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©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com

Rubik’s Cube Unit Study The Geometry, The Art, The Solution

Student Workbook

Written by Bryan Kinkel

©1974 Rubik’s® Used under license Rubik’s Brand Ltd. All rights reserved. www.youcandothecube.com 3

You CAN Do the Rubik’s Cube: Student Workbook Table of Contents

Activity Pages The Man, The Cube, Its Impact

Notes from Student Presentations worksheet 4

Classifying Polygons Classifying Polygons worksheet 5-‐6

Classifying 3-‐dimensional Shapes The Third Dimension worksheet 7 Classifying Polyhedrons worksheet 8 Writing Rules worksheet 9-‐10

Platonic Solids

Testing Net Variations 11 Testing Net Variations worksheet

Square Cut-outs sheet 13

Mix & Map & Match Mix & Map & Match worksheet 12

Solving the Rubik’s Cube Introductory Information 15-‐16 Solution Guide 17-‐25 What’s Next? 26

How To Videos How To Videos worksheet 27

Intro to Speed Solving Intro to Speed Solving worksheet 28 Writing Inverse Algorithms worksheet 29 Writing Mirrored Algorithms worksheet 30 Exploring Advanced Speed Solving Algorithms 31

Creating Rubik’s Art Practice & Pixelate worksheet 32 Building a Mini-‐mosaic worksheet 33 Designing a Rubik’s Mosaic using Gimp worksheet 34 Designing a Rubik’s Mosaic using Twist the Web worksheet 35

Making a 2x2 Rubik’s Cube Making a 2x2 Rubik’s Cube worksheet 36 Cut-‐outs for a 2x2 sheet 37 Making a 2x2 Solution Guide worksheet 39


4

Notes from Student Presentations

Write down three interesting facts learned from each of the different presentations. Who is Ernö Rubik? 1)

2)

3)

How did the Rubik’s® Cube come to be? 1)

2)

3)

What impact has the Rubik’s Cube had over the years? 1)

2)

3)

What else? (fun & random facts about the Rubik’s Cube) 1)

2)

3)

What do you think was the greatest influence in young Ernö’s life that lead him to develop the Rubik’s Cube?

What feature(s) made the Rubik’s Cube unique?

Why do you think the Rubik’s Cube became such a cultural phenomenon?

What impresses you the most about Ernö Rubik and his cube?


Classifying Polygons

Find the following definitions.

Two Dimensional –

Line Segment –

Polygon –

Congruent –

Regular polygon –

Irregular polygon –

Convex polygon –

Concave polygon –

Give the prefixes for the following numbers. Ex: Decade means 10 years, and century means 100 years.

3 =

7 =

11 =

4 =

8 =

12 =

5 =

9 =

20 =

6 =

10 = Deca

100 = Centi

What does the suffix –gon mean?


6

Fill in the blanks using the shapes above. shape # concave/convex regular/irregular # of sides name

ex) convex irregular 3 triangle

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

5 ex 1

4

3

2

8 7 10

9 6


The Third Dimension

Define the following words.

Space figure –

Polyhedron –

Parts of a Polyhedron: Face –

Edge –

Vertex –

Special Polyhedrons

Prism -‐ Platonic Solid -‐ Pyramid -‐ Others:

Concave, Truncated, etc...

Name the shape that is both Name the shape that is both a prism and a platonic solid: a pyramid and a platonic solid:


8

Classifying Polyhedrons Visit each of the shape stations. At each station, attem

pt to correctly nam

e each space figure (using two w

ords). Then docum

ent the number of faces, edges, and vertices of each shape.

Leave the last column blank. It w

ill be used later in the lesson.


Writing Rules

Prisms 1) Count the faces, edges, and vertices of the following prisms.

Name n faces vertices edges triangular prism 3 rectangular prism 4 pentagonal prism 5

*n represents the number of sides ofone of the bases

2) Find the patterns in the table above and use it to make a prediction.

Name n faces vertices edges hexagonal prism 6

3) Draw a hexagonal prism and check your prediction.

4) Write function rules that describe the shortcuts in finding the number of faces, edges, andvertices of a prism. (Vertices is already completed as an example)

Faces: F =

Edges: E =

Vertices: V = 2n or V = 2·∙n

Pyramids 1) Count the faces, edges, and vertices of the following pyramids.

Name n faces vertices edges triangular pyramid 3 rectangular pyramid 4 pentagonal pyramid 5

*n represents the number of sides ofthe base


10

2) Find the patterns in the previous table and use it to make a prediction.

Name n faces vertices edges hexagonal pyramid 6

3) Draw a hexagonal pyramid and check your prediction.

4) Write function rules that describe the shortcuts in finding the number of faces, edges, andvertices of a prism.

Faces: F =

Edges: E =

Vertices: V =

Platonic Solids

What shortcuts could be used to count the faces, edges, and vertices of these regular polyhedrons? Write a description and try it.

Euler’s Formula

Go back to the Classifying Polyhedrons worksheet. Label the blank column “F + V”, and in each row add the number of faces to the number of vertices and record that in the new column. When you get done with that, compare column “F + V” with column “E”, and write down a function rule describing the pattern.


Testing Net Variations

Polyhedrons may have more than one net. Your goal is to compile a list of unique nets that, when folded, make a regular hexahedron (cube). A net of a cube must consist of six congruent squares. Cut out the six squares on the next page, and use them along with a Rubik’s® Cube to create and test different possible net variations.

Here are a couple of examples of how to do this: 1) Make a formation with the squares.2) Set a Rubik’s Cube on one of the squares.3) As you flip the Rubik’s Cube onto the other squares,

document which colors contact each square.4) If a formation ends with each color being used

exactly once, then it is a net.

Example 1 Example 2

How many different net variations can you find?

*Note: The following nets are examples of what would be considered the same net. Nets arecongruent if one net can be made by rotating and/or reflecting the other.

Green Red

Y

B

W

R G O

Y

W

R G

O G

Green

White

This one is a net of a cube because each color has been used exactly once.

This one is NOT a net of a cube because each color was not used exactly once. Green showed up twice (Blue zero times).


12

Mix & Map & Match Part 1:

1) Scramble your Rubik’s® Cube.

2) Using crayons, markers or colored pencils, map the scrambled Rubik’s Cube onto a templatednet.

3) Hand in your net and Rubik’s Cube. The nets will be shuffled and displayed, and the Rubik’sCubes will be randomly given back out.

4) Find the net that corresponds to your new Rubik’s Cube.

5) Once you have matched the net and cube, verify the net’s accuracy (use rolling method).

6) Then, using a different templated net, map the Rubik’s Cube again.

7) Compare the two nets. What is similar about them? What is different about them?

Part 2: 8) Find a partner. Without your partner seeing, make exactly one 90° turn on one of the faces of

your Rubik’s Cube (keep track of what face you turned by looking at the color on the middlesquare). Your partner will do the same.

9) Trade puzzles with your partner. Using their nets, try to figure out which face was turned.*Do your best to make sure that your Rubik’s Cube gets realigned to match its net when done.

Part 3: 10) The teacher is going to collect all of the nets (from parts 1 & 2), shuffle them, label them

1,2,3,…, and then display them for the class to view.

11) Walk around and examine the different nets. Try to find the pairs of nets that display the samemixed up Rubik’s Cube. Make a list of your matches by writing down their numbers.

12) Compare your list with classmates to determine the answer key.

Part 4: 13) Find and take the pair of nets that matches the scrambled Rubik’s Cube you have. Then find a

partner again.

14) Put your Rubik’s Cube and one of your nets out of view of your partner.

15) Cut out the six squares of the other net, shuffle them up, and then hand the cut-‐up net to yourpartner. They will do the same.

16) See if you can figure out how to configure the six squares so that they will make a net thatmatches your partner’s mixed up cube.

17) When done, ask for their cube, and use it to see if you were successful (use rolling method).


Squares cut-‐outs sheet for testing net variations


Meeting the Cube

In a previous lesson, we discovered that a cube has 6 faces, 12 edges, and 8 corners (vertices). Take a look at this disassembled Rubik’s® Cube and notice those same numbers. The 6 faces are all connected together as one piece called the core. Since they are connected, it means that the alignment of the face colors cannot be changed. The white face will always be opposite of the yellow face, red opposite orange, and green opposite blue.

There are 12 pieces in the middle group. These are our edge pieces. Notice how each of the edge pieces has two colored tiles on it. That is because an edge of a polyhedron connects two faces. Therefore these 12 edge pieces will always be set between two face components of the core.

On the left side of the picture, we find our corner pieces. Notice how there are 8 of them, and that each of them has 3 colored tiles. This is because a cube has 8 corners (vertices), and each of a cube’s corners connects 3 edges of 3 faces.

Now, when you look at an assembled cube, you may start to see how it is held together. Focus on a piece that has 2 tiles on it (an edge). It is between exactly 2 centers (face pieces). Then focus on a piece that has 3 tiles on it (a corner). It is held in place by 3 edge pieces, and is diagonal from 3 centers.


16

Labeling the Cube

In the lessons to follow on solving the Rubik’s® Cube, we will need to be able to read algorithms. An algorithm is a set of steps one should follow to complete a task. Before we get introduced to these algorithms, we need to label the Rubik’s Cube by the ways that we can turn the different faces.

We are going to label the six faces according to the way that we are holding the cube. As we hold the puzzle, the side that is facing us will be referred to as the Front Face. Then from there, the side to our right will be identified as the Right Face; and to our left, the Left Face. The top and bottom of the cube will be labeled the Up Face and Down Face. The face hidden from our view, opposite the Front Face will be described as the Back Face. The algorithms will use abbreviations to save space by using F, R, L, U, D, and B in place of Front, Right, Left, Up, Down, and Back.

The next thing that we need to identify is which way we want to rotate a particular face. The algorithms we will learn utilize apostrophes (‘) to indicate a 90° counterclockwise turn. If a step in an algorithm does not have an apostrophe, then we are to make a 90° clockwise turn. These clockwise and counterclockwise turns are based on viewing that particular face (imagine a clock mounted on that face). In the diagram below we can see the difference between a clockwise rotation of the Right Face compared to a clockwise rotation of the Left Face.

Lastly, we may see a 2 in an algorithm. If we do, it will be paired with one of the letters F, R, L, U, D, or B. This will stand for turning that particular face twice (180°). Since a 180° turn of a face clockwise or counterclockwise will end in the same state, you can turn it whichever way you prefer.

Here is an example of reading an algorithm: F U2 F’ D’ R

If we were to follow this algorithm, we would turn the front face 90° clockwise, then turn the upper face 180° clockwise (or counter), then turn the front face 90° counterclockwise, then turn the down face (bottom) 90° counterclockwise, and finally turn the right face 90° clockwise.

L U

R

D

F B

R F


Layer One – Making a Cross (Plus Sign) The white layer does not need to be solved first, but I will recommend that you start with that face until you develop some consistency with solving the Rubik’s® Cube. To make the cross, first locate the white center and hold the cube so that it is the upper face. Then you need to locate the 4 edge pieces that have a white sticker/tile on them (you are not concerned about the corners with white yet). These 4 white stickers/tiles will be in one of four places: Already on top! Along the equator Along a longitude On the bottom (down face)

*Position O is where we are going to move the white tile to, but before we do that, we want to make sure that we aren’t replacing it with a white tile already on top. 1) If there is a white tile in position O, turn U (upper face) until a nonwhite tile is in that position. If it is already nonwhite, skip to step 2. 2) Turn L’ (left face counterclockwise)

1) If there is a white tile in position O, turn U (upper face) until a nonwhite tile is in that position. If it is already nonwhite, skip to step 2. 2) Turn R (right face clockwise)

Along the equator

Hold the cube so that the white center is on top, and a white tile is on the front face (side facing you).

If the white tile is on the left:

If the white tile is on the right:


Layer One – Making a Cross (Plus Sign) continued

Along a longitude

Hold the cube so that the white center is on top, and a white tile is on the front face.

If the white tile is on the top:

If the white tile is on the bottom:

1) In this particular case there will never be a white tile in position O. 2) Turn F U’ R

1) If there is a white tile in position O, turn U until a nonwhite tile is in that position. 2) Turn F’ U’ R

On the bottom

Hold the cube so that the white center is on top, and the white tile that is on the bottom (down face) is in the position nearest you.

1) If there is a white tile in position O, turn U until a nonwhite tile is in that position. 2) Turn F2 (front face 180°)


Layer One – Permuting the Cross

We have made the cross! Now we need to permute the cross. This means that we need our cross to match up with the core. In other words, the four edge pieces that our making up our white cross need to also match the center tiles on the front, right, left, and back sides (assuming white is still on top).

When examining how well our cross is permuted (seeing if any of the four edge pieces match on the sides), we will notice one of four cases: none of them match, one of them matches, two of them match, or all four of them match. The following algorithms require that exactly two of the edges are matching (if all four are matching, we get to skip this step). The good news is that by simply by turning the upper face, we can always obtain two or four matching edges.

1) Get two or four matching edges.a. If no edges are matching, turn U, or U’, or U2. One of those three moves will result in two or fouredges matching.

b. If one edge is matching, turn U, or U’, or U2. One of those three moves will result in two or fouredges matching.

c. If two edges are matching, skip to step 2.d. If four edges are matching, skip to Solving the Corners.

2) Once we have our two matching edges, we need to swap (permute) the other two. We will do thisone of two ways:

Permute – to change the order of; rearrange

If the two edges are adjacent: If the two edges are opposite:

R U’ R’ U R R U2 R’ U2 R


20

Layer One – Solving the Corners

Now that our cross (on the upper face) is matching on all four sides, we need to solve the four white corners. Each piece of this puzzle has a specific spot it needs to go, so we need to be careful which white tile we put in each corner. Take a look at the two diagrams:

In both cases a white tile is on top, however there is only one piece that truly goes in that spot. Since we are trying to put a corner piece in that connects the white, red, and blue faces, the corner piece needs to comprise of those same colors. The following algorithm that we will use affects many pieces, however it will swap the following two corners (without scrambling our cross):

1) Find a corner piece in layer three that contains a white tile. Turn D (down face) until the corner pieceis directly below the spot it belongs.

2) Turn R’ D’ R D. This willput the corner piece in the rightspot, but may not put white ontop. If not, do the same algorithmtwo or four for times until it is inthe right spot and white is on top.

3) Repeat these steps for the other three corners containing a white tile.

*If piece containing a white tile is in layer one and not in the right spot, R’ D’ R D will bring thecorner piece to layer three. From there, go back to step 1.

Good Bad


Layer Two – Solving the Edges Layer two comprises of four center pieces and four edge pieces. Our goal is to find the four correct edge pieces and get them in their correct spots (the correct way). 1) Flip the cube upside-‐down (making D the white face). 2) Look for one of the four edges in the (new) top layer. Turn U until that piece matches up with the center (core) on the side. 3) Determine if the piece will need to go to the left, or to the right when going into layer two. 4) Pick the needed case, and follow the corresponding algorithm. If the edge needs to go right: If the edge need to go left: U R U’ R’ U’ F’ U F U’ L’ U L U F U’ F’ 5) Repeat these steps for the three other edges. *If a needed edge piece is already in layer two (but not solved), you will need to do one of the two algorithms to remove it. Then go back and follow the above steps.

*In this particular diagram, the edge piece would need to go to the right because the piece consists of red and green.

U’ L’ U L U F U’ F’


Layer 3 – Making a Cross Layer 3, or Last Layer, is actually referring to the top layer with the method we are using (since we flipped the cube over after working on layer 1). When solving this layer, you may notice a similar theme (making a cross, permuting the cross, solving the corners) even though we will see some new algorithms. To make a yellow cross (without scrambling layers 1 or 2) we will need only one algorithm. Looking at the yellow face (ignore corners), we should see one of four cases: Case 1 Case 2 Case 3 Case 4 No yellow edges Two adjacent Two yellow edges Completed showing yellow edges opposite each other Cross! Determine which case you are starting at and skip to that step. *The color of the front face is irrelevant. What matters is the placement of the yellow tiles. 1) Turn F R U R’ U’ F’. This will bring you to case 2, two adjacent yellows. However, we will to either: spin the cube 180°, or turn U2 to be set up for case 2. 2) Turn F R U R’ U’ F’. 3) Turn F R U R’ U’ F’. 4) Done!


Layer 3 – Permuting the Cross

Now that we have the yellow cross, we need to match it to the core (just like with layer 1). So, similar to before, we are going to turn U until either two or four edges are matching. Below are the three casesthat you will encounter and what you should do.

1) Turn U until either two or four edges are matching. Then go to the corresponding case.

Case 1: The two matching edges are adjacent. Use the algorithm to the right.

Case 2: The two matching edges are opposite each other. You will essentially do the same algorithm as case 1, but twice and with one step in the middle.

Case 3: All four edges are matching; done!

R U R’ U R U2 R’ U

R U R’ U R U2 R’ U

R U R’ U R U2 R’ U U2


24

Layer 3 – Permuting the Corners

Just like with layer one, we are now going to solve the corners. However, since we don’t want to scramble layers 1 or 2, we are going to break up solving the corners into two parts: first we will permute the corners (get them in their right spots), and then we will orient the corners (turn them so the yellow tiles are facing up). So for now, we don’t care if the corner pieces are yellow tile up. We want to see how many of the corner pieces are in their right spots. When we go to check, we will either see: none of the corner pieces in their right spots, one in the right spot, or all four in their right spots. This will lead us to one of four cases.

1) Determine which of the following cases your cube is in, and then follow the algorithm.

Case 1: One corner piece is in the right spot. Spin the entire cube in your hand until that corner piece on the front face in the upper-‐right position. Then use the algorithm to the right. *Depending on how your cube is scrambled,you may need to do the algorithm twice.

The algorithm that we are using swaps the positions of the three corners that are out of place.

Cases 2 & 3: None of the corner pieces are in the right spots. We are going to utilize the same algorithm as in case 1. Your cube should match one of the following:

Case 4: All of the corner pieces are in their right spots. Done!

U R U’ L’ U R’ U’ L

U R U’ L’ U R’ U’ L

U R U’ L’ U R’ U’ L U R U’ L’ U R’ U’ L

U R U’ L’ U R’ U’ L spin the cube so that the Left Face becomes your Front Face.

spin the cube so that the Right Face becomes your Front Face.


Layer 3 – Orienting the Corners

All of the corners pieces are in the right spots, but now we might need to turn some of them. If none of them need to be oriented (turned), then we are done! Otherwise, we will need to orient two, three, or all four of the corners.

We are going to refer to the upper-‐right position of the Front Face as the fixing spot.

1) Pick a Front Face that already has ascrambled corner in the fixing spot. Usethe given algorithm either two or fourtimes (until the particular corner is fixed).*Layers 1 & 2 will appear to bescrambled, but if you continue to followthe steps, layers 1 & 2 will fix themselves.

2) Turn U until the next scrambledcorner is in the fixing spot. *Do notchange your Front Face.

3) Repeat steps 1 & 2 until all four corners are fixed.

Congratulations!

Orient – to relate something to its surroundings; align; turn

R’ D’ R D


“I Can Solve the Rubik’s® Cube! What’s Next?” 1) How fast can you solve the Rubik’s® Cube? Grab a stop watch and find out. You could also go to http://www.cubetimer.com. This site is nice for testing your speed. Besides a timer, this site also gives scrambling instructions and keeps track of your times/attempts. 2) Can you solve the Rubik’s Cube from memory? If not, while practicing, try to notice (focus on) patterns in what you are doing rather than the letters of the algorithm. This could make it easier to remember the turns that need to be made. 3) Can you “unsolve” the Rubik’s Cube? See if you can solve the Rubik’s Cube in a way that you replicate some of the designs below. All of these designs can be replicated without learning any new algorithms. Front Back Front Back


How To Videos

Partner’s name:

Stage assigned:

□ Meeting the Cube

□ Reading Algorithms

□ Layer 1 – Making a Cross (Plus Sign)

□ Layer 1 – Permuting the Cross

□ Layer 1 – Solving the Corners

□ Solving Layer 2

□ Layer 3 – Making a Cross

□ Layer 3 – Permuting the Cross

□ Layer 3 – Permuting the Corners

□ Layer 3 – Orienting the Corners

What algorithms/information is needed for this stage?

How many Rubik’s® Cubes are needed for video, and how should they be set up?

What things need to be said, and who is going to say each part?

Is every case covered? In other words, when the viewer gets to this stage, will this video walk them through the stage no matter what their starting point?


Intro to Speed Solving Part 1: The algorithms that we use have cycles. This means that if you do the same algorithm over and over again, the puzzle will eventually go back to the state it started in prior to the repeated steps. Examples: Layer 3 – Permute the Cross (U R U R’ U R U2 R’ U) has a cycle of 3.

Layer 1 – Orient the Cross (R U’ R’ U R) has a cycle of 12.

Layer 2 algorithm U R U’ R’ U’ F’ U F has a cycle of 15.

Starting with a solved Rubik’s Cube, test examples 1 & 2 by repeating each given algorithm the specified number of times. When you are done, the puzzle should be returned to the solved state. Part 2: Determine the cycle values of the following algorithms:

1) Layer 1 – Solving the Corners (R’ D’ R D) has a cycle of ______.

2) Layer 3 – Orienting the Cross (F R U R’ U’ F’) has a cycle of ______.

3) Layer 3 – Permute the Corners (U R U’ L’ U R’ U’ L) has a cycle of ______. Part 3: Make up your own algorithm consisting of 3-‐5 turns. The algorithm can be completely random; it does not need to contribute to solving the cube. Once you have made your algorithm, test it to determine its cycle value.


Writing Inverse Algorithms Inverse operations can be seen in writing “return” directions off of a map, or solving an algebraic equation (as seen below). Inverse operations have us “undo” everything that has been done. In other words, inverse operations make us do the opposite of each step AND in the reverse order. Here is an example (this is not one of the learned algorithms): Original algorithm: L, U’, R’, U Inverse algorithm: U’, R, U, L’

1) Layer 1/3 – Solving the Corners: R’ D’ R D Write the inverse algorithm:

2) Layer 3 – Permute the Cross: U R U R’ U R U2 R’ U Write the inverse algorithm:

3) Layer 3 – Permute the Corners: U R U’ L’ U R’ U’ L Write the inverse algorithm:

4) When done, compare answers with a neighbor. Then test out your new algorithms.

5) When will these algorithms be beneficial?

Looking at the equation 5 = #$%&'

Following the order of operations Following inverse operations Start at x End at what x equals 1) multiply by 3 3) divide by 3 2) add 2 2) subtract 2 3) divide by 4 1) multiply by 4 Get an answer of 5 Start at the answer 5


30

Writing Mirrored Algorithms

Mirrored (or reflective) operations can be seen by observing movements in a mirror. Let’s compile some observations by simulating mirrors. Find a partner and stand facing each other. Indicate who will be the “model” and who will be the mirror. The model will perform a couple of movements, and the partner will act as if they were the reflection in the mirror.

Movements by model Movements by mirror Wave with left hand What hand is Mirror waving with? Pat your head w/right hand Is Mirror patting head? With what hand? Put hands behind your back Where are Mirror’s hands? Slowly rotate right arm clockwise Which direction is Mirror’s arm rotating? Turn body right, rotating 90° clockwise Which way did Mirror rotate? Lower your chin, then raise it When did Mirror’s chin go down? Up?

What we hopefully noticed, was that with mirrored operations, up is still up, down is still down, front is still front, and back is still back. We may have also noticed that right becomes left, left becomes right, clockwise becomes counterclockwise, and counterclockwise becomes clockwise.

Here is an example: Original algorithm: L, U’, R’, U

Mirrored algorithm: R’, U, L, U’

1) Layer 1/3 – Solving the Corners: R’ D’ R DWrite the mirrored algorithm:

2) Layer 3 – Permute the Cross: U R U R’ U R U2 R’ UWrite the mirrored algorithm:

3) Layer 3 – Permute the Corners: U R U’ L’ U R’ U’ LWrite the mirrored algorithm:

4) When done, compare answers with a neighbor. Then test out your new algorithms.

5) When will these algorithms be beneficial?


Exploring Advanced Speed Solving Algorithms

1) What does acronym OLL stand for? What does OLL mean? (search the internet)

2) What does acronym PLL stand for? What does PLL mean? (search the internet) In order to use them, we will need more abbreviated notations. Here are the notations we have used and some of the new letters/sets that may come up in more complex algorithms:

• F (front) – the side facing toward, as viewed by the solver • B (back) – the side that is opposite the front, as viewed by the solver • L (left) – the side to the left of the front, as viewed by the solver • R (right) – the side to the right of the front, as viewed by the solver • U (up) -‐ the side on top, as viewed by the solver • D (down) – the side on bottom, as viewed by the solver

*The six letters above assume a 90° clockwise rotation. • 2 (two) – turn the given face twice • ‘ (apostrophe) – turn counterclockwise • f (front two faces) • b (back two faces) • l (left two faces) • r (right two faces) • u (upper two faces) • d (downward two faces) • x (rotate entire cube) – D will become F • y (rotate entire cube) – R will become F • z (rotate entire cube) – U will become R

3) Solve layers 1 and 2 of a scrambled Rubik’s Cube, but not the last layer. Then go to one of the following sites (top preferred):

• http://www.cubezone.be/oll.html • https://ruwix.com/the-‐rubiks-‐cube/rubiks-‐cube-‐solution-‐with-‐advanced-‐

friedrich-‐method-‐tutorial/orient-‐the-‐last-‐layer-‐oll/ • https://www.speedsolving.com/wiki/index.php/OLL

Match up your mixed-‐up third layer to the same scenario from the site’s list. Then follow the provided algorithm to orient the last layer.

4) When successful in part 3, go to one of the following sites (top preferred): • http://www.cubezone.be/pll.html • https://ruwix.com/the-‐rubiks-‐cube/rubiks-‐cube-‐solution-‐with-‐advanced-‐

friedrich-‐method-‐tutorial/permutate-‐the-‐last-‐layer-‐pll/ • https://www.speedsolving.com/wiki/index.php/PLL

Match your puzzle to the same scenario on the site and follow the corresponding algorithm.

x

y

z


32

Practice & Pixelate

1) Try to make one of the faces of your Rubik’s® Cube match each of the following configurations.(It is not possible to have them all completed at the same time.)

a b c

2) Describe how you went about the process of making these configurations. What techniques,strategies, or algorithms did you use to complete your tasks?

3) Design a template (or blueprint) for a mini-‐mosaic using only white, yellow, green, blue, red, andorange to design a picture or pattern. Each individual square may consist of only one color.


Building a Mini-‐mosaic

Take your picture/design from the previous page and translate it onto these templates that represent faces of 9 different Rubik’s Cubes.

Cut out your 9 mini-‐templates. Hand a mini-‐template and a Rubik’s Cube to eight different students. The nine of you will each solve a particular portion of the mosaic. Then collect all nine cubes back up and place them together to display your mini-‐mosaic. Take a picture, and then let the next student distribute the Rubik’s Cubes and their mini-‐templates.


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Designing a Rubik’s Mosaic using Gimp

You are going to learn how to turn a picture into a Rubik’s Cube mosaic using the program GIMP 2 to do photo editing. The steps below will have two components: objective (what you want to happen) and task (how you are going to do it).

1) Objective: Get a picture to turn into a mosaic.Task: Take a picture, or find one online, and download it to the desktop of your computer.

2) Objective: Get the picture into the GIMP 2 program.Task: Open the program GIMP 2 and upload the picture by going to FILE → OPEN. Then selectthe picture from the desktop.

3) Objective: Cut out the desired portion of the picture selected.Task: Crop the image by using the rectangle select tool, found on the left panel, to outline thedesired portion of the picture. Then click IMAGE → CROP TO SELECT.

4) Objective: Reconfigure how many pixels are in the picture to align with the number of Rubik’sCubes there are to use for the mosaic.Task: Reformat the resolution by first going to IMAGE → SCALE IMAGE. With a calculator,recalculate a new Width by using the original width shown, the original height shown, and thenumber of Rubik’s Cubes available.

𝑛𝑒𝑤 𝑤𝑖𝑑𝑡ℎ = 9 ∙ 𝑛 ∙ 𝑤

ℎType the new width into the width textbox and press enter. *The height should auto correct when typing in the new width. Then click scale.

5) Objective: Get a better look at your picture by zooming in.Task: Zoom in by using the zoom tool found on the left panel.

6) Objective: You are only allowed to use 6 colors, so you are going to reduce the number of colorsused in the picture.Task: Change the # of colors by opening IMAGE → MODE → INDEXED, replacing 256 with 6, andthen clicking convert.

7) Objective: You need to specify which 6 colors are to be used (red, orange, yellow, green, blue,white).Task: Change the 6 current colors to that of the Rubik’s Cube by first opening WINDOWS →DOCKABLE DIALOGS → COLORMAP. To replace each color, click on one at a time and replacethe HTML notation with one of the following:

Red: f60000 Orange: ff9c00 Yellow: ffff00

Green: 00d900 Blue: 3737b3 White: ffffff

*Determine which colors will be switched with which before typing in the HTMLs.

8) Optional: transfer image to Microsoft Paint for more detailed editing. To do this, click onSELECT → ALL, then click ctrl + c. Open up Paint and paste in the image by clicking ctrl + v.


Designing a Rubik’s Mosaic using Twist the Web

Using Gimp 2, we were able to walk through how a picture is transformed into a template for a Rubik’s Cube mosaic. Now we will use a web-‐based program that might be seen as a quicker method.

1) Take or find a picture, and download it onto your computer’s desktop.

2) Open the picture in Microsoft Paint to crop it. Leave only the rectangular portion of the picturedesired for the mosaic. *Make sure to eliminate any white border around your picture, if thereis any.

3) Resave it to your desktop.

4) In Google Chrome, go to www.mosaic.twisttheweb.com.

5) Drag your picture from the desktop into the dotted box on the website.

6) Enter the number of Rubik’s Cubes that you have available for your mosaic. Then click on theright arrow

7) Optional: On this next page, you can alter the threshold of the colors by moving the slider. Tomove past this, click the right arrow.

8) On the “Assembly Guide” page, I recommend selecting Blocks of 3x3 cubes. Even though thiswill use more paper, it is better for Mosaic Builders who are less experienced. Then click theright arrow.

9) Finally, print or save the mosaic template.


Making a 2x2 Rubik’s® Cube You are going to make a 2x2 Rubik’s Cube.

1) Cut out 24 squares (found on the Cut-‐Outs page). Color them: 4 yellow, 4 blue, 4 orange, 4 red, and 4 green (leaving 4 white). The squares should measure about 1” x 1”.

2) Tape the squares onto the corner pieces of the corresponding sides.

*Do not do any taping on the edge pieces.

3) You now have a 2x2 Rubik’s Cube. Turn it slowly to begin, as some pieces of paper may catch. If so, curl those edges upward.

4) Mix it up and then solve it. Have some tape nearby just in case some of your squares come

loose.

5) What are the similarities and differences between solving the 3x3 and the 2x2? Similarities:

Differences:


Cut-‐outs for a 2x2 Rubik’s Cube

Color 4 of these squares yellow, color 4 blue, color 4 orange, color 4 red, color 4 green, and leave 4 of them white. Then cut all 24 of the squares out.


Making a 2x2 Solution Guide

Let’s make a solution guide for the 2x2 Rubik’s Cube. U

se our 3x3 guide, and cross out any step that applies to an edge piece. W

e now have a guide for

solving the 2x2 Rubik’s Cube. Try it out.

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