Are You InKLEINed - 4 Solitaire?

Presented by:

Matt Bach Ryan Erickson Angie Heimkes Jason Gilbert Kim Dressel

History of Peg Solitaire

Invented by French Noblemen in the 17th Century, while imprisoned in the Bastille

The game used the Fox & Geese Board that was used by many games in Northern Europe prior to the 14th Century

Fox and Geese Board

May have originated from Iceland

The game is 2 player Consists of 1 black token and

13 white tokens The Fox must capture as

many geese as he can so they can’t capture him

The Geese must maneuver themselves so they can prevent the fox from escaping.

This is a 19th Century version of Peg Solitaire

Puzzle Pegs

Puzzle-Peg

A 1929 version of Peg Solitaire

Jewish Version

Made at Israel in 1972 with instruction printed in Hebrew.Very identical to the previous versions

Teasing Pegs

This game has an alternative called French Solitaire.

Hi-Q

Felix Klein

We are modeling peg solitaire on the Klein 4-Group named after him.

Born in Dusseldorf in 1849

Studied at Bonn, Got Tingen, and Berlin

Fields of Work

Non-Euclidean geometry

Connections between geometry and group theory

Results in function theory

More about Felix Klein

He intended on becoming a physicist, but that changed when be became Plucker’s assistant.

After he got his doctorate in 1868, he was given the task of finishing the late Plucker’s work on line geometry

At the age of 23, he became a professor at Erlangen, and held a chair in the Math Department

In 1875, He was offered a chair at the Technische Hochschule at Munich where he taught future mathematicians like Runge and Planck.

Rules of Peg Solitaire Rule 1: You can only move a peg in the following directions: North, South, East, and West.

Rule 2: During a move, you must jump over another peg to the corresponding empty hole.

Rule 3: To win, you must only have one peg remaining on the board

Example Game (Cross)

Initial Configuration 1st Move

Cross (1st & 2nd Move)

Cross (2nd & 3rd Move)

Cross (3rd & 4th Move)

Cross (4th & 5th Move)

You Win!!!

Other Peg Solitaire Games

ArrowDiamond

Double ArrowPyramid

FireplaceStandard

GROUPS

1. Binary Operation a*b G for all a, b G

2. Associative (a*b)*c = a*(b*c) for all a, b, c G

3. Identity a*e = e*a =a for all a G

4. Inverses a*b = b*a = e  

Let G be a nonempty set with operation *

a, b, c are elements of G

e is the identity element of G

G is a GROUP if it has:

SPECIAL PROPERTIES

If the group has the property :

a*b = b*a

then the group is called ABELIAN

A group is called CYCLIC if an element aG such that G = { nZ}

na

KLEIN 4 GROUP

It has two special properties1. Every element is its own

inverse2. The sum of two distinct

non zero elements is equal to the third element

The Klein 4 Group is the direct sum of two cyclic groups.

Z Modules

Configuration Vectors Move Vectors

and contains values described by lattice points {-1, 0, 1, 2, -3}

(0,0) (1,0) (0,1) (–1,0) (0,-1)

An integer module is similar to a vector space.

In our case, contains:

BZ

BZ

Move Vectors

ije Equations are represented in the following way: is a configuration with a peg in the (i,j)th position. Moves are made by adding and subracting these vectors.

1,,1,,

1,,1,,

,1,,1,

,1,,1,

jijijiji

jijijiji

jijijiji

jijijiji

eeeu

eeed

eeel

eeer

Module Homomorphism Properties

The mapping must satisfy these properties:

1. (a + b) = (a) + (b) 2. (ca) = c(a) 

A KERNEL of a homomorphism from a group G to another group is the set:

{xG| (x) = e} The kernel of is denoted as Ker

TESSELLATION

A mapping of the Klein 4 Group onto the board

Definition of Feasibility

The dictionary defines feasibility as follows: Can be done easily; possible without

difficulty or damage; likely or probable.

Peg Solitaire Feasibility Problem

Objective:1. We want to prove whether a certain

board configuration is possible.2. We must prove there is a legal sequence

that transforms one configuration into another.

3. Use the 5 Locations Thm and the Rule of Three to solve the feasibility problem.

How the Feasibility Problem Works

Given a Board B and a pair of configurations (c,c') on B, determine if the pair (c,c') is feasible.

The Solitaire Board

The board is a set of integer points in a plane C and C' are tessellations or configuration vectors of the board C' is “1 – C” or the opposite of C

The Solitaire Board is defined as follows:

ZxZZB 2

The Five Locations Theorem

Dr. Arie Bialostocki

Prove: If a single peg configuration is achieved, the peg must exist in one of five locations

Prerequisites

English style game board

Game begins with one peg removed from the center of the board

General rules apply

Game Ending Configuration

Five locations in which a single peg board configuration can be achieved

Klein 4 Group

Additive Cyclic Group

I. Every element is it’s own inverse

II. The sum of any two distinct nonzero elements is equal to the third nonzero element

Board Tessellation

Assign x, y, z values to a 7x7 board starting in row 1 and column 1

Map from left to right, top to bottom

Remove the four locations from each of the four corners to produce a board tessellation

Adding Using Tessellation

By Klein 4 properties I and II, the sum of any x + y + z = 0

Therefore, adding up the individual pegged locations based on the tessellation, the total board value initially = y

Calculating After Move

For any move, the sum of two elements from x, y, z is replaced by the third element

According to property II of Klein 4 groups, this substitution does not affect the overall sum of the board

Peg Must Be Left In Y

Therefore, a single peg can only be left in a y location

However, because of the rules of symmetry, six of these eleven locations must be removed

Five Locations Remain

Therefore, only five locations remain and Dr. Bialostocki’s Five Locations Theorem holds.

Notion for Scoring Let },,,{:

2ecba

2

2 is the Klein 4 - Group

Abelian group with the following properties

a + a = b + = c + c = e

a + b = c, a + c = b , b + c = a

is the Klein Product Module 2

2

Classic Examples

Define two maps

Define two maps

2

2

2

21:, gg

g1

g2

How did they get that?

)3(mod2)(

)3(mod1)(

)3(mod0)(:),(1

jiifc

jiifb

jiifajig

)3(mod2)(

)3(mod1)(

)3(mod0)(:),(2

jiifc

jiifb

jiifajig

Game Configurations

A single peg or “basis vector” is represented by the following:

)0,.....1,0,......0,0(eij

emptyfilled

Score Map(A module homomorphism- a linear like

map)

For any board , the score map can be defined by the following notation:

2B

2

2:

B

As shown by the previous examples

)),(),,(()(21

jiji ggeij

Thus the score of Bc

)),(),,(()(21

),(

jijic ggcBji

ij

An Example

= board vector that has a peg in (0,0) and is empty every where else.e 0,0

( ) = 1*e 0,0))0,0((,)0,0((

21gg

= 1 * (a , a)= (a , a)

The Board Score

B = English 33- boardC = e 0,0

(B)= ( ) + (1 - ) e 0,0 e 0,0

= (a , a) + (a , a) =(a + a, a + a)

=(e , e)

g1

Note: B

ijm),()( eemij

LetWe can show that For example,

),()(

),(),()(

),(),(),()(

)()()()(

0,0

0,0

0,0

0,10,00,10,0

,1,1

ee

cccc

ccaabb

r

r

r

eeer

eeer jiijjiij

 

Rule of Three

A necessary condition for a pair of configuration (c, c) to be feasible is that (c - c) = (e, e), namely, c - c er().

Proof:Suppose (c, ) is feasible.

c

Then c = c +

k

i

i

m1

(c) = [c + ]

k

i

i

m1

k

i

i

m1m

i

(c) = (c) + ( )mi

(c)= (c) + (e,e)

(c)= (c) + (e,e)

(c) - (c) = (e,e)

(c - c) = (e,e)

Proposition 2

Let B be any board. A necessary condition for the configurations pair (c, ) to be feasible, with = 1 - c the complement of c, is that the board score is (B) = (e,e).

c

c

Proof:Assume(c, ) is feasible. = 1 - c

c

c

By the Rule of Three c - Ker(), i.e.

c

( c - ) = (e,e)

c

( c) - ( ) = (e,e) ( c) = (e,e) + ( )

( c) = ( )  

c

c

c

Proof Continued:

However:

(B) = ( c) + ( )

c

(B) = ( c) + ( c)

(B) = (e,e)

Conclusion

By using the Five Locations Theorem, and the Rule of Three, we have shown how it is possible to come up with the winning combinations in peg solitaire, and have shown why they work

Possible Questions

Can this model be applied to other games? How many solutions are there to the Peg

Solitaire Game? Is there a general algorithm for solving

central solitaire?

References Dr. Steve Deckelman “An Application of Elementary Group Theory to

Central Solitaire”– by Arie Bialostocki

“Solitaire Lattices”– by Antoine Deza, Shmuel Onn

Websites– http://bio.bio.rpi.edu/MS99/WhitneyW/advance/klein.ktm– http://library.thinkquest.org/22584/temh3043.htm– http://physics.rug.ac.be/fysica/Geschiedenis/mathematicians/Klein

F.html– http://www.ahs.uwaterloo.ca/~museum/vexhibit/puzzles/solitaire/s

olitaire.html