Are You InKLEINed 4 Solitaire

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Are You InKLEINed - 4 Solitaire?
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Presented by

• Matt Bach
• Ryan Erickson
• Angie Heimkes
• Jason Gilbert
• Kim Dressel

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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

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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 cant capture him
• The Geese must maneuver themselves so they can
  prevent the fox from escaping.

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Puzzle Pegs
This is a 19th Century version of Peg Solitaire
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Puzzle-Peg
A 1929 version of Peg Solitaire
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Jewish Version
Made at Israel in 1972 with instruction printed
in Hebrew. Very identical to the previous versions
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Teasing Pegs
This game has an alternative called French
Solitaire.
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Hi-Q
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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

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Fields of Work

• Non-Euclidean geometry
• Connections between geometry and group theory
• Results in function theory

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More about Felix Klein

• He intended on becoming a physicist, but that
  changed when be became Pluckers assistant.
• After he got his doctorate in 1868, he was given
  the task of finishing the late Pluckers 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.

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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

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Example Game (Cross)
1st Move
Initial Configuration
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Cross (1st 2nd Move)
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Cross (2nd 3rd Move)
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Cross (3rd 4th Move)
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Cross (4th 5th Move)
You Win!!!
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Other Peg Solitaire Games
Arrow Diamond
Double Arrow Pyramid
Fireplace Standard
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GROUPS
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

• Binary Operation ab ? G for all a, b ? G
• Associative (ab)c a(bc) for all
  a, b, c ?G
• Identity ae ea a for all a? G
• Inverses ab ba e  

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SPECIAL PROPERTIES

• If the group has the property
• ab ba
• then the group is called ABELIAN
• A group is called CYCLIC if ? an element a?G such
  that
• G ? n?Z

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KLEIN 4 GROUP

• It has two special properties
• 1. Every element is its own inverse
• 2. 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.

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Z Modules

• An integer module is similar to a vector space.
• In our case, contains

• 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)

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Move Vectors

• 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.

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Module Homomorphism Properties

• The mapping must satisfy these properties
• ?(a b) ??(a) ?(b)
• ?(ca) c?(a) 
• A KERNEL of a homomorphism ? from a group G to
  another group is the set
• x?G ?(x) e
• The kernel of ? is denoted as Ker ?

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TESSELLATION
A mapping of the Klein 4 Group onto the board
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Definition of Feasibility

• The dictionary defines feasibility as follows
• Can be done easily possible without difficulty
  or damage likely or probable.

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Peg Solitaire Feasibility Problem

• Objective
• We want to prove whether a certain board
  configuration is possible.
• We must prove there is a legal sequence that
  transforms one configuration into another.
• Use the 5 Locations Thm and the Rule of Three to
  solve the feasibility problem.

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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.

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The Solitaire Board
The Solitaire Board is defined as follows

• 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

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The Five Locations Theorem

• Dr. Arie Bialostocki
• Prove If a single peg configuration is
  achieved, the peg must exist in one of five
  locations

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Prerequisites

• English style game board
• Game begins with one peg removed from the center
  of the board
• General rules apply

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Game Ending Configuration

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

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Klein 4 Group

• Additive Cyclic Group
• I. Every element is its own inverse
• II. The sum of any two distinct nonzero elements
  is equal to the third nonzero element

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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

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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

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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

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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

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Five Locations Remain

• Therefore, only five locations remain and Dr.
  Bialostockis Five Locations Theorem holds.

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Notion for Scoring

• Let

is the Klein 4 - Group
is the Klein Product Module

Abelian group with the following properties
? a a b c c e
? a b c, a c b , b c a
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Classic Examples

• Define two maps

Define two maps
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How did they get that?
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Game Configurations

• A single peg or basis vector is represented by
  the following

?
filled
empty
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Score Map(A module homomorphism- a linear like
map)

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

As shown by the previous examples
Thus the score of
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An Example

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

? ( ) 1
1 (a , a) (a , a)
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The Board Score

• B English 33- board
• C

? (B) ? ( ) ? (1 - )
(a , a) (a , a)
(a a, a a)
(e , e)
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Note
Let
We can show that For example,

•  

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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(?).

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Proof
Suppose (c, ) is feasible.
Then c? c
? (c?) ? c

? (c?) ? (c) ( )
??
? (c?) ? (c) ?(e,e)
? (c?) ? (c) (e,e)
? (c?) - ? (c) (e,e)
? (c? - c) (e,e)
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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).

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Proof
Assume(c, ) is feasible. 1 - c
By the Rule of Three c - ? Ker(?), i.e.
?( c - ) (e,e)
?( c) - ?( ) (e,e)
?( c) (e,e) ?( )
?( c) ?( )  
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Proof Continued

• However

?(B) ?( c) ?( )
?(B) ?( c) ?( c)
?(B) (e,e)
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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

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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?

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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/mathe
  maticians/KleinF.html
• http//www.ahs.uwaterloo.ca/museum/vexhibit/puzzl
  es/solitaire/solitaire.html