolling of the cubes, the dealing of some cards, and the setting of a whole number as the Goal for that shake. A shake ends with at least one player attempting to write a Solution that both equals the Goal and correctly uses the cubes on the playing mat.

B. The following equipment is needed to play the game.1. 16 cards: each card contains a unique combination of zero to four dots colored blue

(B), red (R), green (G), or yellow (Y). No card contains more than one dot of any color. At the start of a shake, some of these cards are dealt face up to form the Universe for that shake.Comment Players should make sure all 16 cards are in the game, with no duplicates. One of the cards

is blank.

2. 18 cubes: these consist of the following groups.a. 3 digit cubes: each face has one of the digits 1 through 5.


b. 8 color cubes: each face has a dot colored B, R, G, or Y. Each dot names the set of all cards in the Universe that contain a dot of that same color.

c. 4 operation cubes: each face has one of the symbols U, ∩, –, or '.(i) U means the union of two sets.

Example B U G is the set of cards in the Universe that are either B or G.

(ii) ∩ means the intersection of two sets.Example R ∩ Y is the set of cards that are both R and Y.

(iii) – means set subtraction.Example B – Y is the set of cards that are B but not Y.

(iv) ' means the complement of a set.Example G' (often read “green prime”) is the set of cards that are not G.

d. 3 Restriction cubes: each face has one of the symbols V, Ʌ, =, or C.(i) V names the set of all the cards in the Universe for the shake.(ii) Ʌ names the set of no cards (the null or empty set).(iii) = and C are special operators used to make mathematical statements about the

cards in the Universe. Each such statement is called a Restriction. (See Section VI-B below.)Comment The = and C symbols are not used in the Elementary Division. Instead, before the cubes are rolled, the Goal-setter first sets out either two V and one Ʌ cube or one V and two Ʌ cubes. Then the remaining cubes are rolled.


Comment Many games have a section labeled “Resources.” However, any reference in these rules to the “playing mat” or the “mat” does not include the Resources section.

4. A one-minute sand timer: this is used to enforce time limits.5. A challenge block: This is a cube or similar object and not a flat object like a coin. It

should not be so large that two players can grab it simultaneously.AOS - 73

C. Players may use only pencils or pens, blank paper, and (for Adventurous On-Sets) variation sheets. No prepared notes, books, tables, calculators, cell phones or other electronic devices may be used except that players’ paper may contain preprinted Universe charts on which the cards that are dealt may be marked.

Comment The chart a player uses may not have sets pre-shaded or pre-marked in any way. (See Appendix B for samples.)



rolled cubes form the Resources for the shake.1. A shake begins as soon as the timing for rolling the cubes and dealing the cards is

started or the cubes are rolled or the first card is dealt.2. During a shake, no player may turn over a cube or obstruct the other players’ view of

any cube. (See Section IX-C.)3. In Elementary Division, the three Restriction cubes are not rolled. Instead the Goal-

setter first sets out either two V and one Ʌ cube or one V and two Ʌ cubes. Then the remaining cubes are rolled.

B. While the Goal-setter rolls the cubes, the player to the right of the Goal-setter shuffles and deals the cards.1. In Elementary, Middle, and Junior Divisions, at least six but no more than 12

cards must be dealt.2. In Senior Division, at least 10 but no more than 14 cards must be dealt.

After the cards are dealt and positioned in a manner agreeable to all players, no one may touch them or in any way obstruct the other players’ view of them until Solutions are checked. (See Section VII-C.) However players may look at the cards that were not dealt (usually for purposes of making their charts)Comment The dealer may not take back a card that has been dealt unless the number of cards

exceeds the maximum allowed in the division. In that case, the extra card(s) must be removed from the Universe.

C. In Adventurous On-Sets, after the cubes have been rolled and the cards have been dealt but before the Goal is set, each player must select a variation from the appropriate list in Section XIII of these rules. A variation is a special rule that, if it conflicts with any of the regular tournament rules, supersedes those rules.1. The Goal-setter makes the first selection, then the player to the left of the Goal-

setter, then the third player if there is one.a. Each player has 15 seconds to make a variation selection.b. To begin a shake, the Goal-setter has one minute to roll the cubes. At the end of

this minute, she has 15 seconds to select a variation. However, if the Goal-setter selects a variation before the minute for rolling the cubes expires, the next player has the rest of that minute plus 15 seconds to select a variation. If the second player also selects a variation before that minute expires, the third player (if there is one) has the rest of that minute plus 15 seconds to select.

c. A player selects a variation by circling its name in the list for that shake. This list is located on the reverse side of the scoresheet or on a separate sheet. For certain variations (e.g., required cube, wild cube, Double Set), the player must also fill in a blank to indicate which cube is required or wild, which set counts double, and so on.

AOS - 74

2. If a player selects a variation that has no effect on the shake, a variation that conflicts with one already chosen for the shake, or a variation that has already been chosen for the shake, the player loses one point and must pick another variation. If, on the second try, the player still does not select an appropriate variation, he loses another point and may not pick a variation for that shake.If a player’s illegal variation selection is not pointed out before the next player selects a legal variation or a legal Goal is set (whichever comes first), the player making the illegal selection is not penalized. However, the illegal variation is ignored for the shake.Examples It is illegal to choose U Required when no U cube was rolled, Y Wild when no Y cube is in

Resources, or (in Middle, Junior, and Senior) No Null Restrictions when no = or C cube was rolled (and no Wild Cube has been chosen).

3. In two-player matches in Elementary, Middle and Junior Divisions, the player who is not the Goal-setter must select two variations for the shake. In Senior Division, any player may pick two variations for any shake in either a two- or three-player match.A Player picking two variations must select both within the 15 second time limit (See Section XI-A-1-b).

III. Setting the GoalA. The player who rolls the cubes must set a Goal by transferring the cube(s) of the Goal from

Resources to the Goal section of the playing mat.B. A Goal consists of at least one and at most three digit cubes that form an expression that

names a whole number.1. If more than one cube is used to set the Goal, the way the cubes are placed in the

Goal determines the Goal’s value.a. The sum of two numbers is indicated by placing the cubes in a horizontal line (side

by side).b. The product of two numbers is indicated by placing the cubes in a vertical line.c. The negative of a number is indicated by placing the cube so that its numeral is

upside-down.

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The following are the only legal configurations for the Goal:Goal Meaning Goal Meaning

A

A + B A x B x C

A + B + C (A x B) + C

A x B A x (B + C)

Comment Any digit cubes not used in the Goal must be placed in Forbidden since they are not used in Solutions.

2. Once a digit cube touches the Goal section of the mat, it must be used in the Goal.a. The Goal-setter indicates the Goal has been set by saying “Goal.”b. The Goal-setter may rearrange or regroup the cubes in the Goal section until she



cube (but not = or C) from Resources to Forbidden then set the Goal.2. A Goal-setter who is leading in the match may not make a bonus move.

If the Goal-setter makes a bonus move while leading in the match and an opponent points out the error before the next player moves or someone legally challenges, the cube in Forbidden is returned to Resources. The Goal-setter also receives a one-point penalty.

D. If the Goal-setter believes no Goal can be set that has at least one correct Solution (see Section VII), he may declare “No Goal.” Opponents have one minute to agree or disagree with this declaration.1. If all players agree, that shake is void and the same player repeats as Goal-setter


Resources was rolled. For example, there are three 1’s, the operations are all U signs, and each color appears on at least four cards. (Even in this case, the Goal-setter might be able to pick a variation like wild cube that would allow a Goal to be set.)

(b) Players receive no points for the void shake.(c) If the Goal-setter makes a Bonus move, he is committed to setting a goal and may not declare “No Goal”

2. An opponent who does not agree with the “No Goal” declaration indicates disagreement by picking up the challenge block (see Section V-B) and challenging the “No Goal” declaration. She then has two minutes to write a legal Goal and a correct Solution. If there is a third player, he also can choose to write a legal Goal and a correct Solution.Note: The Challenger (and third party, if he agrees) can use as many cubes from Resources as needed.

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three sections of the playing mat (Required, Permitted, Forbidden) or challenge the last Mover.The move of a cube is completed when it touches the mat. Once a cube is legally moved to the mat, it may not be moved again during the shake. (Exception: When the Shift from Permitted variation is played – see Section XIII below.)

C. If you are not leading in the match, then on your turn you may take a bonus move before making a regular move.1. To make a bonus move, the Mover must say “Bonus,” then move a cube from

Resources to Forbidden, then another cube to Forbidden, Permitted or Required.Comments(a) If you do not say ‘Bonus’ before moving the cube to Forbidden, the move does not count as a bonus



2. If the player in the lead makes a bonus move and an opponent points out the error before another player makes a legal move or challenge, the Mover must return the second cube played on that turn to Resources. The Mover also receives a one-point penalty.Comments(a) Players tied for the lead may make Bonus moves.(b) Players often call “Bonus” and move two cubes simultaneously to Forbidden. If the player did not call “Bonus”, he may return either of the two cubes to Resources.

D. In Middle, Junior and Senior Divisions, no = or C cubes may be played to Forbidden until four or fewer cubes remain in Resources.Comments(a) V and Ʌ may be played to Forbidden.(b) Allowing an = or C to be played to Forbidden with four or fewer cubes left in Resources is intended to

cover those rare situations where a player would have no choice but to make a Solution possible with one more cube. For example, there may be a color cube and two = or C cubes left in Resources. If the color cube and either Restriction cube is needed for a Solution, then the Mover would have no choice but to play one of the three cubes to Required or Permitted. The exception above allows the player to move either the = or C cube to Forbidden to avoid a Now Challenge.

V. ChallengingA. Whether or not it is your turn, you may challenge another player who has just completed a

move or set the Goal. The two main challenges are Now and Impossible.Note Players may also challenge a “No Goal” call, see Section III-D-2.


Comments(a) If the Goal is not a legal configuration (see Section III-B-1) or the Goal equals a negative number, an

opponent should challenge Impossible.(b) A Player who challenges “Never” will be considered to have challenged “Impossible”. There will be no

penalty for saying “Never” instead of “Impossible”.2. By challenging Now, a player claims that a Solution can be written using the cubes

on the mat and one cube (or no additional cubes) from Resources.AOS - 77

a. A player may challenge Now only if there are at least two cubes in Resources.If a player challenges Now with fewer than two cubes in Resources, the challenge is invalid and is set aside. The player who called Now also receives a one point penalty.Comment If only one cube remains in Resources and no one challenges Impossible, then a Solution



B. A challenge block is placed equidistant from all players. To challenge, a player must pick up the block and say “Now” or “Impossible.”If a player picks up the block, then decides not to challenge (without saying “Now” or “Impossible”), the player accepts a one-point penalty and play continues. A player who picks up the block and makes a challenge against himself is also penalized one point, and the challenge is set aside. Comments(a) The purpose of the block is to determine who is the Challenger in a shake. (b) Touching the challenge block has no significance. However, players may not keep a hand or finger on,

over, or near the block for an extended period of time. (See Section IX-C.)(c) A player must not pick up the challenge block for any reason except to challenge. For example, don’t pick


VI. The Parts of a SolutionA. Set-Name part: this part consists of one legal Set-Name. A Set-Name is legal if it specifies

a set of cards in the Universe and does not contain any symbol or group of symbols that is undefined in On-Sets.= and C cubes may not be used in the Set-Name of the Solution even if they are in Required. If the Set-Name part of the Solution contains an = or C symbol, it is automatically wrong.Examples of Set-Names B, R', G U Y, (R ∩ B) – Ʌ, (V – G)' U RComments(a) A Set-Name written on paper may contain pairs of grouping symbols such as parentheses, brackets or

braces even though these do not appear on the cubes. These symbols indicate how the Solution-writer would physically group the cubes if the Solution were built with the cubes.

(b) The Solution-writer must not write “= Goal” after the Set-Name. Doing so makes the Solution incorrect.

B. Restriction part: this part consists of one or more Restrictions.1. A Restriction is a rule that is applied to the cards in the Universe. Any card that does

not satisfy the Restriction is temporarily removed from the Universe while that Solution is being checked. After all Restrictions in the Solution have been applied to the Universe, the Solution-writer’s Set-Name is worked out using the cards that remain in the restricted Universe.Comments(a) Any cards removed while checking a Solution are returned to the Universe before another Solution is

checked.(b) Any Restrictions in a Solution apply to the Universe only for that Solution.

2. There are three types of Restrictions. Any set used in each type must be represented by a legal Set-Name.a. Subset Restriction: this type has the form Set 1 C Set 2. A card in the Universe does

not satisfy a subset Restriction if it is in Set 1 but not in Set 2.AOS - 78

Examples B C R', G – Y C Ʌ, B – R C (G ∩ V)', B U Y C B U Y

b. Equals Restriction: this type has the form Set 1 = Set 2. A card in the Universe does not satisfy the Restriction if it is in one of the two sets but not in the other.Examples B = R, G – Y = V, (B ∩ G)' = Y – R, R = R

c. Chain Restriction: this type has two or more = or C cubes in it.(i) Restrictions of the following form are defined, where A, B, and C are sets.

A C B C C (meaning A C B and B C C) A = B = C (meaning A = B and B = C) A C B = C (meaning A C B and B = C) A = B C C (meaning A = B and B C C)

(ii) Restrictions of the following form also are permitted, where each one is worked out from left to right like those above.A C B C C C D, A = B = C = D, A C B C C = D, A = B C C C D, and so on.

3. In a Restriction, no pair of parentheses (or other symbols of grouping) may enclose an = or C symbol. However, a player may put parentheses around one side of a Restriction, like this: B C (R U G). Comment A common error is putting parentheses around part of a chain Restriction, like this where A,

B, and C are sets: (A C B) C C, A = (B = C ), and so on. Such parentheses make the chain meaningless just as the parentheses in the algebraic equation (2x=3)+7 make it meaningless. Also these parentheses are inappropriate: (B = R)' However, this does not mean that parentheses may not be used at all in Restrictions. Parentheses may legitimately be placed within any Set-Name in a Restriction, as in the following examples.(R U B) – G = V, B = (G U R)' C V, R' = B C (R – Y) U V

Notice in these examples that no pair of parentheses encloses an = or C.

VII. Writing and Checking SolutionsA. After a valid challenge, at least one player must write a Solution.



3. After any challenge in a three-player match, the Third Party must decide by the end of the two minutes for writing Solutions whether to agree with the Challenger or the Mover. If the player with whom the Third Party agrees must write a Solution, then the Third Party must also write a Solution.Comment To indicate his intention on the challenge, the Third Party may:(a) state whether or not he will present a Solution;(b) indicate which party, Mover or Challenger, the Third Party is “joining” (agreeing with) on the


does not present a Solution, she is assumed to be joining the player who is not writing a Solution. (Challenger on an Impossible or Mover on Now).

B. To be correct, a Solution must satisfy the following criteria:1. The Solution contains a valid Set-Name part.2. Middle, Junior, Senior only: The Solution contains a Restriction part if there are one

or more = or C cubes in Required.If no = or C cubes are in Required but some are in Permitted or Resources, the Solution may include a Restriction part.

3. The Solution equals the Goal. That is, the number of cards selected from the Universe by the Set-Name equals the Goal.

AOS - 79

If the Solution includes one or more Restrictions, these must be applied to the Universe before the Set-Name is worked out. If there are two or more Restrictions, they may be applied to the Universe in any order.Comments(a) Unlike Equations, the Solution-writer must not write = Goal after the Set-Name.(b) Mid/Jr/Sr: With the Absolute Value variation (see Section XIII below), the Goal may have more than

one value. Then any Solution must equal one of the legal values of the Goal.

4. The Solution uses the cubes correctly.a. The Solution contains at least two cubes.

Example In Middle, Junior and Senior Divisions, the following Solution satisfies this rule:

Restriction: B = BSet-Name: B

The Solution contains three cubes even though the Set-Name contains only one.

b. Every cube in Required is used in the Restriction part (if there is one). These same cubes (except any = or C) must also be used in the Set-Name.

c. Each cube in Permitted may be used in the Restriction part (if there is one). These same cubes (except any = or C) may also be used in the Set-Name.

d. The Solution uses no cube in Forbidden.Comment Since several Resource cubes may show the same symbol, it is possible to have a U in

Forbidden that must not be used in the Solution at the same time that there is a U in Required that must be used.

e. After a Now challenge, the Solution must contain at most one cube from Resources.f. After an Impossible challenge, any cubes in Resources are considered to be in

Permitted and therefore may be used in the Solution.5. In Adventurous On-Sets, the Solution satisfies all conditions imposed by the variations

selected for that shake. (See Section XIII for the list of variations.)Examples(a) If the variation “– Required” has been chosen, each Solution must contain a – sign.(b) If the Two Operations variation has been chosen, then the Set-Name part of every Solution must

contain at least two operation symbols.

6. Every legal interpretation of the Solution equals the Goal.a. An ambiguous Solution is one that has more than one legal interpretation. Such a

Solution is incorrect if an opponent shows that one of the interpretations does not equal the Goal.

b. The only defined order of operations in On-Sets is that the ' operation takes priority over all other operations (U, ∩, –, and special operations defined by variations). Consequently, a Solution may be ambiguous if the writer does not use parentheses (or other symbols of grouping such as brackets or braces) to indicate the order of operations. ‘ is a unary operation, therefore, a player may not insert parenthesis to split a ‘ from a Set-Name.

C. After the time for writing Solutions has expired, each Solution that is presented must be checked for correctness.1. After a challenge in a three-player match (and before any Solution is presented), the

Third Party must indicate by the end of the two minutes for writing Solutions whether he is presenting a Solution.Comment To indicate his intention on the challenge, the Third Party may(a) state whether or not he will present a Solution;(b) indicate which party, Mover or Challenger, the Third Party is “joining” (agreeing with) on the

challenge. This can be done verbally or by pointing to the party.(c) present or not present a Solution when the time limit for writing Solutions expires.

AOS - 80

In any case, the Third Party may not retract his decision once he has indicated whether or not he will present a Solution.

2. All Solutions must be presented before any is checked.a. Once a player presents a Solution to the opponent(s), she may make no further

corrections or additions even if the time for writing Solutions has not expired.b. Each Solution-writer must indicate the Solution to be checked. A writer who

forgets to indicate the Solution must do so when asked.3. Opponents have two minutes to check each Solution. When more than one Solution

must be checked, they may be checked in any order. In a three-player match, both opponents must check a player’s Solution during the same two minutes. No other Solution should be checked during this time.Comment When both players in a two-way match present Solutions after the last cube has been moved

(see Section VIII below), only one Solution should be checked at a time.

4. Within the time for checking a Solution, opponents must accept or reject the Solution. If the Solution is rejected, an opponent must show that it violates at least one of the criteria in Section VII-B. A Solution is correct if no opponent shows that it is incorrect.After a challenge in a three-player match, a player who does not present a Solution for a shake scores 2 if he accepts another player’s Solution as correct even if that Solution is subsequently proven wrong by the other checker.Comment Players must not physically move the cubes in Required, Permitted and Resources to form the

Solution since this causes arguments over where each cube was played.

5. A player who claims an opponent’s Solution does not equal the Goal must give at least one of the following reasons:a. The Goal has no legal interpretation.

Examples(a) The Goal is in the shape of a backwards L, which is not a legal configuration.(b) The Goal equals a negative number.

b. The Solution equals a value that is not a legal value of the Goal. (The only time when the Goal might have more than one value would be in Mid/Jr/Sr when the Absolute Value variation is in effect – see Section XIIIB.)(i) Checkers must make an effort to determine whether the Solution equals the Goal

before rejecting the Solution. This can usually be done by applying the Solution to the Universe and turning over cards (if they disobey the Restriction) and/or selecting out the cards that are included in the Set-Name.

(ii) The checker can give a general argument that the Solution does not equal the Goal.Examples(a) The Goal is 0, and the Solution clearly does not give the null set.(b) Mid/Jr/Sr: The Goal is 5, and the Restriction removes all but 4 (or fewer) cards in the

Universe (with no Double Set in Jr/Sr).

c. The Solution may be grouped so that it does not equal any value of the Goal. If an opponent believes there is an interpretation of a Solution that does not equal the Goal, that opponent must copy the Solution on his paper and add symbols of grouping to create a wrong interpretation. If this revised Solution does not equal the Goal, the Solution is incorrect. However, each checker has only one opportunity to prove ambiguity. If there is a second checker, the checkers may either work together to prove ambiguity or work separately. If working separately, the second checker may simultaneously and independently try to prove ambiguity. When both checkers are ready (or one is ready and the other has nothing to show), follow the same procedure used for checking the original Solution. That is, each attempt at proving ambiguity is

AOS - 81

checked. If either shows a legal interpretation of the Solution such that the Solution does not equal the Goal, then the original Solution is incorrect. If each attempt at proving ambiguity fails, the Solution must be accepted as correct. That is, once the Solution-writer starts checking the attempt(s) at proving ambiguity, no further objections to the Equation are allowed.Comments(a) In the case where the checkers work separately to prove ambiguity, if the time for checking the

Solution runs out, either or both checkers may take an additional minute (paying the one-point penalty to do so). If only one checker wishes to take the additional minute, the other checker may make no further changes to his revision of the Solution. If he tries to do so, then he incurs the one-point penalty also.

(b) Just as two players writing Solutions after a challenge or forceout may not communicate with each other, so two checkers attempting to prove ambiguity separately may not communicate while doing so.

(c) Each checker working separately has only one opportunity to prove ambiguity. Similarly, checkers working together have just one joint chance to prove ambiguity.

(d) While only one checker is attempting to prove ambiguity, the other checker may continue to check other aspects of the Solution.

(e) If each checker separately trying to prove ambiguity is ready with his revision of the Solution before the time for checking expires, no -1 penalty is enforced during the time the original Solution-writer checks any attempts at proving ambiguity.

Examples(a) The Set-Name B U G – R is ambiguous and may be interpreted by an opponent as

(B U G) – R or as B U (G – R). If the interpretation the opponent selects does not equal the Goal, the Solution is incorrect.

(b) R U G' is not ambiguous. It must be interpreted as R U (G') since ' takes priority over U.Comment Some variations (such as Y Wild) allow certain cubes to be used for other symbols. If a

Solution-writer wishes a cube to stand for anything other than what is on the cube, she must indicate clearly and unambiguously in writing what each such cube represents. (See Appendix A for a list of suggested ways of doing this.)

d. A symbol or group of symbols in the Solution has no defined meaning.Examples(a) The Set-Name is R U 'B or R Ʌ G.(b) Mid/Jr/Sr: The Restriction is R U (B = G) – R.

e. A variation is applied wrongly or not at all.Examples(a) With – Wild, a Solution uses a – for one symbol and another – for a different symbol.(b) With Two Operations, the Solution contains only one operation.


or Permitted or challenge Impossible. When the cube has been moved, each player has two minutes to write a Solution.The last cube in Resources may not be moved to Forbidden. If a player does so, any challenge that is made is set aside and the cube is returned to Resources. There is no penalty unless the player’s time to move expires. (See Section XI.)

B. An opponent may challenge Impossible against the player who moved the last cube from Resources to Required or Permitted, provided the challenge is made by the end of the first minute for writing Solutions. If the challenge is made, the Mover (and the Third Party, if siding with the Mover) has the rest of the original two minutes to write a Solution.Comment Any Now challenge against the player moving the last cube is invalid as is any Impossible

challenge made after the first minute for writing Solutions. In both cases, the challenge is set aside. A player challenging Now in this case receives a one point penalty.

IX. Illegal Procedures

AOS - 82

A. Any action that violates a procedural rule is an illegal procedure. A player charging illegal procedure must specify clearly (within 15 seconds) the exact nature of the illegal procedure.1. If a move is an illegal procedure, the Mover must return any illegally moved cube(s) to

their previous position (usually Resources) and, if necessary, make another move.The Mover must be given at least 10 seconds to make this correction, unless the original move was made after the 10-second countdown (see Section XI-A-3 below), in which case the time-limit rule (Section XI-A) is enforced. In general, there is no direct penalty except that the Mover may lose a point if she does not legally complete her turn during the time limit.Examples of illegal proceduresMoving out of turn, moving two cubes without calling “Bonus” before the first cube touches the mat in Forbidden, moving the last cube in Resources to Forbidden, or (in Middle, Junior and Senior) moving = or C to Forbidden with more than four cubes left in Resources.

2. If the move is not an illegal procedure, the cube stands as played.Comment There is no penalty for erroneously charging illegal procedure. However, see Section C below if a player does so frequently.


Example Suppose the player in the lead makes a bonus move. Before anyone notices the illegal procedure, the next mover moves (or a valid challenge is issued). Then the illegal bonus move stays in Forbidden without penalty.

C. Certain forms of behavior interfere with play and annoy or intimidate opponents. If a player is guilty of such conduct, a judge will warn the player to discontinue the offensive behavior. Thereafter during that round or subsequent rounds, if the player again behaves in an offensive manner, the player may be penalized one point for each violation after the warning. Flagrant misconduct or continued misbehavior may cause the player’s disqualification for that round or all subsequent rounds. Judges may even decide to have the other two opponents replay one or more shakes or the entire round because play was so disrupted by the third party. In some cases, judges may order the shake replayed by all three players.Examples This rule applies to use of a cell phone, constant talking, tapping on the table, humming or

singing, loud or rude language, keeping a hand or finger over or next to the challenge block, making numerous false accusations of illegal procedure, and so on. It also includes not playing to win but rather trying only to ruin the perfect scores of one or both opponents (for example, by erroneously challenging Now or Impossible at or near the beginning of each shake so that both opponents will score 5 for the round), saying one variation but circling another, constantly charging illegal procedure erroneously, counting down the 10-second warning in an obnoxious manner, etc.

X. Scoring a ShakeA. After a challenge, a player is correct according to the following criteria.


2. That player did not have to write a Solution (someone else did), and no opponent wrote a correct Solution.

B. After a challenge, points are awarded as follows.1. Any player who is not correct scores 2.2. Any player who is correct scores 6, unless that player is the Third Party agreeing with


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minute time limits are enforced with the timer. If a player fails to meet a deadline, he loses one point and has one more minute to complete the task. If he is not finished at the end of this additional minute, he loses his turn or is not allowed to complete the task.Note: In Elementary and Middle Divisions, each one-point penalty must be approved by a judge initialing the scoresheet.1. The time limits are as follows:

a. rolling the cubes and setting the Universe 1 minuteb. making a variation selection

This time limit does not begin until after the one minute for rolling the cubes and setting the Universe.

15 seconds

c. setting the Goal 2 minutesd. first turn of the player to the left of the Goal-setter 2 minutese. all other regular turns (including any bonus moves) 1 minutef. stating a valid challenge after picking up the challenge

block15 seconds

g. deciding whether to challenge Impossible when no more cubes remain in ResourcesIf the Impossible challenge is made, any time (up to a minute) the Challenger took deciding to challenge counts as part of the two minutes for writing a Solution.

1 minute

h. writing a SolutionDuring this time, the Third Party (if there is one) must decide whether to present a Solution after a Now or an Impossible challenge. At the end of these two minutes they must present their Solution.

2 minutes

i. deciding whether an opponent’s Solution is correct 2 minutes


3. If a player does not complete a task before sand runs out for the time limit, she must be warned that time is up. An opponent must then count down 10 seconds loud enough for the opponent to hear. The one-point penalty for exceeding a time limit can be imposed only if the player does not complete the required task by the end of the countdown.The countdown must be done at a reasonable pace; for example, “1,010; 1,009; 1,008…,”An exception to this rule occurs when a player picks up the Challenge Block but does not state a valid challenge within the 15-second time limit. If the player does not wish to challenge, he loses one point and play continues.

B. A round lasts a specified amount of time (usually 30 minutes). When that time is up, players are told not to start any more shakes.

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Players have five minutes to finish the last shake. After these five minutes, players still involved in a shake in which no challenge has been made and one or more cubes remain in Resources will be told: “Stop, don’t move another cube — this is the end of the round. Each player has two minutes to write a correct Solution that may use any of the cubes remaining in Resources. Any player who presents a correct Solution scores 4 points for that shake; an incorrect Solution scores 2.”

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B. When a round ends, each player must sign (or initial) the scoresheet and the winner (or one of those tied for first) turns it in. If a player signs or initials a scoresheet on which his score is listed incorrectly and there is evidence that there was intent to deceive and the error was not a simple oversight, then do the following:1. If the error gives the player a lower score, he receives the lower score.2. If the error gives the player a higher score, he receives 0 for that round.

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Two-Player Matches Pointsfirst place 6

two-way tie for first 5three-way tie for first 4

XIII. Adventurous VariationsA. Elementary Variations (grade 6 and below)1. R e q u ir e d C ub e The Solution must contain a cube. The player selecting this variation

specifies which non-digit symbol from the Resources fills the blank in the previous sentence.Comment If, say, Required – is chosen along with B Wild, a B cube used as – does not satisfy the Required

Cube variation.

2. W il d Cu b e The cube may represent any symbol on the cubes except a digit. The cube must stand for the same symbol everywhere it occurs in the Solution. The player selecting this variation specifies which cube from the Resources is wild. The wild cube may not be a digit. Each Solution-writer must specify in writing the interpretation of the wild cube if it stands for anything other than itself in his Solution.Comments(a) If both B Wild and B Required are chosen, a B cube must be in the Solution but may stand for another

symbol.(b) See Appendix A for examples of ways to indicate what a wild cube stands for in a Solution.

However, if B is wild but used as B, this need not be indicated.

3. U an d ∩ I n t e rc h a n g e a b le Any U may represent U or ∩, and any ∩ may represent∩ or U.Comments(a) U and ∩ need not be used consistently. In a Solution, one U (or ∩) may be used as U and another U (or

∩) used as ∩.(b) Any wild cube used as U or ∩ gains the full interchangeable power granted U and ∩ by this variation.(c) If U (or ∩) Wild and U-∩ Interchangeable are both chosen for a shake, then, if U (or ∩) is used just for itself or

∩, it need not be used consistently. However, if U (or ∩) is used for any symbol other than U or ∩, then it must represent that same symbol throughout the Solution.

(d) Since this variation makes U and ∩ “wild” in only a limited way, players are not required to indicate in writing where in the Solution a U stands for ∩ or a ∩ stands for U. They should simply write the symbol they want mathematically.

(e) If U Wild is also called, this does not mean ∩ cubes are wild and vice-versa.

4. V a n d Ʌ I n t e rc h a n g ea b le Any V may represent V or Ʌ, and any Ʌ may represent Ʌ or V.Comment The comments above for U and ∩ Interchangeable, substituting V for U and Ʌ for ∩, apply here.

5. T w o Ope r a ti on s Each Solution must contain at least two operation symbols. The operation symbols are U, ∩, –, and '.Comments(a) If a wild cube is also chosen, a wild cube used as an operation counts as an operation symbol. On the

other hand, any wild operation cube not used as an operation does not count as an operation symbol.(b) A Solution like R U B U V satisfies this variation. The variation does not require two different operation

symbols in the Solution.

6. M u lti p le O p e r a ti o n s Every operation sign in Required, Permitted, or Resources may be used multiple times in any Solution.Comments(a) After an Impossible challenge, any operation sign in Resources may be used many times in any Solution.

After a Now challenge, if the one cube allowed from Resources is an operation cube(or a wild cube used as an operation), it may be used multiple times.

(b) With this variation, an operation cube is not used to represent another symbol. So, players may simply write an operation sign multiple times in Solutions without any additional comment.

7. Sh i f t f r o m Pe r m i t t e d On your turn, you may transfer a cube in Permitted to either Required or Forbidden. This move takes the place of your regular move.Comments(a) If not in the lead, you may make a bonus move from Resources to Forbidden before transferring a cube out

of Permitted as your regular move.(b) You may never shift a cube from Permitted to Forbidden as a Bonus move.(c) Once the last cube in Resources has been moved to Required or Permitted, no more cubes from

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Permitted may be shifted.

B. Middle Variations (grade 8 and below)1. R e q u ir e d C ub e The Solution must contain a cube. The player selecting this

variation specifies which non-digit symbol from the Resources fills the blank in the previous sentence.Comment(a) If, say, Required – is chosen along with B Wild, a B cube used as – does not satisfy the Required Cube

variation.(b) If a player selects = or C Required, this variation is satisfied by using the required cube in a Restriction. If

the required cube is a color, V or Ʌ, or an operation symbol, the variation is satisfied by using that symbol in either a Restriction or the Set-Name. However, in the latter case, if the required symbol is played to Required, then, as usual, it must be in both a Restriction (if one is made) and the Set-Name.

2. W il d Cu b e The cube may represent any symbol on the cubes except a digit. The cube must stand for the same symbol everywhere it occurs in the Solution. The player selecting this variation specifies which cube from the Resources is wild. The wild cube may not be a digit. Each Solution-writer must specify in writing the interpretation of the wild cube if it stands for anything other than itself in his Solution.Comments(a) If both B Wild and B Required are chosen, a B cube must be in the Solution but may stand for another








4. V a n d Ʌ I n t e rc h a n g ea b le Any V may represent V or Ʌ, and any Ʌ may represent Ʌ or V.Comment The comments above for U and ∩ Interchangeable, substituting V for U and Ʌ for ∩, apply here.




6. M u lti p le O p e r a ti o n s Every operation sign in Required, Permitted, or Resources may be used multiple times in any Solution.Comments(a) After an Impossible challenge, any operation sign in Resources may be used many times in any Solution.


(b) With this variation, an operation cube is not used to represent another symbol. So players may simply write an operation sign multiple times in Solutions without any additional comment.

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Permitted may be shifted.(d) You may never shift an = or C cube from Permitted to Forbidden (even when there are four or fewer cubes

in Resources).

8. No N u ll R e stricti on s Each Restriction must remove at least one card from the Universe. In a chain Restriction, this variation is satisfied if any part of the chain removes a card.Comment If a Solution includes more than one Restriction, each must remove at least one card regardless of

the order in which they are applied to the Universe.

9. Ab s o lu t e V a lue Any upside-down cube(s) in the Goal may be interpreted as right- side-up by a Solution-writer.Examples(a) The Goal 3 (upside-down 2) may be interpreted as 1 or 5.

3(b) The Goal (where the 2 and 1 are upside-down) may equal 5 or 7. The 2 must be interpreted as

right-side up in order to create a legal (non-negative) value.

C. Junior Variations (grade 10 and below)SPECIAL RULE: The following three variations are in effect for all shakes.

1. M u lti p le O p e r a ti o n s Every operation sign in Required, Permitted, or Resources may be used multiple times in any Solution.

Comments(a) After an Impossible challenge, any operation sign in Resources may be used many times in any Solution.


(b) With this variation, an operation cube is not used to represent another symbol. So, players may simply write an operation sign multiple times in Solutions without any additional comment.






3. V a n d Ʌ I n t e rc h a n g ea b le Any V may represent V or Ʌ, and any Ʌ may represent Ʌ or V.Comment The comments above for U and ∩ Interchangeable, substituting V for U and Ʌ for ∩, apply here.-----

4. R e q u ir e d C ub e The Solution must contain a cube. The player selecting this variation specifies which non-digit symbol from the Resources fills the blank in the previous sentence.Comment(c) If, say, Required – is chosen along with B Wild, a B cube used as – does not satisfy the Required Cube

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variation.(d) If a player selects = or C Required, this variation is satisfied by using the required cube in a Restriction. If

the required cube is a color, V or Ʌ, or an operation symbol, the variation is satisfied by using that symbol in either a Restriction or the Set-Name. However, in the latter case, if the required symbol is played to Required, then, as usual, it must be in both a Restriction (if one is made) and the Set-Name.

5. W il d Cu b e The cube may represent any symbol on the cubes except a digit. The _____ cube must stand for the same symbol everywhere it occurs (Restriction(s) and Set-Name). The player selecting this variation specifies which cube from the Resources is wild. The wild cube may not be =, C, or a digit. Each Solution-writer must specify in writing the interpretation of the wild cube if it stands for anything other than itself in his Solution.Comments(a) If both B Wild and B Required are chosen, a B cube must be in the Solution but may stand for another






7. No N u ll R e stricti on s Each Restriction must remove at least one card from the Universe. In a chain Restriction, this variation is satisfied if any part of the chain removes a card.Comment If a Solution includes more than one Restriction, each must remove at least one card regardless of

the order in which they are applied to the Universe.



Permitted may be shifted.(d) You may never shift an = or C cube from Permitted to Forbidden (even when there are four or fewer cubes

in Resources).

9. Doub le Se t Each card in the Universe that is contained in the set will count double for all Solutions. The player selecting this variation specifies which nonempty set of cards that does not equal the Universe counts double. The set must be named using an expression consisting of at most four symbols (not counting grouping symbols).ExamplesA player selecting this variation may choose to double B, R', G ∩ Y, (B – R)', V – B', and so on. Players may not Double Sets like B U R U G, Y – (B ∩ G)', B' – R', V – (R – B), and so on. Also if a player selects R – Y as the doubled set but there are no cards in R – Y, the player is penalized one point and must select another variation. Similarly, if a player selects B U R' as the Double Set and every card in the Universe is B or R', the player loses a point and must pick another variation.Comment In Senior Division, if a player specifies a Double Set using -, that – means regular subtraction, even if a subsequent player calls Symmetric Difference. However, if Symmetric Difference is called first, than any – in a Double Set called by a subsequent player (or the same player) means Symmetric Difference.

10. R e qu ir ed /F o r b i d d e n C a rd The player selecting this variation either specifies one card in the Universe that must be in the Set-Name of any Solution or specifies one card in the Universe that must not be in the Set-Name of any Solution.Comments

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(a) The player states the required or forbidden card orally and records the card in a blank on the variation selection sheet. For example, “BRG”, “RY”, “blank”, and so on.

(b) If Blank Card Wild (see below) is chosen along with, say “BR forbidden,” then the blank cardmay not be made the forbidden card (BR) for any Solution.

11. B l an k C a rd W i ld Each Solution-writer must specify in writing which colors, if any, are on the blank card.Comments(a) This variation may be chosen only if the blank card has been dealt.(b) If the blank card remains blank for a Solution, the Solution-writer does not need to specify this.(c) Suppose Double Set and Blank Card Wild are both chosen with, say, B the Double Set. If a player chooses

to put a B dot on the blank card, the blank card counts double for that player’s Solution.(d) If Required Card is also chosen with the blank card required, that variation is satisfied if the blank card is in

the Set-Name even if the Solution-writer puts one or more colors on the blank card.(e) Blank Card Wild and Blank Card Forbidden are in conflict. So if Blank Card Wild is selected, a player then

choosing Blank Card Forbidden is penalized one point and vice-versa.

12.Ab s o lu t e V a lue Any upside-down cube(s) in the Goal may be interpreted as right- side-up by a Solution-writer.Examples(a) The Goal 3 (upside-down 2) may be interpreted as 1 or 5.

3(b) The Goal (where the 2 and 1 are upside-down) may equal 5 or 7. The 2 must be interpreted as

right-side up in order to create a legal (non-negative) value.

D. Senior Variations (grade 12 and below)Players may choose any of the Junior variations (except for the three that are in effect for every shake) plus the following.13. S y mm e tric D i f f e r e n ce The – symbol means “symmetric difference”: that is, A – B

equals (A – B) U (B – A), where these last two – signs mean set subtraction.If – Wild has already been selected for the shake, no player may select Symmetric Difference for that shake. Similarly, if Symmetric Difference has been chosen, no player may select – Wild. (In either case, the player selecting the second of the two conflicting variations receives a -1 penalty.)Comments(a) If Wild Cube is also chosen, any wild cube used as – means symmetric difference, not set subtraction.(b) In Solutions, players simply write the – sign with the understanding that it means symmetric difference.

14. T w o So luti o n s Each Solution-writer must write two Solutions; the set named by the second Solution must contain at least one card that is not in the set named by the first Solution.When this variation is in effect, players have three minutes to write Solutions and three minutes to check each player’s pair of Solutions. Also no Now challenges may be made with fewer than three cubes left in Resources. Furthermore, when the Mover plays the last cube from Resources to Required or Permitted, the other players have two minutes to challenge Impossible.Comments(a) An Impossible challenge should be made against a Goal of 0 since it is impossible to satisfy the Two

Solutions variation in this case. Similarly, a Goal equal to the number of cards in the Universe is impossible. With Double Set, the Goal can legitimately be larger than the number of cards. However, if the Goal is such that all cards in the Universe must be in any Solution, then an Impossible challenge should be made.

(b) In determining the rules a player’s two Solutions must follow, it is helpful to think of the Solutions as if they were presented by different players. Each Solution must use the cubes correctly and obey all the variations for the shake.

(c) After a Now challenge, Solution A of a player may use one cube from Resources and Solution B of that player may use a different Resource cube (or no cube).

(d) A wild cube may stand for one symbol throughout one of a player’s Solutions and another symbol throughout the other Solution.

(e) With Blank Card Wild, a Solution-writer may put one set of dots (or no dots) on the blank card for one Solution and another set of dots on the blank card for the other Solution. However putting a different set of dots on the blank card does not make it a different card for the second Solution. Thus, if the first Solution

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yields a set consisting of cards 1, 2, 3 and the blank card and the second Solution produces cards 1, 2, 3, and the blank card (with a different set of colors on it), these Solutions do not satisfy this variation. Similarly, if the Goal is 1, a player may not write one Solution that produces the blank card with a certain set of dots on it (or no dots) and a second Solution that produces the blank card with a different set of dots on it.

(f) Suppose Required Card is chosen and the Goal is 1. An opponent should challenge Impossible since it is impossible to write two Solutions that produce a different card and satisfy the Required Card variation.

(g) With Absolute Value, one Solution may equal one interpretation of the Goal, and the other Solution may equal the other interpretation.

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On-Sets AppendixHere are examples of ways of removing ambiguities in On-Sets solutions.Division Variations Examples Default

EMJS Wild Cube R∪G or B-V Wild Cube = itself: It is also B ∪ sufficient to indicate in one place in

the solution what the Wild Cube represents. It is understood to represent the same symbol throughout(Restriction&Set Name)

EMJS ∪,∩ interchangeable ∪=∪, ∩=∩ Write the symbol you want in eachplace in the solution. No other indication is necessary.

EMJS V,Λ interchangeable V=V, Λ=Λ Same as above

EMJS Multiple Operation Write the operation sign as many times as you need. You do not need indicate Multiple Operation.

S Blank Wild Blank card stays blank.

The Seven Proper On-Sets Goals

A A B A B C

A A + B A + B + C

A

B

A

B

C

A

B C

A

B C

A X B A X B X C (A X B) + C (B + C) X A

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Official Preprinted On-Sets ChartsA player may use these preprinted charts, or similar charts. Any other preprinted materials are not allowed.

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Michigan Leagues of Academic Games Elementary On-Sets Variations

1. Required cube The Solution must contain a ___ cube. The player selecting this variation specifies which non-digit symbol from the Resources fills the blank in the previous sentence.

2. Wild cube The ___ cube may represent any symbol on the cubes except a digit. The ___ cube must stand for the same symbol everywhere it occurs in the Solution. The player selecting this variation specifies which cube from the Resources is wild. The wild cube may not be a digit. Each Solution-writer must specify in writing the interpretation of the wild cube if it stands for anything other than itself in his Solution.

3. U and Ω interchangeable Any U may represent U or Ω, and any Ω may represent Ω or U.

4. V and /\ interchangeable Any V may represent V or /\, and any /\ may represent /\ or V.

5. Two operations Each Solution must contain at least two operation symbols. The operation symbols are U, Ω, –, and '.

6. Multiple operations Every operation sign in Required, Permitted, or Resources may be used multiple times in any Solution.

7. Shift from Permitted On your turn you may transfer a cube in Permitted to either Required or Forbidden. This move takes the place of your regular move.

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Michigan Leagues of Academic Games Middle On-Sets Variations


2. Wild cube The ___ cube may represent any symbol on the cubes except a digit. The ___ cube must stand for the same symbol everywhere it occurs (Restriction(s) and Set-Name). The player selecting this variation specifies which cube from the Resources is wild. The wild cube may not be =, C, or a digit. Each Solution-writer must specify in writing the interpretation of the wild cube if it stands for anything other than itself in his Solution.

3. U and Ω interchangeable Any U may represent U or Ω, and any Ω may represent Ω or U.

4. V and /\ interchangeable Any V may represent V or /\, and any /\ may represent /\ or V.

5. Two operations The Set-Name of each Solution must contain at least two operation symbols. The operation symbols are U, Ω, –, and '.

6. Multiple operations Every operation sign in Required, Permitted, or Resources may be used multiple times in any Solution (Set-Name or Restriction or both).


8. No null Restrictions Each Restriction must remove at least one card from the Universe. In a “chain” Restriction this variation is satisfied if any part of the chain removes a card.

9. Absolute value Any upside-down cube in the Goal may be interpreted as rightside-up by a Solution-writer.

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Michigan Leagues of Academic Games Junior On-Sets Variations

SPECIAL RULE: The following three variations are in effect for all shakes.1. Multiple operations Every operation sign in Required, Permitted, or Resources may be used

multiple times in any Solution (Set-Name or Restriction or both).2. U and Ω interchangeable Any U may represent U or Ω, and any Ω may represent Ω or U.3. V and /\ interchangeable Any V may represent V or /\, and any /\ may represent /\ or V.



6. Two operations The Set-Name of the Solution must contain at least two operation symbols. The operation symbols are U, Ω, –, and '.



9. Double set Each card in the Universe that is contained in the ___ set will count double for all Solutions. The player selecting this variation specifies which nonempty set of cards that does not equal the Universe counts double. The set must be named using an expression consisting of at most four symbols (not counting grouping symbols).

10. Required/Forbidden card The player selecting this variation either specifies one card in the Universe which must be in the Set-Name of any Solution or specifies one card in the Universe which must not be in the Set-Name of any Solution.

11. Blank card wild Each Solution-writer must specify in writing which colors, if any, are on the blank card. This variation may be chosen only if the blank card has been dealt.


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Michigan Leagues of Academic Games Senior On-Sets Variations

SPECIAL RULE: The following three variations are in effect for all shakes.1. Multiple operations Every operation sign in Required, Permitted, or Resources may be used

multiple times in any Solution (Set-Name or Restriction or both).2. U and Ω interchangeable Any U may represent U or Ω, and any Ω may represent Ω or U.3. V and /\ interchangeable Any V may represent V or /\, and any /\ may represent /\ or V

.



6. Two operations The Set-Name of the Solution must contain at least two operation symbols. The operation symbols are U, Ω, –, and '.



9. Double set Each card in the Universe that is contained in the ___ set will count double for all Solutions. The player selecting this variation specifies which nonempty set of cards that does not equal the Universe counts double. The set must be named using an expression consisting of at most four symbols (not counting grouping symbols).

10. Required/Forbidden card The player selecting this variation either specifies one card in the Universe which must be in the Set-Name of any Solution or specifies one card in the Universe which must not be in the Set-Name of any Solution.

11. Blank card wild Each Solution-writer must specify in writing which colors, if any, are on the blank card. This variation may be chosen only if the blank card has been dealt.


13. Symmetric difference The – symbol means “symmetric difference;” i.e., A – B = (A – B) U (B – A), where these last two – signs are the usual set subtraction.

14. Two Solutions : Each Solution-writer must write two Solutions; the Set-Name of the second Solution must contain at least one card that is not in the Set-Name of the first Solution.

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WFF ‘N PROOF® Tournament Rules 2017-18I. Starting a Match (Round)

A. Two- or three-player matches will be played. A match is composed of one or more shakes. A shake consists of a roll of the cubes and the play of the game ending with at least one player attempting to write a solution. A solution is a set of WFFs (see section II-B) (premises) and rule names (see section III) (rules) such that the goal can be inferred from those WFFs as premises by the rules named. It must be accompanied by a proof which justifies it.

B. The following equipment is needed to play the game.1. 28 cubes: there are fourteen uppercase letters (C, A, K, E, N, R) and fourteen

lowercase letters (p, q, r, s, i, o).2. A playing mat: this contains four sections:

(a) Goal: cubes played here form the Goal.(b) Permitted: any or all cubes played here may be used in any Solution.(c) Forbidden: any or all cubes played here cannot be used in any Solution.(d) Required: all cubes played here must be used in any Solution, in either the premises or rules

portion of the solution, and must be an essential part of the solution (see Section VII-B-4)

3. A one-minute sand timer: this is used to enforce time limits.4. A challenge block: This is a cube or similar object and not a flat object like a coin. It

should not be so large that two players can grab it simultaneously.C. Players may use only pencils or pens, and blank paper. No prepared notes, books, tables,

calculators, cell phones, or other electronic devices may be used.D. The Goal-setter for the first shake is determined by rolling an uppercase letter. On each

subsequent shake, the Goal-setter is the player immediately to the left of the previous Goal-setter.To determine the first Goal-setter, each player rolls an uppercase letter. The player rolling the letter closest to the start of the alphabet sets the first Goal. Players tied roll again until the tie is broken.

II. Symbols and Well Formed Formulas (WFFs)A. There are four types of cubes used in the tournament version of Wff ‘N Proof

1. Sentence variables consist of p, q, r, & s. They are the most basic WFFs and form the building blocks of more complicated WFFs

2. CAKE letters consist of C, A, K, & E. They are used to connect p, q, r, & s which are sentence variables to form WFFs and with rules cubes to form rules.

3. Negation, the N cube, is used to make WFFs (see II-B-3-b) or with rules cubes to make rules.

4. Rules Cubes only consist of R, i, & o. They are used to form rules. CAKE letters are also used with i and o to make rules.

B. The sentence variables and the CAKE letters are used to form Well Formed Formulas, or WFFs.1. The Goal (the conclusion of the proof), any premises presented, and every line of any

proof presented must be legal WFFs.2. The most basic WFFs are the sentence variables - p, q, r, s3. Longer WFFs are formed using the CAKE letters along with the sentence variables

(a) C, A, K, or E are placed in front of two smaller WFFs to make a longer WFFExamples: Kpr (The K connects the p and the r to form a K WFF)

Cqs (The C connects the q and the s to form a C WFF)Esp (The E connects the s and the p to form an E WFF)

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Aqq (The A connects the q and the q to form an A WFF)CKprAsp (The C connects the Kpr WFF to the Asp WFF to form a C WFF)AEqsp (The A connects the Eqs WFF to the p WFF to form an A WFF)KAqKqsEpp (The K connects AqKqs with Epp to form a K WFF)

(b) N is placed in front of a smaller WFF to make a WFFExamples: Nq

NsNNpNKprKNprCNpNs

III. RulesA. In the Elementary Game of WFF ‘N Proof and the Middle Game of WFF ‘N Proof, the

following rules are used: Ko, Ki, Rp, Ai, Co, Eo, EiNote: None of these rules require sub-proofs, and as a result sub-proofs are not allowed in this Game.

B. The Regular Game of WFF ‘N Proof uses the following rules in addition to the rules mentioned before: Ci, Ao, Ni, No, & R Note: As these rules require sub-proofs, sub-proofs are allowed in the Middle, Junior and Senior Divisions.

C. Even if it is used multiple times in a proof, each rule must only appear once in a solution.Example:

Note in the proof above, Ko is used twice in the proof, but is only allowed once in the solutionD. In addition to being used to form the Rp (Repeat) rule, and as the R (Reiterate) rule in

Regular WFF, the R cube may vary and stand for any rule.1. The R cube takes the place of the entire rule (i.e. Ci, Ko).2. If multiple R cubes are used, each R may stand for a different rule3. The use of the R cube as a rule must be indicated in the solution. The most common

way to do this is to write R and put the rule in parenthesis after the R: i.e. R(Ci)

IV. Starting a Shake and Setting the GoalA. To begin a shake, the Goal-setter rolls all 28 (14 uppercase, 14 lowercase) cubes. The

symbols on the top faces of the rolled cubes form the Resources for the shake.1. A shake begins as soon as the timing for rolling the cubes is started or the cubes are

rolled.2. During a shake, no player may turn over a cube or obstruct the other players’ view of any

cube. (See section IX-C)B. The player who rolls the cubes must set a Goal by transferring the cube(s) of the Goal

from Resources to the Goal section of the playing mat.C. A Goal consists of at least one and at most seven cubes which form a WFF. Once a cube touches the Goal section of the mat, it must be used in the Goal.

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1. The Goal-setter indicates the Goal has been set by saying “Goal Set.”2. The Goal-setter may rearrange or regroup the cubes in the Goal section until he says

“Goal Set.”3. If the time runs out to set the Goal or the setter turns the timer, it has been set.4. The Goal may not be changed once it has been set.

D. Before moving the first cube to the Goal section of the mat, the Goal-setter may make a bonus move.1. To make a bonus move, the Goal-setter must say “Bonus,” then move one cube from

Resources to the forbidden section (See section I-B-2) prior to placing cubes on the Goal section.

2. A Goal-setter who is leading in the match may not make a bonus move.If the Goal-setter makes a bonus move while leading in the match and an opponent points out the error before the next player moves or someone legally challenges, the cube that has been forbidden is returned to Resources. In the Middle, Junior and Senior Division, the mover also receives a one-point penalty.Note – it is legal to bonus while in the lead in Elementary WFF.

V. Moving CubesA. After the Goal has been set, play progresses in a clockwise direction (to the left).B. On their turn, each player must either move a cube from Resources to one of the

sections of the playing mat (Required, Permitted, Forbidden) or challenge the last Mover.The move of a cube is completed when it touches the mat. Once a cube is legally moved to the mat, it stays where it was played for the duration of the shake.

C. If you are not leading in the match, then “on your turn you may take a bonus move before making a regular move.”1. To make a bonus move, the Mover must say “Bonus,” then move one cube from

Resources to Forbidden (See section I-B-2).Comments(a) If you do not say ‘Bonus’ before moving cube to forbidden, the move does not count as a bonus

move but as a regular move. You are not entitled to play a second cube.(b) When making a bonus move, the first cube must be placed in forbidden. The second cube may be

moved to Forbidden, Permitted, or Required.

2. If the player in the lead makes a bonus move and an opponent points out the error before another player makes a legal move or challenge, the Mover must return the second cube played on that turn to Resources. The mover also receives a one-point penalty.Note: If both cubes were placed in forbidden at the same time, the Mover chooses which to return to Resources.


a move including setting the Goal. The only two legal challenges are Now and Impossible.1. By challenging Impossible, a player claims that no correct Solution and Proof can be

written regardless of how the cubes remaining in Resources may be played.Comments(a) If the Goal is not a WFF an opponent should challenge Impossible. Examples of such

Goals are pKr, CCCss, Apo, Epqrs.(b) Occasionally it is obvious before the Goal-setter completes the Goal that no Solution is possible.

However, opponents must still wait until the Goal-setter indicates the Goal is finished before challenging. You may not pick up the challenge block and “reserve” the right to challenge when the

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Goal is completed.(c) A Player who challenges “Never” will be considered to have challenged “Impossible”. There will be

no penalty for saying “Never” instead of “Impossible”.2. By challenging Now, a player claims that a Solution with a Proof can be written using

the cubes on the mat and one cube (or no additional cubes) from Resources.a. A player may challenge Now only if there are at least two cubes in Resources.

If a player challenges Now with fewer than two cubes in Resources, the challenge is invalid and is set aside. The player who called Now also receives a one point penalty.Comment If only one cube remains in Resources and no one challenges Impossible, then a Solution



B. A challenge block is placed equidistant from all players. To challenge, a player must pick up the block and say “Now” or “Impossible.”A player who picks up the block and makes an invalid challenge is penalized one point, and the challenge is set aside. Examples of invalid challenges are (a) challenging yourself in Regular Wff (you were the last Mover) – in Elementary Wff ‘N Proof and Middle Wff ‘N Proof, the challenge is set aside and there is no -1 penalty, (b) challenging Now before any cubes have been played to the mat. If a player picks up the block, then decides not to challenge (without saying “Now” or “Impossible”), the player accepts a one-point penalty and play continues.Comments(a) The purpose of the block is to determine who is the Challenger in a shake. This includes situations such

as when two players wish to challenge at the same time in a three-player match.(b) Touching the challenge block has no significance. However, players may not keep a hand, finger, or

pencil on, over, or near the block for an extended period of time. (See section IX-C)(c) A player must not pick up the challenge block for any reason except to challenge.

VII. Writing and Checking Solutions and ProofsA. After a valid challenge, at least one player must write a Solution and a Proof.

1. After a Now challenge, the Challenger must write a Solution and a Proof. (The Mover may not present a Proof and Solution.)

2. After an Impossible challenge, the Mover must write a Solution and a Proof. (The Challenger may not present a Proof and Solution.)

3. After any challenge in a three-player match, the Third Party must decide whether to agree with the Challenger or the Mover. If the player with whom the Third Party agrees must write a Solution and a Proof, then the Third Party must also write a Solution and a Proof.

B. To be correct, a Solution must include premises that are WFFs (see section II-B), and legal rules (see section III) and must be accompanied by a Proof.1. The Solution must be separate from the Proof in such a way that it is clear that it is not

part of the Proof. Solutions can be written above, below, left, or right of the Proof.2 . The Proof must follow the following guidelines:

(a) The top line of the proof should list the premises and the goal, separated by an arrow.i.e. p,r Kpr

(b) All premises in the solution must be listed first in the proof. Each premise must be listed on a separate line.

(c) Premises are often followed in proofs with an “s” which stands for suppose or supposition. However, no player shall be penalized for failure to indicate supposition.

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(d) Each proof must include at least one rule in addition to the Premise.(e) Sub-proofs must be indented. The lines that delimit sub-proofs must be written.(f) Each sub-proof must have one and only one premise.(g) Each WFF below the premise(s) must be justified by writing one of the legal rules (see section III). The

only exception to this is that the R (Reiterate) rule may be combined with exactly one other rule in forming a reason for a WFF in a sub-proof.

(h) When using the R rule, it is necessary to indicate in writing the number of times that the WFF has been reiterated (i.e. if a WFF has been reiterated from an exterior proof through two levels of sub proofs, R,R would be used).

(i) It is common to number lines of a proof and to refer to these step numbers when writing rules in proof. However, no player shall be penalized for failure to number steps or to refer to steps by number when writing the rule for each step. The following three Proofs with Solutions show how to number steps.

(j) Examples of acceptable proofs: Proof without a sub-proof:

Proof with sub-proofs:

Proof showing standard way to indicate wild cubes and cubes in Required:

3. The Solution must use the cubes correctly.(a) The Solution contains at least two cubes.(b) The Solution uses all the cubes in Required (see section VII-B-4).(c) The Solution uses no cubes in Forbidden.

Comment Since several Resource cubes may show the same symbol, it is possible to have a q in Forbidden which must not be used in the Solution at the same time that there is a q in Required which must be used.”

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(d) The Solution may use one or more cubes in Permitted.(e) After a Now challenge, the Solution must contain at most one cube from Resources.(f) After an Impossible challenge, any cubes in Resources may be used in the Solution.

4. Any cubes in Required must be used in the Solution in such a way that removal of those cubes from the solution would cause it to be incorrect.(a) To prove Non-Essential, the solution checker must write a new solution minus one or more Premises

and/or Rules that contained a cube that was put in Required and a Proof to support this new solution.(b) No cubes may be added to a solution when writing a proof of non-essentiality.(c) Multiple premises and / or rules may be proven to be non-essential with the same proof(d) The proof of non-essentiality cannot split the premises or rules of the original proof, or rearrange the

cubes to make new premises or rules.(e) Cubes in Required should be indicated in a solution. The most common way to do this is to put an

arrow below the cubes that are in Required. Not writing arrows does not make the solution incorrect.If the cubes in Required are not indicated, the solution may be ambiguous if the solution contains two of the same cube where one cube is in Required and the other is in Permitted. In that case the solution checker may chose to prove either cube non-essential.Example: A solution of Kpr / Ko, Ki is presented, where one of the K cubes is in Required, and two are in Permitted. If the K in Required is not pointed out in the solution, the solution checker may choose to prove which cube in the solution is marked as from Required section and prove it non-essential.

(e) Example of non-essential proof and solutionIf the following proof and solution was presented:

The solution checker could prove that the p cube was non-essential in the solution by presenting the following proof and solution:

C. After the time for writing Solutions has expired (or when all Solution-writers are ready), each Solution and Proof that is presented must be checked for correctness.1. After a challenge in a three-player match (and before any Solution and Proof are

presented), the Third Party must indicate by the end of the three minutes for writing Solutions whether she is presenting a Solution and Proof. The Third Party may not retract her decision once she has indicated whether or not she will present a Solution and Proof.Comment - To indicate her intention on the challenge, the Third Party may:

(a) state whether or not she will present an Solution;(b) indicate which party, Mover or Challenger, the Third Party is “joining” (agreeing with) on the

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challenge. This can be done verbally or by pointing to the party.(c) present or not present a Solution and Proof when the time limit for writing expires.

2. All Solutions with Proofs must be presented before any is checked.(a) Once a player presents a Solution and Proof to the opponent(s), she may make no further corrections

or additions even if the time for writing has not expired.(b) Each Solution-writer must indicate the Solution to be checked, including the proof that accompanies

the Solution. A writer who forgets to indicate the Solution and Proof must do so when asked.

3. Opponents have three minutes to check each Solution and Proof. When more than one Solution must be checked, they may be checked in any order. In a three-player match, both opponents must check a player’s Solution and Proof during the same three minutes. No other Solution should be checked during this time. Any proofs and solutions of non-essentiality must be written within the same three minutes.

4. Within the time for checking a Solution and Proof, opponents must accept or reject the Solution and Proof. If it is rejected, an opponent must show that it violates at least one of the criteria in section VII-B. A Solution and Proof is correct if no opponent shows that it is incorrect.After a Challenge in a three-player match, a player who does not present a Solution and Proof for a shake scores 2 if he accepts another player’s Solution and Proof as correct even if that Solution and Proof subsequently proven wrong by the other checker.Comment - Players must not physically move the cubes in Required, Permitted, and Resources to form the Solution being checked. This causes arguments over where each cube was played.

5. A player who claims an opponent’s Solution is not correct must give at least one of the following reasons (or cite one of the reasons in section VII-B).(a) The Goal is not a WFF.(b) The Solution contains a premise that is not a WFF.(c). The Solution writer has used a rule incorrectly in their proof(d) A cube in Required is non-essential in the solution writer’s proof. (see section VII-B-4)(e).The Solution uses a rule, or rules that are not allowed in that division(f) The solution writer does not have the necessary cubes to complete their solution(g) The Solution is not accompanied by a correct proof.

Proofs may be incorrect for any of the following reasons:1. A WFF other than a premise does not have a Rule listed to justify it2. The proof does not conclude with the goal3. Two or more WFFs are written on the same line.4. One line has an incorrectly written WFF5. A Rule is written incorrectly6. In Regular WFF ‘N Proof, a sub-proof is not indented7. There are two or more Rules use to justify a WFF. The lone exception to this is the R (reiteration)

rule may be used in Regular WFF in combination with other rules.

D. A Player may not ask the judge if a Solution equals the Goal. If asked, a Judge should simply state that he can not answer that question. A player should be more specific about why they think a Solution is incorrect, asking, for example, procedural questions or questions about the proof, such as, “Was the Co rule used correctly,” or “In line 3 of the proof is that a Wff?”.


or Permitted or challenge Impossible. When the last cube has been moved, each player has three minutes to write a Solution and Proof.The last cube in Resources may not be moved to Forbidden. If a player does so, any challenge that is made is set aside and the cube is returned to Resources. There is no penalty unless the player’s time to move

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expires. (See Section XI.)B. An opponent may challenge Impossible against the player who moved the last cube from

Resources to Required or Permitted, provided the challenge is made by the end of the first minute of the three minutes for writing Solutions and Proofs. If the challenge is made, the Mover (and the Third Party, if siding with the Mover) has the rest of the original three minutes to write a Solution.Comment Any Now challenge against the player moving the last cube is invalid as is any Impossible

challenge made after the first minute for writing Solutions. In both cases, the challenge is set aside. A player challenging Now in this case receives a one point penalty.

IX. Illegal ProceduresA. Any action which violates a procedural rule is illegal procedure. A player charging illegal


their previous position (usually Resources) and, if necessary, make another move.The Mover must be given at least 10 seconds to make this correction, unless the original move was made after the ten-second countdown (see section XI-A-3 below), in which case the time limit rule (section XI-A) is enforced. In general, there is no direct penalty except that the Mover may lose a point if he does not legally complete his turn during the time limit.Examples of illegal proceduresMoving out of turn, moving two cubes without calling “Bonus” before the first cube touches the mat in a forbidden section. (see section I-B-2)

2. If the move is not illegal procedure, the cube stands as played.Comment There is no penalty for erroneously charging illegal procedure. However, see section

IX-C if a player does so frequently.

B. An illegal procedure is insulated by a legal action (for example, a move or challenge) by another player so that, if the illegal procedure is not corrected before another player takes a legitimate action, it stands as completed.Example Suppose the player in the lead makes a bonus move. Before anyone notices the illegal

procedure, the next mover moves (or a valid challenge is issued). In this case, the illegal bonus move stays without penalty.


language, keeping a hand or finger over or next to the challenge block, making numerous false accusations of illegal procedure, and so on. It also includes not playing to win but rather trying only to ruin the perfect scores of one or both opponents (for example, by erroneously challenging Now or Impossible at or near the beginning of each shake so that both opponents will score 5 for the round), constantly charging illegal procedure erroneously, counting down the 10-second warning in an obnoxious manner, etc.

X. Scoring a ShakeA. After a challenge, a player is correct according to the following criteria.1. That player had to write a Solution and Proof and did so correctly.

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If the Third Party agrees with the person who must write a Solution and Proof , the Third Party must write a correct Solution and Proof, also.

2. That player did not have to write a Solution and Proof (someone else did), and no opponent wrote a correct Solution and Proof.Exception: After a Challenge in a three-player match, a player who does not present a Solution and Proof for a shake scores 2 if he accepts another player’s Solution and Proof as correct even if it is subsequently proved wrong by the other checker.

B. After a challenge, points are awarded as follows.1. Any player who is not correct scores 2.2. Any player who is correct scores 6, unless that player is the Third Party agreeing with


Impossible, points are awarded as follows:1. Any player who writes a correct Solution and Proof scores 4.2. Any player who does not write a correct Solution and Proof scores 2.


XI. Time LimitsA. Each task a player must complete has a specific time limit (listed below). The one, two, and

three-minute time limits are enforced with the timer. If a player fails to meet a deadline, he loses one point and has one more minute to complete the task. If he is not finished at the end of this additional minute, another one-point penalty is imposed and he loses his turn or is not allowed to complete the task.Note: In Elementary and Middle Divisions, each one-point penalty must be approved by a judge initialing the score sheet.1. The time limits are as follows.

(a) rolling the cubes and setting the Goal 2 minutes(b) first turn of the player to the left of the Goal-setter 2 minutes ( c ) all other regular turns (including any bonus moves) 1 minute(d) stating a valid challenge after picking up the challenge block 15 seconds(e) writing a Solution 3 minutes

During this time, the Third Party (if there is one) must decide whether to present a Solution after a Now or an Impossible challenge.

(f) deciding whether an opponent’s Solution and Proof is correct 3 minutes


3. A player who does not complete a task before sand runs out for the time limit must be warned that time is up. An opponent must then count down 10 seconds loud enough for the opponent to hear. The one-point penalty for exceeding a time limit may be imposed only if the player does not complete the required task bythe end of the countdown.The countdown must be done at a reasonable pace; for example, “1010, 1009, ..., zero.”An exception to this rule occurs when a player picks up the Challenge Block but does not state a valid challenge within the 15-second time limit. If the player does not wish to challenge, he loses one point and play continues.

B. A round lasts a specified amount of time (usually 35 minutes). After 30 minutes, players are told not to start any more shakes. Players have five minutes to finish the last shake. After these five minutes, players still involved in a shake in which no challenge has been made and one or more cubes remain in Resources will be told: “Stop, don’t

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Second place 4

move another cube – this is the end of the round. Each player has three minutes to write a correct Solution and Proof that may use any of the cubes remaining in Resources.” Any player who presents a correct Solution and Proof scores 4 points for that shake; an incorrect Solution and Proof scores 2.

XII. Scoring a MatchA. Each player is awarded points for the match based on the sum of his scores for the shakes

played during that match according to the following tables.


However, if there is evidence of intent to deceive and the error was not a simple oversight, then do the following.1. If the error gives the player a lower score, she receives the lower score.2. If the error gives the player a higher score, she receives 0 for that round.

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Third place 2

WFF ‘N Proof – A Game of Symbolic Logic- Appendix

Creating a WFFSingle Letter WFF – p, q, r, s 2 Letter WFFs – Np, Nq, Nr, Ns

3 Letter WFFs – Use a CAKE letter and 2 single WFFs after the CAKE letter – Cpq,

Ars, Kqr, Eps

3 or more letter WFFs – CAKE letters connect WFFs together to make longer WFFs – KrAsp, EKrqs, AEspKsr

First 7 Rules (Basic WFF)K out Ko Kw1w2 --------- w1 or w2 K in Ki w1, w2 ---------- Kw1w2 or Kw2w1 Repetition Rp w1 ---------- w1 (any WFF-- any WFF) A in Ai w1 or w2 ------- Aw1w2 C out Co Cw1w2, w1 ----- w2 E out Eo Ew1w2 ------ Cw1w2 or Cw2w1 E in Ei Cw1w2, Cw2w1 ------ Ew1w2 or Ew2w1

Next 5 Rules (Regular WFF): These rules use sub-proofs

C in Ci w1 w2 (in a sub-proof)------ Cw1w2 A out Ao Aw1w1, w1 w3(sub-proof), w2 w3(sub-proof), w3 N out No Nw1 w2, Nw2(sub-proof) ------ w1 N in Ni w1 w2, Nw2(sub-proof) ------ Nw1Reiteration R Brings a WFF from outside a sub-proof to inside a sub-proof.

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MLAG LINGUISHTIK Tournament Rules 2017-18

INTRODUCTORY STATEMENT Every effort will be made to accommodate the physically/sensory impaired student; however, it is the responsibility of the student to inform judges and to provide any special items needed for play.

LT1 GAME MANUAL REFERENCE

The following tournament version of the Advanced Games Instructions, explained on pages 6-13 in the LinguiSHTIK Games Manual, will be played at all levels.

LT 2 OBJECT OF THE GAME

The object of LinguiSHTIK is to make a 4-10 letter word using cubes from the game mat. The word must satisfy the demands made in the course of play and must be used in a sentence type, classified by pattern, structure or purpose that is designated by the first player.

LT 3 MATERIALS ALLOWED

The LinguiSHTIK Scoring Chart, the LinguiSHTIK Order of Play Sheet, the LinguiSHTIK General Demand Sheet (in elementary and middle divisions only), a supply of blank Demand Sheets, the LinguiSHTIK Game Mat, and all 23 LinguiSHTIK cubes are the only supplies and materials allowed in the game. Absolutely forbidden are the grammar books, dictionaries, the LinguiSHTIK Games or Judges Manual, the LinguiSHTIK Rules and Dictionary of Terms. (Caution: 1 cubes in some newer games contain 4 orange cubes; only 3 will be used in tournament play. 2 red cubes must contain the letter U; some games have C’s instead.)

Players may bring to the table only BLANK paper (lined or unlined) and writing implements (pens or pencils). BEFORE the round begins, players should check the papers of their opponents to make sure that all papers are BLANK. Once a round begins, any player may write anything on her/his own paper.

LT 3A JUDGING RULE

In Elementary and Middle Division, a judge must initial any -1 penalty.

LT 4 OFFICIAL REFERENCES

Dictionary: Merriam Webster’s online Unabridged Dictionary (dictionary.eb.com)1. Judges are reminded to check the Addenda when checking the veracity of the word.2. Words are not considered “foreign” if they are listed as an entry in the official dictionary. An entry includes the definition of the word. Foreign words listed without a definition will not be acceptable.

Grammar: Elements of Language, 6th Course published by Holt, Rinehart Winston (Elements of Language shall be considered the primary reference with the remaining two to serve as secondary sources to expand upon the Elements or when Elements does not address an issue.)Prentice-Hall Grammar and Composition, Levels 1-6, The Plain English Handbook

Judging: The LinguiSHTIK Judges’ Manual as revised in 2011. This compendium also addresses many of the arcane grammatical questions which are not directly addressed by traditional grammars. It is available free at www.academicgames.org

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http://www.academicgames.org/

THESE REFERENCE SOURCES WILL BE THE FINAL AUTHORITY ON ALL WORDS AND GRAMMAR USED.

LT 5 WHO GOES FIRST?

To determine who initiates the game, each player rolls a cube of the same color. The player, who rolls a letter closest to the beginning of the alphabet, becomes Player One for the first shake. If there is a tie, players involved will roll again until the tie is broken. To start a new shake proceed in a clockwise manner, to Player One’s left. Thus Player Two in the first shake becomes Player One in the second shake.

LT6 SENTENCES CLASSIFIED BY PATTERN, STRUCTURE, AND PURPOSE

To start a shake, Player One rolls the cubes, orders them in a group called Resources, and designates a sentence pattern, structure, or purpose. Players should write the sentence types in the designated section of their Demand Sheets. Allowable sentence types are listed below. See the Dictionary of Terms for further explanation.

SENTENCE PATTERNS will be restricted as follows:

ELEMENTARYS-V S-V-DO S-V-IO-DOS-LV-PN S-LV-PA INVERTED

MIDDLES-V S-V-DO S-V-IO-DO INVERTEDS-V-DO-OC(noun) S-V-DO-OC(adj) S-LV-PN S-LV-PA

JUNIOR, SENIORS-V S-V-DO S-V-IO-DO INVERTEDS-V-DO-OC(n) S-V-DO-OC(adj) S-LV-PN S-LV-PAS-V-Retained DO S-V-Retained IOS-V-Retained OC(noun) S-V-Retained OC (adjective)

SENTENCE STRUCTURES will be the same for all divisions:SIMPLE COMPOUND COMPLEX COMPOUND-COMPLEX

SENTENCE PURPOSE will be the same for all divisions:DECLARATIVE IMPERATIVE INTERROGATIVE EXCLAMATORY

LT 7 SENTENCE SPILLOVER CONFUSION

The sentence patterns listed under LT 6 are basic forms which do not change with the addition of single word modifiers, phrases, or dependent clauses. No structure change occurs by the addition of single word modifiers or phrases; however, the addition of clauses may change the structure of the sentence. (SEE DICTIONARY OF TERMS for more information and examples.)

LT 8 HOW TO MAKE A DEMAND

After Player One has stated the sentence designation, the next two moves and some later ones are Demands. In making a Demand, a player selects a green or black cube and places it on the section of the mat designated

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as Demands and simultaneously states his Demand. (SEE LT 9, LT 10, LT 15, LT 16, and LT 17 for explanations of Demands.)

Each player must write his/her Demand on the Demand Sheet and on their individual notebook paper. A player making a Demand is highly encouraged to check that all of the players have written down the correct Demand, particularly when it involves a letter. The Demand Sheet is the source for judges when answering questions about each shake. It does not replace a player’s responsibility to write down the demands on his own.

When the cube touches the mat, it is assumed to be played and may not be retracted; therefore, a player may not put the cube down in the Demands section and slide it over into the other section of the mat. A black or green cube in the Demands Section of the mat may not be used as one of the letters to be formed.

LT 9 TYPE DEMANDS

Player Two makes the second move which must be a Type Demand. Permissible Type Demands for all divisions are as follows:

1. Noun 2. Pronoun 3. Verb 4. Adjective5. Adverb 6. Preposition 7. Conjunction 8. Interjection

LT 10 FUNCTION DEMAND

Player Three must make a Function Demand unless the Type Demand is Interjection in which case Player Three may make a General Demand or place a cube on the playing mat. Permissible Function Demands are as follows:

ELEMENTARY DIVISION

NOUN: 1. subject 2. direct object 3. indirect object4. predicate noun 5. object of the preposition6. appositive* 7. noun used as an adjective

PRONOUN: 1. subject 2. direct object3. indirect object 4. predicate noun5. object of preposition 6. appositive*

* FORBIDDEN: Demanding that an appositive be restrictive

VERB: 1. main verb 2. infinitive 3. auxiliary

ADJECTIVE: 1. noun modifier 2. pronoun modifier3. adjacent adjective 4. predicate adjective

ADVERB: 1. verb modifier 2. adjective modifier 3. adverb modifier

PREPOSITION: 1. introductory word in an adjective phrase2. introductory word in an adverb phrase

The addition of dependent clauses or phrases will not affect the sentence pattern. When dealing with sentence structure, observe the rules governing simple, compound, complex, and compound-complex. The addition of

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clauses may change the structure of the sentence. (SEE DICTIONARY OF TERMS for more information and examples.)

FORBIDDEN: Compound Prepositions (SEE Dictionary of Terms)

CONJUNCTION: 1. subordinator 2. conjunctive adverb

FORBIDDEN: Correlative Conjunctions

INTERJECTION: NONE : The second demand is a general demand.NOTE ON INTERJECTIONS: A word may used as an interjection if the official dictionary lists the word as an interjection or lists the word as “used interjectionally”.

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MIDDLE DIVISION

NOUN: 1. subject 2. direct object 3. indirect object4. predicate noun 5. object of the preposition6. appositive* 7. noun used as an adjective8. objective complement

PRONOUN: 1. subject 2. objective complement 3. appositive*4. direct object 5. predicate noun 6. indirect object7. object of preposition

* FORBIDDEN: Demanding that an appositive be restrictive

VERB: 1. main verb 2. verbal 3. infinitive4. gerund 5. participle 6. auxiliary

ADJECTIVE: 1. noun modifier 2. pronoun modifier3. adjacent adjective* 4. predicate adjective5. objective complement

*SEE Dictionary of Terms




CONJUNCTION: 1. subordinator 2. conjunctive adverbFORBIDDEN: Correlative Conjunctions


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JUNIOR AND SENIOR DIVISIONS

NOUN: 1. subject 2. direct object3. indirect object 4. predicate noun5. object of the preposition 6. appositive7. noun used as an adjective 8. objective complement 9. retained direct

object 10. retained indirect object11. retained objective complement

PRONOUN: 1. subject 2. objective complement 3. appositive4. direct object 5. predicate noun 6. indirect object7. object of preposition 8. retained direct object9. retained indirect object 10. retained objective complement

VERB: 1. main verb 2. verbal 3. infinitive4. gerund 5. participle 6. auxiliary

ADJECTIVE: 1. noun modifier 2. pronoun modifier3. adjacent adjective* 4. predicate adjective5. objective complement 6. retained objective complement

*SEE Dictionary of Terms




CONJUNCTION: 1. subordinator 2. conjunctive adverbFORBIDDEN: Correlative Conjunctions


LT 11 PROPER ORDER OF PLAY

1) Player One states Sentence Designation (pattern, structure, or purpose)2) Player Two states the Type Demand (part of speech for the word)3) Player Three states the Function Demand

When the start of the game does not proceed in the proper order, then one of the players should declare Illegal Procedure and see that the proper procedure is followed. A player who does not make the proper demand in the right order must retract his demand and make a proper one. NO PENALTY IS INVOLVED UNLESS the player fails to make the proper demand in the one minute time limit allowed for making a demand. (SEE LT 13- TIME LIMITS AND PENALTIES)

LT 12 CHOICE OF PLAYSA. MOVE A CUBE TO LETTERS: A player may play a cube to the LETTERS Section of the mat with the intention that the letter played may be used to form the designated word at a later time. There is no set order for placing letters on this section of the mat. When a cube touches the mat in LETTERS, it is considered played and may not be retracted.

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B. MAKE A GENERAL DEMAND (SEE LT 15 & LT 16): A player may not move a cube to letters and also make a demand. A player may do one or the other, not both.C. CHALLENGE NOW OR IMPOSSIBLE: SEE LT 19 for explanation of the Challenges.D. PASS: Instead of playing a cube or making a Challenge, a player may PASS, giving up his turn.

SEE LT 25 for a detailed explanation of Pass and Forceout.

LT 13 TIME LIMIT AND PENALTIESAllowable time limits are: 1. Rolling and ordering the cubes and stating sentence designation 1 minute 2. Moving a cube to LETTERS 1 minute 3. Making a Demand 1 minute

PENALTY: In the situations above (1-3), if a player fails to make a play within the time limit, the player suffers a one point penalty (-1) and loses his turn. PENALTY: If Player 1, 2 or 3 has not made the proper demand in the 1 minute time allowed for making a demand, he/she receives a 1 point penalty and is instructed by the judge to “move”.

4. Writing a Solution 3 minutesFORBIDDEN: A Player may not take a -1 penalty in order to add a minute to the solution writing time.

5. Checking an opponent’s Solution 2 minutesPENALTY: In the situations above (4-6), if a player fails to act within the time limit, the player simply forfeits his right to do the indicated activity. THERE IS NO POINT PENALTY.PENALTY: *If a player makes a Challenge Now statement with fewer than three cubes in the Letters section of the mat, that player would receive a -1 penalty, lose his turn, and the challenge would be invalid.

6. Illegal Procedure: Any action which violates a procedural rule is an Illegal Procedure. A Player charging illegal procedure must clearly specify immediately the exact nature of the illegal procedure.Examples: Moving out of turn, making an illegal demand

A. If a move is an illegal procedure, the Mover must return any illegally moved cube to its previous position (usually Resources) and, if necessary, make another move. The Mover must be given at least 10 seconds to make the correction, unless the original move was made after the ten second countdown, in which case the time limit rule is enforced. If the player does not correct the action within the time limit than the player gets a -1 penalty, the action is set aside, and the player loses his turn.

B. If the move is not an illegal procedure, the cube stands as playedC. An illegal procedure is insulated by a legal move by another player so that , if the illegal procedure is not

called or corrected before the another player makes a legal move, it stands as played.

Players’ moves and activities are subject to the time limits stated above. A one minute sand timer is usually used to keep time. In practice, players will usually have more than two minutes to complete what they must do. Players timing an opponent may either flip or not flip the timer, as the case may be, so as to give the opponent the lesser amount of time. If for instance, 15 seconds is left from the previous time limit, let this sand run out, then flip the timer to begin the next player’s one or two minute time limit.

A player being timed must be given a ten second warning after the timer runs out. The player issuing the warning is responsible for being sure that his opponent is aware that a warning has been given. If one of the players does not notice the time has expired, the player being timed must move within ten seconds after someone does notice the expiration of time.

FORBIDDEN: Use of any kind of time-out rule during or between shakes and rounds.

LT 14 LATER DEMANDSDemands, if any, after the second Demand (Function Demand) made by Player Three, may be either General Demands or additional Function Demands.

LT 15 DEMANDS ABOUT THE WORDGeneral Demands must be about the word to be formed, not about the sentence to be written.

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LT 16 ACCEPTABLE GENERAL DEMANDSOnly the items in the following list may be used as General Demands. Items A-G are General Demands for all divisions. The remainder of the General Demands is broken down by division.

A. COLOR WILD: A color is wild in this shake. In the word to be formed, cubes of this color may represent a single letter more than once, or it may represent different letters. For instance, one wild cube may stand for “G” and another wild cube for “E” in the same shake. ONLY ONE COLOR MAY BE WILD IN A SHAKE.B. MUST CONTAIN: The word must contain a certain letter designated by the player making the demand. ONLY ONE LETTER MAY BE DEMANDED IN A SHAKE.C. MUST NOT CONTAIN: The word may not contain the letter designated by the player making the demand. ONLY ONE LETTER MAY BE FORBIDDEN IN A SHAKE.D. LETTER TRANSFER: All occurrences of a letter designated by a player making this demand become the other letter specified by the player making this demand. For instance, “All P’s are X’s”. In this case, “P’s” are entirely eliminated from the shake. Even a wild cube designated as a “P” becomes an “X”. ONLY ONE LETTER TRANSFER IS ALLOWED IN A SHAKE.E. NUMBER OF LETTERS: The word must contain the exact number of letters designated by the player making the demand. No fewer than four and no more than ten letters may be demanded.F. DOUBLE VOWEL: The word must contain a double vowel. This means the word must contain two consecutive vowels of the same letter; for example ee, oo, aa.G. DOUBLE CONOSONANT: The word must contain a double consonant. This means the word must contain tow consecutive consonants of the same letter; for example, tt pp.

The following demands are broken down by divisions and relate to the particular part of speech which was demanded in the type demand. These demands are also found on the Order of Play Sheets for each division.

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ELEMENTARY DIVISION

H. NOUN: 1. singular* 2. plural* 3. collective***Not applicable to noun used as adjective

** When a collective noun is used as an adjective, it cannot be singular or plural

I. PRONOUN 1. singular 2. plural 3. personal4. indefinite 5. possessive

J. VERB 1. singular form 2. plural form 3. linking4. regular 5. irregular 6. simple present tense7. simple past tense 8. simple future tense

K. ADJECTIVE 1. positive degree of comparison2. comparative degree of comparison3. superlative degree of comparison

L. ADVERB 1. positive degree of comparison2. comparative degree of comparison3. superlative degree of comparison

M. PHRASES* The solution word must be in an1. infinitive phrase2. appositive phrase

N. CLAUSES* The solution word must be in a1. dependent clause 2. noun clause3. adjective clause 4. adverb clause

*Only one clause or one phrase may be demanded in a shake.One of each may not be demanded.

O. The word must be a COMPOUND WORD.

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MIDDLE DIVISION

H. NOUN: 1. singular* 2. plural* 3. collective**4. nominative case* 5. objective case*

*May not be used as adjective** A collective noun used as an adjective cannot be singular or plural nor have a case

I. PRONOUN 1. singular 2. plural 3. personal4. indefinite 5. possessive 6. interrogative7. relative 8. nominative case9. objective case 10. demonstrative

J. VERB 1. singular form 2. plural form 3. linking4. regular 5. irregular 6. present participle7. past participle 8. simple tense** 9. perfect tense**10. progressive form** 11. perfect progressive form**12. function for infinitve 13. function for gerund

** Player may choose to designate present, past or future when tense or form is called,not as an additional demand.FORBIDDEN: Demanding that the verb be in the conditional tense.

K. ADJECTIVE 1. positive degree of comparison2. comparative degree of comparison*3. superlative degree of comparison** If these are demanded, the player may also indicate regular or irregular.

L. ADVERB 1. positive degree of comparison2. comparative degree of comparison*3. superlative degree of comparison** If these are demanded, the player may also indicate regular or irregular.

Notes on degrees of comparison: There are some modifiers that have no comparative or superlative forms; they do not vary in degree. These modifiers will be considered positive for the purposes of this game.

M. CLAUSES* The word must be contained in a1. dependent (subordinate) clause 2. adjective clause3. adverb clause 4. infinitive clause5. noun clause

N. PHRASES* The word must be contained in a

1. infinitive phrase 2. gerund phrase 3. participial phrase4. appositive phrase 5. adjective phrase6. adverb phrase 7. prepositional phrase

*NOTE ON LT 16 M & N: The number of times the two previous demands, M & N, can be made is limited to twice in this division. This maximum number represents a combination of both phrases and clauses. IT IS NOT two clauses and two phrases, BUT RATHER a total of two times that a demand may be made that the word be contained in either a clause or a phrase. EXAMPLE: 2 clauses, 2 phrases, or 1 clause and 1 phrase. NOTE: A prepositional phrase includes a preposition, a noun or pronoun called the object of the preposition, and any modifiers of that object. The grammar book states “any modifier that comes between a preposition and its object is part of the prepositional phrase”. Therefore, in a prepositional phrase with two objects, a clause modifying and following the first object is not part of the phrase, just geographically located within it.

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O. THE WORD MUST BE PART OF A DIRECT QUOTE (proper punctuation and capitalization required) SEE Dictionary of Terms for the definition of direct quote.

P. THE WORD MUST BE A COMPOUND WORDSEE the Dictionary of Terms on the difference between a compound preposition and a preposition which is compound.SEE ALSO Compound Word in the Dictionary of Terms.

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JUNIOR AND SENIOR DIVISIONS

H. NOUN: 1. singular* 2. plural* 3. collective**4. nominative case* 5. objective case*

*May not be used if noun used as adjective is the function demand** a collective noun used as an adjective cannot be singular or plural or have a case

I. PRONOUN 1. singular 2. plural 3. personal4. indefinite 5. possessive 6. interrogative7. relative 8. nominative 9. objective10. demonstrative 11. reflexive 12. intensive

J. VERB 1. singular form 2. plural form 3. linking4. regular 5. irregular 6. present participle7. past participle 8. emphatic form* 9. active voice10. passive voice 11. present infinitive12. perfect infinitive13. present perfect infinitive 14. simple tense**15 perfect tense** 16. progressive form**17. perfect progressive form** 18. imperative mood19. transitive 20. intransitive21. function for infinitive 22. function for gerund

* The player may choose to designate past or present.** Player may choose to designate present, past or future when tense or form is called,

not as an additional demand.FORBIDDEN: Demanding that the verb be in the conditional tense.

K. ADJECTIVE 1. positive degree of comparison2. comparative degree of comparison*3. superlative degree of comparison** If these are demanded, the player may also indicate regular or irregular.

L. ADVERB 1. positive degree of comparison2. comparative degree of comparison*3. superlative degree of comparison** If these are demanded, the player may also indicate regular or irregular.

M. CLAUSES* The solution word must be contained in the following clauses:1. dependent (subordinate) clause 2. adjective clause3. adverb clause 4. infinitive clause5. noun clause 6. elliptical (incomplete)*

*SEE Dictionary of Terms for definition of elliptical clause.

N. PHRASES* The solution word must be contained in the following phrases:

1. infinitive phrase 2. gerund phrase 3. participial phrase4. appositive phrase 5. adjective phrase6. adverb phrase 7. prepositional phrase

*NOTE ON LT 16 M & N: The number of times the two previous demands, M & N, can be made is limited to twice in this division. This maximum number represents a combination of both phrases and clauses. IT IS NOT two clauses and two phrases, BUT RATHER a total of two times that a demand may be made that the word be contained in either a clause or a phrase. EXAMPLE: 2 clauses, 2 phrases, or 1 clause and 1 phrase.

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NOTE: A prepositional phrase includes a preposition, a noun or pronoun called the object of the preposition, and any modifiers of that object. The grammar book states “any modifier that comes between a preposition and its object is part of the prepositional phrase”. Therefore, in a prepositional phrase with two objects, a clause modifying and following the first object is not part of the phrase, just geographically located within it.

O. THE WORD MUST BE PART OF: A DIRECT QUOTE (proper punctuation and capitalization required)* or an INDIRECT QUOTE

SEE Dictionary of Terms for the definition of direct quote.

P. THE WORD MUST BE A COMPOUND WORDSEE the Dictionary of Terms on the difference between a compound preposition and a preposition which is compound.

SEE ALSO Compound Word in the Dictionary of Terms

Q. THE WORD TO BE FORMED MUST NOT BE CONTAINED IN:1. adjective clause 2. adverb clause 3. noun clause 4. infinitive clause 5. elliptical clause 6. direct quote7. indirect quote 8. infinitive phrase9. gerund phrase 10. participial phrase11. appositive phrase 12. adjective phrase 13. adverb phrase

NOTE: Dependent clauses and prepositional phrase were intentionally omitted from this demand.NOTE ON LT 16 Q: The number of times this demand, know as the “Must NOT Be Contained IN…” demand, can be limited to once. Therefore, in combination with LT M & N in the Junior/Senior Divisions, it is possible to demand that a word be contained in two clauses or phrases and not be contained in one other.

R. AFTER THE DEMAND HAS BEEN MADE THAT THE WORD MUST BE IN A CLAUSE OR PHRASE, THIS ADDITIONAL DEMAND CAN SPECIFY HOW THAT CLAUSE OR PHRASE IS TO FUNCTION IN THE SENTENCE.

LT 17 HOW TO CHALLENGE

A challenge block is to be placed equidistant from all players at the table. A player challenges by picking up the challenge block and simultaneously stating his challenge. If the challenger does not pick up the challenge block, there is no challenge. If two players challenge at nearly the same time, the player who picks up the challenge block first is the challenger. If two players pick up the challenge block at exactly the same time, in the opinion of the third player, they are both challengers. The player may never challenge if he made the last move. Either of the two players, other than the last mover, may challenge. It does not need to be a player’s move for him to challenge.

LT 18 TYPES OF CHALLENGES

Instead of moving a cube to letters or making a demand, a player may challenge. The types of challenges are:A. CHALLENGE NOW: Using one more cube from Resources (if needed), a player will write a solution. The one more cube may be a letter which he may use in the word, or it may be a black or green cube which may be used to make a demand. If a player makes a demand, he must write the demand as well as the solution within the time limit. If all the cubes needed to make the word are already in the Letters section of the mat, the player does not need to use one more cube from Resources.B. CHALLENGE IMPOSSIBLE: It is impossible, with *only the demands currently in force to make a word which fits all of the demands with the letters available in letters and resources and also to write the designated sentence. *An additional demand may not be made if Challenge Impossible is called.

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A. CHALLENGE NOWThe CHALLENGER must write a solution within the three minutes. In writing the solution, the player may use at most one more cube from Resources. The one more cube may be a letter which he may use in the word, or it may be a black or green cube which may be used to make an additional demand. This additional demand must be written on the paper with the solution.

The MOVER and THIRD PARTY are assumed to be solvers if they write a solution within the three minutes. A player may chose to be Neutral by saying the word “Neutral”. Neutral means that the player is not going to write a solution.

*If a player makes a Challenge Now statement with fewer than three cubes in the Letters section of the mat, that player would receive a -1 penalty, lose his turn, and the challenge would be invalid.

B. CHALLENGE IMPOSSIBLEThe CHALLENGER may not write a solution. The MOVER must write a solution within the three minutes. In writing a solution, the player may use as many letters as needed from letters and/or resources, but the player may not make any further demands. The THIRD PARTY is assumed to be a solver unless he declares neutral within the first minute of the three minute solution writing period. (See LT 19A for Neutral Procedure.)

LT 20 WHAT IS A SOLUTION?

A solution consists of a written sentence which is of the pattern, structure or purpose designated by Player One and contains a word which satisfies all of the demands made of it. If the player writing a solution is making a demand as his last move, this last demand must also be written beside the solution. A solution shall be considered to be presented when a player directly hands his solution to another player thereby indicating that this is his solution. Once the solution is in the hands of another player, it cannot be withdrawn.

As a courtesy, when a player submits a sentence to opponents for evaluation, the player should circle the submitted sentence so that the opponents know which sentence to evaluate. If a player fails to circle when presenting, his opponents should ask that the sentence be circled. There is NO penalty for failing to circle a sentence.

LT 21 ABOUT THE WORD TO BE FORMED

The word which is formed must conform to the following specifications:A. It may not be a contraction, a hyphenated word, or a proper noun. It may not contain an apostrophe.B. It may not be labeled archaic or obsolete in the official dictionary.C. It may not be a foreign word, including letters and currency whose nationality is listed in the official dictionary. (Note: words are not considered foreign if they are listed as an entry in the official dictionary.)D. It may not be a word that is profanity, vulgar or slang in its usage.E. It may not be an abbreviated version of the word.F. It must be used accurately according to its definition in the official dictionary which is the final authority.G. It must be used in the sentence in the way it is normally used. A word cannot be called an adjective simply because the player wishes to use it in that manner. Again, the official dictionary is the final authority on whether a word may be used as the demanded part of speech.H. Rulings will be made in favor of those concerning themselves with the subject matter of the game as opposed to those who have developed a “gimmick”.

LT 22 ABOUT THE SENTENCE

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The sentence to be formed must conform to the following specifications:A. It must be able, in the opinion of the judges, to be justified as reality. The reality of the situation should be provided in the context of the sentence. The sentence will be viewed as presented by the player. There should be no need for any verbal clarification by the player. The words “in my dream” may not be used in the sentence to justify reality.Note: Judges and coaches realize that there may be other creative ways to write sentences (ex. In the cartoon, the coffee pot danced.); however, the sentences will be able to be judged based on the “reality” of what might appear in that cartoon.Reality will be considered as a separate issue from truth. While the sentence, “Bill Clinton is a Republican.” is not true, it is acceptable within the realm of reality.

B. It must be grammatically correct, including subject-verb agreement. A grammar book may be needed to determine correctness. If two grammar books disagree, the judging team will have the final authority.C. It must have all words spelled correctly and utilize proper capitalization.D. It must begin with a capital letter and close with the proper punctuation.E. Any possessive nouns used in the sentence must be properly punctuated.F. Internal punctuation will apply only to possessives, interjections, appositives, nouns of direct address, direct quotes, and conjunctive adverbs.G. It must not be, in the opinion of the judges, unintelligible or cumbersome.H. The sentence to be formed may not exceed 20 words in length.

LT 23 PASS MOVE and FORCEOUT PROCEDURE

The PASS MoveA. A PASS move should be called if a player feels that any move they could make would cause a NOW challenge.B. Calling PASS may not be done until Players 1,2, and 3 have moved to set up the shake by calling a Sentence Pattern, Structure, or Purpose; calling the TYPE demand; and stating a FUNCTION demand.Note: Usually this will occur near the end of the shake.C. The PASS move can be done by each player only once in a shake. D. If one or two players pass but the next decides to move, that player opens himself to a Now or Impossible challenge. Once the third player passes, FORCEOUT is called. Players need to be aware that calling PASS to early in a shake may result in the necessity of putting a cube on the mat later in the game, which may set up another player to challenge.

FORCEOUT ProcedureA. Once all three players have passed, FORCEOUT is called. B. FORCEOUT means that players have three minutes to write solutions using two more cubes from Resources. Neither of the cubes may be used as a demand. Correct solutions would score 4 points. Players with incorrect or no solutions would score 2 points.

LT 24 PLAYER ONE – SPECIAL MOVEPlayer One, who rolls the cubes and designates the sentence type, may be challenged impossible. If the composition of the resources is such that Player One does not think a solution can be made, regardless of the pattern, structure, or purpose, Player One should call CHALLENGE IMPOSSIBLE instead of a pattern, structure, or purpose. He would be challenging the shake, not a last mover. Players would have one minute to agree or disagree about NO SOLUTION being possible. If no player can write a solution, all players score (0) and proceed to the next shake. A player who disagrees with the IMPOSSIBLE declaration would give a pattern, structure, or purpose, and a word in a solution sentence, using letters from resources. (The time allowed for writing the solution sentence would be a total of 3 minutes which includes the 1 minute for agreeing or disagreeing.) A player who writes a correct solution sentence would score 6.

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If he is correct about no word being possible, and no player can write a solution, all players score (0) and proceed to the next shake. A player disagrees with Forceout declaration would give a pattern or structure as the case may be, and a word in such a sentence, using the letters from resources.

LT 25 FOUR AHEAD AT WARNINGIf a player is four or more points ahead of any player when the warning is called (“Do not start another shake: you have five minutes to finish the shake you are on.”) and the leading player calls Challenge Win, and no player has a correct solution, then any Neutral player receives six points instead of the normal four points. (see Situation C on the LinguiSHTIK Scoring Chart.)

LT 26 PLAYER BEHAVIOR

Certain forms of behavior interfere with play and annoy or even intimidate opponents. Some examples are constant tapping on the table, humming or singing, loud or rude language and constantly touching or moving the challenge block. If a player is guilty of such conduct, a judge will warn the player to discontinue the offensive behavior. After issuing this warning, the judge should inform the official in charge of the division and also the warned student’s moderator, if available. Thereafter during that round or subsequent rounds, if the player again behaves in an offensive manner, a three judge panel will consider the situation and may penalize the student one point for each violation after the warning. This panel will consist of the judge who issued the original warning, the chief judge of the division, and the student’s moderator. If any one of those listed is unavailable or if, for example, the judge who issued the warning and the chief judge are the same person, other judges may fill the positions. Flagrant misconduct or continued misbehavior may cause the player’s disqualification by the panel for that round or the entire tournament.

JUDGE’S NOTE: Discussions of students who ask question after question and the judge knows the player is grasping at straws: The judge should stop after several questions and ask the player to delineate the problem or error. If he can’t, the Judge should move on.

LT 27 PENALTY FOR MISSING A SHAKE

If a player misses a shake at the table, the player scores 0 for that shake.

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DICTIONARY OF TERMS

BASIC SENTENCE PATTERNS

1. S-V (Subject-Verb) Examples: Birds sing.

The boy walked. The children ran over the bridge.

2. S-V-DO (Subject-Verb-Direct Object) Examples: Mother bakes cookies.

Beetles have four wings. I usually leave the hardest problems until last.

3. S-LV-PN (Subject-Linking, Verb-Predicate Noun) Examples: That girl should be the captain.

Asia is a continent of extremes. The honor student became a leader.

4. S-LV-PA (Subject-Linking Verb-Predicate) Examples: The honey was very sweet.

Her pies taste delicious. Bob could not be happier.

5. S-V-IO-DO (Subject-Verb-Indirect Object-Direct Object) Examples: The class gave the teacher a party.

Pam wished us a safe trip. Karen dropped Harry a hint.

6. S-V-DO-OC (noun) (Subject-Verb-Direct Object-Objective Complement noun) Examples: We called Detroit our home.

The judges selected Alice Adams Miss Ohio. The sponsor appointed the girl captain.

7. S-V-DO-OC (adj.) (Subject-Verb-Direct Object-Objective Complement adj.) Examples: Our neighbor painted his house purple.

The dog licked the dish clean. Those people thought the clown funny.

8. S-V-Retained DO (Subject-Verb-Retained Direct Object) Examples: The teacher was given the apple by the student.

On her birthday, the mother was given flowers by her children.

9. S-V-Retained IO (Subject-Verb-Retained Indirect Object) Example: Flowers were given the mother by her children.

10. S-V-Retained OC (noun) (Subject-Verb-Retained Objective Complement noun) Example: Marvin was elected president by his friends.

11. S-V-Retained OC (adjective) (Subject-Verb-Retained Objective Complement adj.) Example: Howard considered the girl pretty..

12. INVERTED A sentence is inverted if the verb, or part of it, precedes the subject. The sentence may begin with L - 126

Here or There. For the purposes of this game, the interrogative sentence shall not be considered inverted. Examples: "That is strike three," yelled the umpire.

Here is a letter for you. Under the tree sat a group of hungry picnickers. There goes the parade.

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SENTENCES CLASSIFIED BY STRUCTURE 1. SIMPLE A sentence which contains a single sentence pattern is a simple sentence. The sentence must

contain only one main subject and one main predicate, although these elements may be compounded. A direct or indirect quote is considered a subordinate clause; therefore, its addition to a sentence will create either a complex or compound-complex sentence.

Examples: My father left yesterday. We visited two museums. Winston and Maurice traveled many miles and saw many things.

2. COMPOUND A sentence which contains two or more independent clauses but no dependent clauses is a compound sentence. Each clause must contain both a subject and a verb. The clauses may be connected by: a coordinating conjunction, conjunctive adverb, or a semicolon.

Examples: The war is over; the guns are silent. The story sounds false, but I know it is true. Byron is on a diet; nevertheless, he eats dessert.

3. COMPLEX A sentence which contains only one independent clause and one or more dependent clauses is a complex sentence. A direct quote is considered to be a dependent clause.

Examples: The train stalled because the switches were frozen. We heard that you were going to Cleveland. He said, "I will not eat meat.

4. COMPOUND-COMPLEX A sentence which contains two or more independent clauses and one or more dependent clauses is a compound-complex sentence.

Examples: When Turk caught the ball, he ran for a touchdown, and the crowd

went wild. After the storm was over, we decided to go home, but the snow which had fallen steadily blocked the road. He stood at the podium, and he said, "The only thing we have to fear is fear itself."

SENTENCES CLASSIFIED BY PURPOSE

1. DECLARATIVE A sentence which makes a statement. Example: In 1945 the United Nations had many member countries.

2. IMPERATIVE A sentence which gives a command or makes a request. Examples: Please write me.

Stop talking and open your books.

3. INTERROGATIVE A sentence which asks a question. Example: Which book did you like best?

Note: As defined in the grammar book, an interrogative sentence asks a question and ends with a question mark. A declarative sentence cannot be made interrogative simply by adding a question mark.

4. EXCLAMATORY A sentence which expresses strong emotion.

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Examples: How beautiful it is! I won the grand prize!

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ADJECTIVES:

ADJACENT ADJECTIVES: For the purposes of this game, an adjacent adjective is an adjective which immediately precedes or follows the noun or pronoun which it modifies. Predicate adjectives will not be accepted as adjacent adjectives. Adjective objective complements will not be accepted as adjacent adjectives since they fulfill a specific grammatical function in the sentence. Adjacent adjectives do not fulfill such a function.

Examples: I bought the brown hat for Sarah. The hat, torn and tattered, was found in the gutter. Several large animals were seen in the neighborhood.

(Sentences 2 & 3 each have two adjacent adjectives) ADJECTIVES – DEGREES OF COMPARISON:

Adjectives have three degrees of comparison: positive, comparative, and superlative. There are some modifiers that have no comparative or superlative forms; they do not vary in degree. These modifiers will be considered positive for the purposes of the game.

POSITIVE - the simplest, or plain, form of the adjective Example - quick - He is a quick student.

COMPARATIVE - The form used to compare two items. Example - quicker - John is quicker than Bill.

SUPERLATIVE - The form used to compare three or more items.

Example - quickest - Alfred is the quickest student in the class.

Some adjectives of two syllables and ALL adjectives of three or more syllables have their comparative and superlative degrees of comparison formed by using the words more and most or less and least.

A few adjectives are IRREGULAR and use different words for their comparative and superlative degrees.

Examples: good better best bad worse worst

The preceding information about IRREGULAR also applies to adverbs.

ADJECTIVES – OBJECTIVE COMPLIMENT: An adjective functions as an objective complement when it is the outer or second complement in a S-V-DO-OC sentence pattern. It completes the sense of the direct object. Adjective objective complements shall not be accepted as adjacent adjectives since they fulfill a specific grammatical function in the sentence.

Examples: He certainly considered the tiger angry. Mr. Jones painted his barn brown.

APPOSITIVE An appositive is a noun or pronoun, often with modifiers, set beside another noun or pronoun to further explain or identify it. One word appositives may have a possessive adjective or article in front of them and still be considered one word appositives. When additional modification is present the appositive becomes a phrase. In LING, appositives must be punctuated correctly.

Example: The word peace is abstract. (This one-word appositive is punctuated correctly.) Example: - single appositive - Nancy, my sister, lives in New Jersey.

- phrase - Nancy, an assistant director of nursing, lives in New Jersey.

COMPOUND PREPOSITION A compound preposition is a specific grammatical term which refers to a preposition comprised of two or

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more words written separately but used together as one preposition. Examples - in spite of, in addition to, in front of . . .

There are, however, prepositions which are compound words. This means that they are single words made up of several words but written together as one.

Examples - within, without.

If in the course of play, preposition is the type demand and then the general demand of compound is made, the player is to write a preposition which is a compound word, example 2, and not a compound preposition, example1. It is forbidden that the word be required to be a compound preposition.

COMPOUND WORD

For the game of LinguiSHTIK, a compound word shall be defined as one solidly written word comprised of two or more smaller words which retains the combined meanings of the smaller words of which it is comprised with no parts functioning as a prefix or suffix.

In an open compound, or two-word compound, such as "light bulb," the word "light" will be ruled as a noun used as an adjective.

CLAUSE A clause is a group of words which contains a subject and a verb and is used as part of a sentence. There are two types: independent and subordinate (dependent). A simple sentence is an independent clause. Note: Independent clause may only be demanded in the Elementary Division. When a type of clause is specified in LING, the word to be formed must be in the specified clause.

CONJUNCTIVE ADVERB An adverb which is used to join two independent clauses. It is a coordinator.

Examples: The goods were of the best quality; hence, they were satisfactory. The candidate made a gross error; therefore, he lost all support.

*Note: In LING, when a conjunctive adverb is used in a sentence, the proper punctuation must be used as shown in the examples above. This is true even if conjunctive adverb has not been demanded.

DIRECT QUOTATION For the purposes of this game, a direct quotation must have an attributive statement such as, He said, “

Examples: acceptable - He said, "I put the cows in the barn." unacceptable - "I put the cows in the barn."

Also for the purposes of this game, a direct quotation must contain a subject and a verb and will be considered a noun clause.

Examples: acceptable – She said, “I hit the ball.”unacceptable – He said, “Under the bridge near the barn.”

A direct quotation must be properly punctuated and capitalized.

DOUBLE CONSONANT OR VOWEL A pair of the same two letters which are consecutive.

Examples: miss (double consonant) foot (double vowel)

ELLIPTICAL CLAUSE For the purposes of this game, an elliptical clause shall be considered as a subordinate clause which begins with a subordinating conjunction and has an understood subject and/or auxiliary verb.

Examples: When (he was) running down the street, the boy was careless. He is taller than his uncle (is). When (she was) only a child, Julia saw a great grey owl.

GERUND L - 131

A gerund is a verb form ending in -ing that is used as a noun. It can also be called a verbal noun and may be used in any way that a noun is used. These should not be confused with participles.

Examples: Traveling is fun. (Used as subject) They do not like my singing. (Used as direct object) By studying, you can pass this course. (Used as object of the prep.)

INFINITIVE

An infinitive is a verb form, usually preceded by "to" which is used as a noun or a modifier. It has three tense forms: present, perfect, and present perfect. They may be in either the active voice (which may have two forms: simple or progressive) or the passive voice.

The word "to", called the sign of the infinitive, is sometimes omitted such as in: She made me (to) leave. and Help me (to) do my homework.

In LinguiSHTIK, the word to be formed is the infinitive itself; however, the "to" needs to be written if required by the sense of the sentence.

Examples: To wait for the bus is tiresome. (noun, subject) Everyone wanted to travel. (noun, direct object) We lacked the strength to resist. (adjective,objective complement) We study to learn. (adverb)

PRESENT INFINITIVE Indicates a time the same as or future to that of the main verb.

Examples: They had come to ask you. He is coming to ask you. (future) The children were disappointed because they had hoped to go with you.

PERFECT INFINITIVE Primarily indicates action previous to the time of the main verb.

Examples: I am glad to have been one of his boys. I am glad to have seen that movie.

PRESENT PERFECT INFINITIVE Used to indicate action, or help make a statement about something which occurred at no definite time in the past.

Examples: The students were delighted to have been selected as the recipients of the scholarships.

To have been elected unanimously as the president of the chapter is a great honor.

INFINITIVE FORMS

KIND ACTIVE PASSIVE

PRESENTSIMPLEto ask

PROGRESSIVEto be asking to be asked

PERFECT to have asked to have been askedPRESENT PERFECT to have been asking

INFINITIVE PHRASEAn infinitive phrase consists of an infinitive and any complements or modifiers it may have.

Examples: We have time to walk around the block. To walk around the block is good exercise.

NOUN USED AS ADJECTIVE

A noun may function as an adjective. Although many noun functions apply also to pronouns, THIS ONE

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DOES NOT. Additionally, a noun used as adjective may be neither singular or plural, nor can it be nominative, objective, or possessive case. Since adjectives have no number or case and these nouns function as adjectives, they lose the ability to express number or case.

Note: many pairs of nouns and the nouns used as adjectives that modify them become so common that they are listed in dictionaries as compound nouns. The designation noun used as adjective may include these compound nouns used as two words, but not hyphenated nouns.

Examples: Mr. Smith is a science teacher. Mr. Jones is a criminal lawyer.

NOUN OBJECTIVE COMPLEMENT

When the inner complement in the pattern S-V-DO-OC is a direct object, the outer complement is an objective complement. The objective complement relates to or completes the direct object. It may also be an adjective. (SEE ALSO Adjective Objective Complement)

Examples: The committee declared Joe the winner. We elected Mary chairman of the committee.

PARTICIPLE

A participle is a verb form used as an adjective. It comes in two forms: the present participle which ends in "-ing" and the past participle which often ends in "-ed, -d, -en, or -t." Participles should not be confused with the participle used in a verb phrase or with gerunds. Particular care must be exercised when it comes to the progressive tense. The following are not participles but rather part of the verb phrase - I am frustrated. I was swimming. Finally, participles cannot be used as noun used as adjective.

Examples: The rapidly developing storm kept small boats in port. The broken toys could not be replaced easily.

PHRASE

A phrase is a group of words which functions as a single part of speech and does not contain a subject and verb. When a type of phrase is specified in LING, the word to be formed must be in the specified phrase.

RETAINED OBJECT

Retained objects can only be found in sentences written in the passive voice. They are separate and distinct from any other type of object since they do not meet the defined functions of such objects. For example, a direct object directly receives the action of the verb; indirect objects indirectly receive the action of the verb, objective complements complete the direct object. Since a passive voice sentence has no action, no object can directly or indirectly receive the action. The following sentences are written in the active voice. IO DOA. The boy gave Mary flowers.

DO OC(adj)B. Howard considered Martha pretty.

DO OC(noun)C. We elected Marvin president.

DOD. We sent flowers.

In all cases above, the objects function as defined. There is action and there is a receiver of that action. When, however, these sentences are transformed into the passive voice, there is no longer an action; however, they remain objects. These objects are correctly called retained objects.

A1. Mary was given flowers by the boys. (retained DO) A2. Flowers were given Mary by the boys. (retained IO) B. Martha was considered pretty by Howard. (retained OC-adj.) C. Marvin was elected president by his friends. (retained OC-noun) D. Flowers were sent by us. (NO RETAINED OBJECT)

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You should note in the passive constructions A1 and A2 of the original sentence A that it can be determined whether the direct or the indirect was retained as object. In LinguiSHTIK when the demand of retained direct, indirect, or objective complement is made, it is suggested, but not required, that players write their active voice construction which they transformed into passive voice for their solution on the same paper as the solution itself. This may serve to avoid disagreements in the presentation of the solutions.

SPILLOVER CONFUSION

When dealing with sentence patterns, observe the rules governing subject-verb-complement. When dealing with sentence structures, observe the rules for simple, complex, etc. sentences. The addition of dependent clauses or phrases will not affect the sentence pattern. Additionally, if a pattern is called and the player writes a solution within a compound sentence, the player must be careful to include the word to be formed in the portion of the sentence which meets the required pattern demand.

Example: A direct object must be contained in an S-V sentence. CORRECT - One of the girls who selected a complicated song for the audition fainted after the performance. (EXPLANATION: Song is the direct object in the subordinate clause which is contained within a S-V pattern.) INCORRECT - The girl stood up and she sang a song. (EXPLANATION: Even though there is an S-V pattern in the compound sentence, the direct object was written in that portion which contains an S-V-O pattern.)

SUBORDINATOR A conjunction that begins a subordinate clause (usually an adverb clause), joins the clause to the rest of the sentence, and shows a relationship between the clause and the remainder of the sentence. Some of the subordinating conjunctions can be used as other parts of speech such as pronouns, prepositions, and adverbs.

Examples: whom, that, since, when, after, although, because VERBS:

AUXILIARY: A verb used with the main verb to create tense and mood. If Auxiliary is called, the player forms the helping verb only, not the predicate verb.

Example: The children will laugh at my jokes. He does know the rules.

MAIN VERB: The main verb in a clause that identifies the action or existence of the subject of the clause.

Examples: The boy cried. The girl is laughing. The lady became a doctor. When the bell rang the children walked into the classroom.

REGULAR AND IRREGULAR VERBS: A regular verb forms its past and past participle by adding “-d” or '-ed” to the infinitive form.

Example: He walks. He walked.An irregular verb forms its past and past participle in some other way, usually by changing the spelling or by making no change at all.

Example: He begins the game today. He began the game yesterday.

SIMPLE VERB TENSES: In LinguiSHTIK, the simple tenses of the verb shall be considered to be the plain tense as shown in the following examples. When it is necessary to use an auxiliary verb to form the passive voice, the word to be formed is the main verb.

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SIMPLE VERB TENSESPRESENT PAST FUTURE

Active Passive Active Passive Active Passive

play or plays am, is or are played

played was or were played

will or shall play will or shall be played

give or gives am, is or are given

Gave was or were given

will or shall give will or shall be given

TRANSITIVE AND INTRANSITIVE VERBS A transitive verb is one which has a direct object.

Example: The cat ate the mouse. The judge explained the contest rules.

An intransitive verb is one which has no direct object. All linking verbs are intransitive. Example: The cat ate.

Patiently, the judge explained. The contestants still misunderstood.

VOICE OF THE VERB A verb can be of one of two voices: active or passive.

Active voice is when the subject performs the action of the verb. Passive voice is when the subject is not the performer of the action. Frequently with passive voice the subject will actually be the receiver of the action. (The action is performed on the subject).

Examples: Bill painted the barn. (active) The barn was painted by Bill. (passive) We gave Mary flowers. (active) The flowers were give Mary by us. (passive)

Note:

Not all the details on the Order of Play Sheets are defined in this Dictionary of Terms. The LinguiSHTIK Tournament Rules describe how to play the game and list official references. Grammatical questions can be answered by referring to Elements of Language, 6th Course published by Holt Rinehart Winston. Complete details and examples are available in the AGLOA LinguiSHTIK Judges’ Manual which may be downloaded from www.academicgames.org.

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ORDER OF PLAY SHEET ELEMENTARY DIVISION

PLAYER ONE - Rolls cubes and states a Sentence Pattern, Structure, OR PurposePATTERNS-V S-LV-PNS-V-DO S-LV-PAS-V-IO-DO INVERTED

STRUCTUREsimple complex compound compound-complex

PURPOSEdeclarative interrogative imperative exclamatory

PLAYER TWO - Uses a BLACK or GREEN cube to state a TYPE Demand

NOUN PRONOUN VERB ADJECTIVE ADVERB PREPOSITION CONJUNCTION INTERJECTION

PLAYER THREE - Uses a BLACK or GREEN cube to state aFUNCTION Demand

NOUN - Subject, Direct Object, Indirect Object, Predicate Noun, Object of the Preposition, Appositive, Noun used as adjective

PRONOUN - Subject, Direct Object, Indirect Object, Predicate Noun, Object of the Preposition, Appositive

FORBIDDEN - Demanding an appositive be restrictiveVERB – Main Verb, Infinitive, Auxiliary

ADJECTIVE – Noun Modifier, Pronoun Modifier, Predicate Adjective, Adjacent Adjective

ADVERB – Verb Modifier, Adjective Modifier, Adverb Modifier

PREPOSITION - Introductory word in an Adjective PhraseIntroductory word in an Adverb Phrase

FORBIDDEN: Compound Preposition

CONJUNCTION - Subordinator, Conjunctive AdverbFORBIDDEN: Correlative Conjunction

INTERJECTION - NONE - The second demand is a General Demand

ELEMENTARY GENERAL DEMANDS

A. COLOR WILD B. MUST CONTAINC. MUST NOT CONTAIN D. LETTER TRANSFERE. NUMBER OF LETTERS F. DOUBLE VOWELG. DOUBLE CONSONANT

H. NOUN I. PRONOUN1. singular* 1. singular2. plural* 2. plural3. collective 3. personal *not applicable 4. indefinite to noun used as adjective 5. possessive

J. VERB1. singular 3. linking 6.simple present tense2. plural 4. regular 7.simple past tense

5. irregular 8.simple future tense

K. ADJECTIVE1. positive degree of comparison2. comparative degree of comparison3. superlative degree of comparison

L. ADVERB1. positive degree of comparison2. comparative degree of comparison3. superlative degree of comparison

M. PHRASES*1. infinitive 2. appositive

N. CLAUSES*1. adjective 3. dependent2. adverb 4. noun

*Only one clause or one phrase may be demanded in a shake. One of each may NOT be demanded.

O. The word must be a COMPOUND WORD

Revised August 2017

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ORDER OF PLAY SHEET MIDDLE DIVISION

PLAYER ONE - Rolls cubes and states a Sentence Pattern, Structure, OR PurposePATTERNS-V S-V-IO-DOS-V-DO S-V-DO-OC (noun)S-LV-PN S-V-DO-OC (adj.)S-LV-PA INVERTED




NOUN PRONOUN VERB ADJECTIVE ADVERB PREPOSITION CONJUNCTION INTERJECTION

PLAYER THREE - Uses a BLACK or GREEN cube to state aFUNCTION Demand

NOUN - Subject, Direct Object, Indirect Object, Predicate Noun, Objective Complement, Object of the Preposition, Appositive, Noun used as adjective

PRONOUN - Subject, Direct Object, Indirect Object, Predicate Noun, Objective Complement, Object of the Preposition, Appositive

FORBIDDEN - Demanding an appositive be restrictiveVERB - Predicate, Verbal, Infinitive, Gerund, Participle, Auxiliary

ADJECTIVE – Noun Modifier, Pronoun Modifier, Predicate Adjective, Objective Complement, Adjacent Adjective

ADVERB – Verb Modifier, Adjective Modifier, Adverb Modifier

PREPOSITION - Introductory word in an Adjective PhraseIntroductory word in an Adverb Phrase

FORBIDDEN: Compound Preposition

CONJUNCTION - Subordinator, Conjunctive AdverbFORBIDDEN: Correlative Conjunction

INTERJECTION - NONE - The second demand is a General Demand

MIDDLE GENERAL DEMANDS

A. COLOR WILDB. MUST CONTAINC. MUST NOT CONTAIND. LETTER TRANSFERE. NUMBER OF LETTERSF. DOUBLE VOWELG. DOUBLE CONSONANT

H. NOUN1. singular* 4. nominative case2. plural* 5. objective case3. collective**not applicable to noun used as adjective

I. PRONOUN1. singular 6. demonstrative2. plural 7. relative3. personal 8. nominative case4. indefinite 9. objective case5. interrogative 10. possessive

J. VERB1. singular 8. simple tense*2. plural 9. perfect tense*3. linking 10.progressive form*4. regular 11.perfect

progressive form*5. Irregular 12.function for

infinitive6. past participle 13. function for

gerund7. present participle

* The player my choose to designate present, past, or future when tense or form is called (not as additional demand)

K. ADJECTIVE1. positive degree of comparison*2. comparative degree of comparison*3. superlative degree of comparison*

*If these are demanded, the player may indicate regular or irregular

L. ADVERB1. positive degree of comparison*2. comparative degree of comparison*3. superlative degree of comparison*

*If these are demanded, the player may indicate regular or irregular

M. CLAUSES*1. dependent 4. noun2. adjective 5. infinitive3. adverb

N. PHRASES*1. infinitive 5. adjective2. gerund 6. adverb3. participial 7. prepositional4. appositive

*NOTE ON LT 16 M & N: The number of times the two previous demands, M & N, known as "Must Be Contained In _____" Demands can be made is limited to twice in this division. This maximum number of two (2) represents a combination of both

phrases and clauses. IT IS NOT two clauses and two phrases, BUT RATHER a total of twice that a demand can be made

that the word to be formed be contained in either a phrase or a clause. EXAMPLE: 2

clauses, 2 phrases, OR 1 clause and 1 phrase.

O. The word must be contained in a DIRECT QUOTE

P. The word must be a COMPOUND WORD

Revised August 2017

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ORDER OF PLAY SHEET JUNIOR AND SENIOR DIVISION

PLAYER ONE - Rolls cubes and states a Sentence Pattern, Structure, OR PurposePATTERNS-V S-V-IO-DOS-V-DO S-V-DO-OC (noun)S-LV-PN S-V-DO-OC (adj.)S-LV-PA INVERTEDS-V-Retained DO S-V-Retained OC (noun)S-V-Retained IO S-V-Retained OC (adj)




NOUN PRONOUN VERB ADJECTIVEADVERB PREPOSITION CONJUNCTION INTERJECTION

PLAYER THREE - Uses a BLACK or GREEN cube to state aFUNCTION DemandNOUN - Subject, Direct Object, Indirect Object, Predicate Noun, Objective Complement, Object of the Preposition, Retained Direct Object, Retained Indirect Object, Retained Objective Complement, Appositive, Noun used as adjectivePRONOUN - Subject, Direct Object, Indirect Object, Predicate Noun,Objective Complement, Object of the Preposition, Retained Direct Object, Retained Indirect Object, Retained Objective Complement, AppositiveFORBIDDEN - Demanding an appositive be restrictiveVERB - Predicate, Verbal, Infinitive, Gerund, Participle, Auxiliary*Functions may be called for infinitives and gerundsADJECTIVE – Noun Modifier, Pronoun Modifier, Predicate Adjective, ObjectiveComplement, Adjacent Adjective, Retained Objective ComplementADVERB – Verb Modifier, Adjective Modifier, Adverb ModifierPREPOSITION - Introductory word in an Adjective Phrase

Introductory word in an Adverb PhraseFORBIDDEN: Compound PrepositionCONJUNCTION - Subordinator, Conjunctive AdverbFORBIDDEN: Correlative ConjunctionINTERJECTION - NONE - The second demand is a General Demand

Revised August 2017

LINGUISHTIK SCORING CHART

CHALLENGER: The Player who makes the challenge.SOLVER: A player other than the Challenger who presents

a correct solution.

NEUTRAL: A player other than the Challenger [Challenge Now] OR a player other than the Challenger or Mover [Challenge Impossible] who does not present a solution.

WRONG: A player who presents an incorrect solution when there has been a Challenge, OR a player who either presents an incorrect solution or does not present a solution during a Forceout.

MOVER: A player who makes the last move before aChallenge Impossible

AGREER: A player who agrees to or a Forceoutand presents a correct solution.

SITUATION 6 POINTS 4 POINTS 2 POINTSA. C H A L L EN G E NOW

Challenger has a correct solution

CHALLENGER SOLVER NEUTRALWRONG

B.C H A L L E N G E NOW Challenger DOES NOT have a correct solution, but another player does

SOLVER NEUTRAL CHALLENGER

WRONG

C.C H A L L EN G E NOW NO PLAYER has a correct solution

NEUTRAL(SEE LT 25*)

CHALLENGERWRONG

D. C H A L L EN G E I M P O SSIBLE

NO PLAYER has a correct solution

CHALLENGER NEUTRAL MOVERWRONG

E. C H A L L EN G E I M P O SSIBLE

At least one player has a correct solution

SOLVER CHALLENGER WRONG

NEUTRAL

F. FO R CE O UT ALL PLAYERS agreed

AGREER WRONG

* LT25 - If a player is four or more points ahead of any other player when the warning has been called, and the leading player CHALLENGES NOW, and NO PLAYER has a correct solution, ANY NEUTRAL PLAYER receives six 6 points.

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JUNIOR & SENIOR DIVISIONS - GENERAL DEMANDS(LT 16 A-G are not listed)

H. NOUN: 1. singular* 2. plural* 3. collective 4. nominative case* 5. objective case**May not be used if noun used as adjective is the function demand.

I. PRONOUN: 1. singular 4. indefinite 7. relative 10. nominative2. plural 5. interrogative 8. intensive 11. objective3. personal 6. demonstrative 9. reflexive 12. possessive

J. VERB: 1. singular form 9. past participle 16. present perfect infinitive2. plural form 10. active voice 17. simple tense **3. linking 11. passive voice 18. perfect tense **4. regular 12. transitive 19. progressive form **5. irregular 13. intransitive 20. perfect progressive form **6. imperative mood 14. present infinitive 21. function for infinitive7. emphatic form* 15. perfect infinitive 22. function for gerund8. present participle

*The player may choose to designate present or past.**The player may choose to designate present, past or future

when tense or form is called (not as an additional demand)

FORBIDDEN: Demanding that the verb be in the conditional tense.K. ADJECTIVE: 1. positive degree of comparison

2. comparative degree of comparison*3. superlative degree of comparison*

*If these are demanded, the player may also indicate regular or irregular.L. ADVERB: 1. positive degree of comparison

2. comparative degree of comparison*3. superlative degree of comparison*

*If these are demanded, the player may also indicate regular or irregular.M. CLAUSES*: The solution word must be contained in the following clauses:

1. dependent (subordinate) 3. adverb 5. infinitive2. adjective 4. noun 6. elliptical (incomplete)*

*SEE Dictionary of Terms for definition of elliptical clause.N. PHRASES*: The solution word must be contained in the following phrases:

1. infinitive 3. participial 5. adjective 7. prepositional2. gerund 4. appositive 6. adverb

*NOTE ON LT 16 M & N: The number of times the two previous demands, M & N, known as the "Must Be that ademand may be made that the word be contained in either a clause or a phrase Contained In..." Demands can be made is limited to two times in this division. This maximum number represents a combination of both phrases and clauses. IT IS NOT two clauses and two phrases, BUT RATHER a total of two in any combination.

EXAMPLE: 2 clauses, 2 phrases, or 1 clause and 1 phrase.O. THE WORD MUST BE PART OF: 1.a direct quote (proper punctuation and capitalization required)*

2. an indirect quote *SEE Dictionary of Terms for the definition of direct quote.P. THE WORD MUST BE A COMPOUND WORD

SEE Dictionary of Terms on the difference between a compound preposition and a preposition which is compound. SEE ALSO Compound Word in the Dictionary of Terms.

Q. THE WORD TO BE FORMED MUST NOT BE CONTAINED IN:1. adjective clause 5. elliptical clause 9. gerund phrase2. adverb clause 6. direct quote 10. participial phrase 12. adjective phrase3. noun clause 7. indirect quote 11. appositive phrase 13. adverb phrase4. infinitive clause 8. infinitive phrase

NOTE: Dependent clause and prepositional phrase were intentionally omitted from this demand.NOTE ON LT 16 Q: The number of times this demand, known as the "Must NOT Be Contained In ...." and not be contained in one other. demand, can be used is limited to o n ce . Therefore, in combination with LT 16M & N in the Junior/Senior Divisions, it is possible to demand that a word be contained in two clauses or phrases.

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R. AFTER THE DEMAND HAS BEEN MADE THAT THE WORD MUST BE IN A CLAUSE OR PHRASE, THIS ADDITIONAL DEMAND CAN SPECIFY HOW THAT CLAUSE OR PHRASE IS TO FUNCTION IN THE SENTENCE.

This sheet should be printed on the back of the JR/SR Order of Play Sheet - Revised August 2017

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LINGUISHTIK SCORING CHART

CHALLENGER: The Player who makes the challenge.

SOLVER: A player other than the Challenger who presents a correct solution.

NEUTRAL: A player other than the Challenger [Challenge Now] OR a player other than the Challenger or Mover [Challenge Impossible] who does not present a solution.

WRONG: A player who presents an incorrect solution when there has been a Challenge, OR a player who either presents an incorrect solution or does not present a solution during a Forceout.

MOVER: A player who makes the last move before a Challenge Impossible.

AGREER: A player who agrees to a Forceout and presents a correct solution.

SITUATION 6 POINTS 4 POINTS 2 POINTSA. CHALLENGE NOW

Challenger has a correct solution CHALLENGER SOLVER NEUTRAL WRONG

B. CHALLENGE NOWChallenger DOES NOT have a correct

solution, but another player doesSOLVER

NEUTRALCHALLENGER

WRONG

C. CHALLENGE NOWNO PLAYER has a correct solution NEUTRAL

(SEE LT 25*)CHALLENGER

WRONG

D. CHALLENGE IMPOSSIBLENO PLAYER has a correct solution CHALLENGER NEUTRAL MOVER

WRONGE. CHALLENGE IMPOSSIBLEat least one player has a correct solution SOLVER

CHALLENGERWRONG

NEUTRALF. FORCEOUTALL PLAYERS agreed AGREER WRONG

*LT 25: If a player is four or more points ahead of any other player when the warning has been called, and theleading player CHALLENGES NOW, and NO PLAYER has a correct solution, ANY NEUTRAL PLAYER receives six 6 points.

Revised August 2017

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MLAG PROPAGANDA Tournament Rules 2017-18

PG1 The AGLOA publication, Propaganda: The Definitive Guide, will be used to determine all definitions.

PG2 Players will be seated in three-player groupings. A four-player grouping will be used only when a threesome cannot be formed

PG3 Six examples from each section will be read in each division, with twelve questions in two separate rounds.

PG4 A round ends when all examples have been read. The maximum score for a round is 48 points.

PG5 Each example will be read by a central reader. From the end of the second reading, players will have thirty (30) seconds to select an answer. Players may not answer before the end of the second reading. The penalty for failing to make a decision within the thirty seconds is minus two (-2) points, and that player WILL NOT be allowed to offer an answer for that example.

PG6 In the Junior and Senior Divisions only, some of the examples will be visual, taken from magazines or newspapers or other print material. On the visuals, some words will usually appear. The central reader will read only the words from the visual that should be considered in trying to determine the technique being used. Then, the words will be repeated. If there is no visual, the central reader will simply read the example twice. Also in the Junior and Senior Divisions, some examples, visual or oral, may contain more than one technique from the section being played. In these cases, the example will count for two answers which may be put in either order. In addition, a Junior/Senior example may be read (or shown) in two different rounds because it contains two techniques from different sections. Further, in the Junior and Senior Divisions, the Non Sequitur technique of Section E is expanded to include recognizing four forms of reasoning as listed and explained in the AGLOA publication, Propaganda: The Definitive Guide.

PG7 Players must mark a response to the example that is read by circling their answer and either the word BOLD or CAUTIOUS in the appropriate section on the answer sheet. Players may not change an answer or BOLD/CAUTIOUS once it has been marked. An erasure or scratch out of an answer makes the answer automatically wrong. An erasure or scratch out of BOLD/CAUTIOUS will be treated as not marking either one.

PG8 After all players have answered and revealed, reader states the “correct” answer and players check each other’s Answer Sheets to determine each player’s score. Answering and scoring on each example is as follows:

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CORRECT INCORRECTBOLD +4 -2

CAUTIOUS +2 0NOT CIRCLING BOLD

OR CAUTIOUS +2 -2

NO ANSWER -2

PG9 All divisions will play four (4) sections each year. The sections follow a rotation schedule to ensure all are covered:

2017-18: B-C-D-F2018-19: A-C-D-E2019-20: A-B-C-F2020-21: B-C-D-E

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MLAG PRESIDENTS Tournament Rules 2017-18

PZ1 The following version of PRESIDENTS is played at all levels.

PZ2 A Presidents chart (gazetteer) listing the number, picture, and name of each president plus date and place of birth, date and place of death, dates in office, and political party is the only reference permitted to be used by each player.

PZ3 Players play in groups of three or four for purposes of scorekeeping. Scores of all players in a group are kept on a Score Sheet at the table. Each player has an individual answer/wager sheet.

PZ4 Ten questions are read each round (eight for Minor Division). Each question consists of three clues — worth 6, 4 or 2 points — with each clue being increasingly revealing. Each question is read aloud by a central reader; for each round, the answers will fall into a specified range. In Minor, Elementary and Middle Divisions, the range will always be the complete 12

presidents for that round (1-12 and 13-24 this year; 25-34 and 35-45 in 2018-19). In Junior/Senior the reader will tell all players a range of about eleven Presidents (e.g., 7 to 17)

within which the correct president will be.

PZ5 Each clue consists of one or more sentences written in the first person, as if the president were stating the information (e.g., “During my term, I…”). Suggested guidelines for these statements are: 6-point clue: This statement should be relatively obscure, yet specific enough to limit the

answer to just one possible president. It should require intensive knowledge of American History and/or personal facts about the president.

4-point clue: This statement should give more information, perhaps including more history and/or personal facts. The Presidents chart often might be used to help narrow the range but not uniquely identify the President.

2-point clue: This statement should make the choice obvious. It should include something unique from the Presidents chart or something in the president's era that is obvious or a very well-known fact about the president.

For 2017-18, in Elementary and Middle Divisions, the 4-point or 2-point clues may contain something about First Ladies. In the Junior and Senior Divisions, the 4- or 2-point statement may contain something about First Ladies and/or Election Opponents and/or one of the special U.S. Leaders listed below (1-24 only):Group 1 – Clara Barton, Jefferson Davis, Frederick Douglas, Robert E. Lee, Lucretia Mott, Sitting Bull, Elizabeth Cady Stanton, Tecumseh, Sojourner Truth

PZ6 The reader will read the clue and, after a short pause, read it again. Players will then have approximately 30 seconds to mark an answer.

PZ7 As players try to determine the identity of a president, they should consider the ENTIRE set of clues rather than one isolated incident or fact.

PZ8 Using a non-erasable pen only, each player may circle one and only one answer per question. The answer may be marked only after the second reading of any one of the three clues. WHEN a player answers determines how many points that player receives if correct. Players may not change an answer once it has been marked. An erasure or scratch out makes the answer automatically wrong.

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PZ9 To answer, a player circles the number of the president on the answer side of the answer/wager sheet and either 6, 4, or 2 (depending on when the answer was written) on the wager side. The answer/wager sheet is then placed answer side DOWN on the appropriate 6, 4, or 2 space of a marked 8.5 x 11 mat. Other players confirm that a player's 6, 4, or 2 matches the location on the mat and corresponds to the clue that has just been read. Once a player places the answer/wager sheet on the mat, that player may NOT touch it or write on it again during that question. Once a player places the answer/wager sheet on the mat, that player may NOT touch it or write on it again during that question. Any answers that are erased, crossed out or changed will be judged incorrect, unless initialed by a judge, who must ensure the answer was changed before the correct answer was revealed.

PZ10 Thirty seconds after the 2-point clue has been read twice, the reader tells players to reveal their answers, then gives the president's name and number. Answers are checked and verified by the players at each table.

PZ11 Those players who have a correct answer win the number of points determined by WHEN they answered (6, 4, or 2). Those players who have an incorrect answer score 0 (zero) for that question. IF A PLAYER HAS FAILED TO CIRCLE THE 6, 4, OR 2, A CORRECT ANSWER SCORES TWO (2) POINTS ONLY.

PZ12 The ultimate winner in each Division is the team that has the most points after all questions have been played.

PZ13 Questions are not asked about any President first elected or sworn in during the current school year.

PZ14 MIDDLE and ELEMENTARY DIVISIONS:For the 2017-18 school year, these players are asked questions about Presidents 1-24 only.

Rotation of themes and U.S. Leaders in future yearsThemes:2017-18: First Ladies (all divisions), Election Opponents (Jr/Sr only)2018-19: Cabinet Members (all divisions), Vice Presidents (Jr/Sr only)2019-20: Scandals (all divisions), Domestic Affairs (Jr/Sr only)2020-21: Slogans (all divisions), Occupation and Posts (Jr/Sr only)2021-22: Presidential Quotes (all divisions), Foreign Affairs (Jr/Sr only)2022-23: Presidential Firsts (all divisions), Foreigh Affairs (Jr/Sr only)U.S. Leaders – Jr/Sr Only2017-18: Group 1 (Pres 1-24): Clara Barton, Jefferson Davis, Frederick Douglas, Robert E. Lee,

Lucretia Mott, Sitting Bull, Elizabeth Cady Stanton, Tecumseh, Sojourner Truth

2018-19: Group 4 (Pres 25-45): Cesar Chavez, John Foster Dulles, Martin Luther King, Jr., Henry Kissinger, Douglas MacArthur, George Marshall, Thurgood Marshall, Sandra Day O’Connor

2019-20: Group 3 (Pres 1-24): Henry Clay, Benjamin Franklin, William Lloyd Garrison, Alexander Hamilton, John Jay, John Marshall, Daniel Webster

2020-21: Group 2 (Pres 25-45): Susan B. Anthony, William Jennings Bryan, Clarence Darrow, Eugene V. Debs, W.E.B. Dubois, Huey Long, Gloria Steinem

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MLAG CURRENT EVENTS Tournament Rules 2017-18CE1 The following version of CURRENT EVENTS is played at all levels.

CE2 Players play in groups of three or four for purposes of scorekeeping. Scores of all players in a group are kept on a Score Sheet at the table. Each player has an individual answer/wager sheet.

CE3 A total of 25-30 questions are played as follows:Round 1 — Lightning Round 15-18 questions Round 2 — Wager Round 10-12 questions

Questions are multiple-choice with four alternatives marked A,B,C or D. Only one of the four alternatives is correct as determined by reliable resources. Participants must use a non-erasable ink pen in recording all answers.

CE4 Current Events Questions refer to the most recent year, January 1 through December 31 – 2017.

CE5 Reference books are NOT permitted at the table. Questions are taken from current atlases, almanacs, reliable Internet-based sources, published reviews of the year and other major publications as well as the Information Please website. Question topics include: Sports, Advances in Technology, Business, Entertainment, Disasters, Criminals and Punishment, Obituaries, Places in the News, Science, Space and Medicine and World Leaders.

CE6 During the Lightning Round, students will be asked questions with assigned values of 2,4, or 6 points.

CE7 During the Wager Round, a central reader announces a category. Each player begins the round with 0 points. Before each question is read aloud, each player writes a wager of 2, 4, or 6 on his/her wager/answer sheet based on the category the central reader announces. All wagers at a table are revealed simultaneously, then recorded on a common score sheet before the question is read.

CE8 For both rounds of CURRENT EVENTS, the central reader reads aloud the question and the four alternative answers. The reader may read the question and alternatives twice and only twice. At the end of the second reading, the question and choices are projected to the players via a central screen.

CE9 From the end of the second reading, each player has about 30 seconds to circle her/his answer on the wager/answer sheet.

CE10 If a player is unsure of an answer or wishes not to answer on a question, he/she may abstain from answering during the Wager Round. To abstain, a player must NOT circle an answer choice (A-B-C-D), but circle ABS on the wager/answer sheet before the correct answer is revealed. Each player may abstain no more than twice per round. If a player abstains on a question a third or more times, the player loses his/her wager for that question and receives the highest negative score (-3)

CE11 Scoring for the Lightning Round:Wagering is not permitted and there are NO abstentions during this round. The value of each questions (2,4 or 6 points) is announced by the central reader prior to

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the reading of the questions. If the player’s choice is correct, the assigned point value is awarded. If the player’s choice is incorrect, then no points are awarded. Players cannot lose points for incorrect answers in the Lightning Round

CE12 Scoring for the Wager Round:a) If a player's answer agrees with the reader's, that player wins his/her wager (6,4, or 2)b) If a player's answer disagrees with the reader's, he loses HALF his wager. (-3,-

2 or -1)c) If a player abstains, the player neither gains nor loses points, provided he has

not exceeded the abstention limit of two (see CE10).d) If a player is not at the table to answer a question, the player scores -4

for that question.

CE13 Play proceeds until all questions have been dealt with in a round. The ultimate winner in a Division is determined by the total number of points in both CURRENT EVENTS rounds.

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MLAG THEME Tournament Rules 2017-18TH1 The following version of THEME is played at all levels.

TH2 Players play in groups of three or four for purposes of scorekeeping. Scores of all players in a group are kept on a Score Sheet at the table. Each player has an individual answer/wager sheet.

TH3 A total of 25-30 questions are played as follows:Round 1 — Lightning Round 15-18 questions Round 2 — Wager Round 10-12 questions

Questions are multiple-choice with four alternatives marked A,B,C or D. Only one of the four alternatives is correct as determined by reliable resources. Participants must use a non-erasable ink pen in recording all answers.

TH4 Reference books are NOT permitted at the table. Questions are taken from reference books, i.e. historical reference texts on the chosen topic, reliable Internet-based sources, and other basic text references.

TH5 The Theme for 2017-18 is Greek and Roman Mythology. See TH14 for full listing of subtopics and scope.

TH6 During the Lightning Round, students will be asked questions with assigned values of 2,4, or 6 points.

TH7 During the Wager Round, a central reader announces a category before reading the question. Each player begins the round with 0 points. Before each question is read aloud, each player writes a wager of 2, 4, or 6 on his/her wager/answer sheet based on the category the central reader announces. All wagers at a table are revealed simultaneously, then recorded on a common score sheet before the question is read.

TH8 For both rounds of THEME, the central reader reads aloud the question and the four alternative answers. The reader may read the question and alternatives twice and only twice. At the end of the second reading, the question and choices are projected to the players via a central screen.

TH9 From the end of the second reading, each player has about 30 seconds to circle her/his answer on the wager/answer sheet.

TH10 If a player is unsure of an answer or wishes not to answer on a question, he/she may abstain from answering during the Wager Round. To abstain, a player must NOT circle an answer choice (A-B-C-D), but circle ABS on the wager/answer sheet before the correct answer is revealed. Each player may abstain no more than twice per round. If a player abstains on a question a third or more times, the player loses his/her wager for that question and receives the highest negative score (-3)

TH11 Scoring for the Lightning Round:Wagering is not permitted and there are NO abstentions during this round. The value of each questions (2,4 or 6 points) is announced by the central reader prior to the reading of the questions. If the player’s choice is correct, the assigned point value is awarded. If the player’s choice is incorrect, then no points are awarded.

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Players cannot lose points for incorrect answers in the Lightning Round

TH12 Scoring for the Wager Round:a) If a player's answer agrees with the reader's, that player wins his/her wager (6,4, or 2)b) If a player's answer disagrees with the reader's, he loses HALF his wager. (-3,-

2 or -1)c) If a player abstains, the player neither gains nor loses points, provided he has

not exceeded the abstention limit of two (see CE10).d) If a player is not at the table to answer a question, the player scores -4

for that question.

TH13 Play proceeds until all questions have been dealt with in a round. The ultimate winner in a Division is determined by the total number of points in both THEME rounds.

TH14 The Theme Questions for 2017-18 refer to the following: Greek and Roman Mythology

I. Major Greek Gods and Goddesses of Mount Olympus

Parentage/origination stories, powers/expertise/symbol, Romans names, relationships with other gods, children

Greek God or Goddess Roman NameZeus Jupiter/JovePoseidon NeptuneHades PlutoAphrodite VenusApollo ApolloAres MarsArtemis DianaAthena MinervaDemeter CeresDionysus BacchusHera JunoHermes MercuryHephaestus VulcanHestia VestaPersephone Proserpina

II. Greek HeroesParentage/origination stories, powers/expertise/symbol, Romans names, relationships with other gods, children

Greek Hero Mythological Creature or EventAchilles Trojan WarAtalanta Calydonian BoarBellerophon PegasusCadmus DragonDaedalus Labyrinth, Flight of Icarus

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Heracles 12 LaborsJason Golden FleeceOdysseus OdysseyOrpheus LyreParis Trojan WarPerseus MedusaTheseus Minotaur

III. Categories of the Game

Questions will come from the parentage/origination stories and other events identified above concerning these 15 Gods/Goddesses and 12 Heroes. Categories for questions in both the lightning and wagering rounds will be as follows:

Category DescriptionImmortality Suits Me Questions will seek the identity of a god or

goddessI Can be Your Hero Questions will seek the identity of a heroYou A-Muse Me Questions will ask who was inspired,

encouraged or guided by whomCreatures, Monsters & Beasts Questions will seek the identity of a

creature, monster, or beastIt’s Just a Myth Questions will concern the lives &

adventures of gods, goddesses & heroesWhere in the World am I? Questions will concern famous places in

Greek mythology or modern-day locations named for characters or places in Greek mythology.

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