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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