Are You In KLEIN ed - 4 Solitaire?

1. Are You InKLEINed - 4 Solitaire?

2. Presented by: • Matt Bach • Ryan Erickson • Angie Heimkes • Jason Gilbert • Kim Dressel

3. 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

4. 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.

5. Puzzle Pegs This is a 19th Century version of Peg Solitaire

6. Puzzle-Peg A 1929 version of Peg Solitaire

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

8. Teasing Pegs This game has an alternative called French Solitaire.

9. Hi-Q

10. 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

11. Fields of Work • Non-Euclidean geometry • Connections between geometry and group theory • Results in function theory

12. 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.

13. 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

14. Example Game (Cross) 1st Move Initial Configuration

15. Cross (1st & 2nd Move)

16. Cross (2nd & 3rd Move)

17. Cross (3rd & 4th Move)

18. Cross (4th & 5th Move) You Win!!!

19. Other Peg Solitaire Games Arrow Diamond Double Arrow Pyramid Fireplace Standard

20. 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 a*b  G for all a, b  G • Associative (a*b)*c = a*(b*c) for all a, b, c G • Identity a*e = e*a =a for all a G • Inverses a*b = b*a = e  

21. 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 aG such that G = {  nZ}

22. 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.

23. Z Modules • Anintegermoduleissimilarto 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)

24. 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.

25. 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 

26. TESSELLATION A mapping of the Klein 4 Group onto the board

27. Definition of Feasibility The dictionary defines feasibility as follows: • Can be done easily; possible without difficulty or damage; likely or probable.

28. 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.

29. 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.

30. 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:

31. The Five Locations Theorem • Dr. Arie Bialostocki • Prove: If a single peg configuration is achieved, the peg must exist in one of five locations

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

33. Game Ending Configuration • Five locations in which a single peg board configuration can be achieved

34. 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

35. 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

36. 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

37. 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

38. 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

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

40. 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

41. Classic Examples Define two maps Define two maps

42. How did they get that?

43. Game Configurations A single peg or “basis vector” is represented by the following:  filled empty

44. 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

45. An Example = board vector that has a peg in (0,0) and is empty every where else.  ( ) = 1* = 1 * (a , a) = (a , a)

46. The Board Score B = English 33- board C =  (B)=  ( ) +  (1 - ) = (a , a) + (a , a) =(a + a, a + a) =(e , e)

47. Note: Let We can show that For example,

48. 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().

49. 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)

50. 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).