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Mathematical and Computational Analysis of Chomp

Mathematical and Computational Analysis of Chomp. Salvador Badillo -Rios and Verenice Mojica. Goal.

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Mathematical and Computational Analysis of Chomp

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  1. Mathematical and Computational Analysis of Chomp Salvador Badillo-Rios and VereniceMojica

  2. Goal The goal of this research project was to provide an extended analysis of 2-D Chomp using computational and mathematical means in order to provide a pattern that may aid in finding the winning strategy for all board sizes.

  3. GameDescriptionGeometric 2-D Chomp: • Two-player game • Players take turns choosing a square box from an m x n board • The pieces below and to the right of the chosen cell disappear after every turn

  4. GameDescriptionGeometric 2-D Chomp: • Two-player game • Players take turns choosing a square box from an m x n board • The pieces below and to the right of the chosen cell disappear after every turn Player 1 makes a move

  5. GameDescriptionGeometric 2-D Chomp: • Two-player game • Players take turns choosing a square box from an m x n board • The pieces below and to the right of the chosen cell disappear after every turn Player 2 makes a move

  6. GameDescriptionGeometric 2-D Chomp: • Two-player game • Players take turns choosing a square box from an m x n board • The pieces below and to the right of the chosen cell disappear after every turn Player 1 makes a move

  7. GameDescriptionGeometric 2-D Chomp: • Two-player game • Players take turns choosing a square box from an m x n board • The pieces below and to the right of the chosen cell disappear after every turn Player 2 makes a move

  8. GameDescriptionGeometric 2-D Chomp: • Two-player game • Players take turns choosing a square box from an m x n board • The pieces below and to the right of the chosen cell disappear after every turn Player 1 makes a move

  9. GameDescriptionGeometric 2-D Chomp: • Two-player game • Players take turns choosing a square box from an m x n board • The pieces below and to the right of the chosen cell disappear after every turn Player 2 loses!

  10. GameDescriptionNumeric 2-D Chomp: • Players take turns choosing a divisor of a given natural number, N • They are not allowed to choose a multiple of a previously chosen divisor • The player to choose 1 loses N = 24*33 = 432

  11. GameDescriptionNumeric 2-D Chomp: • Players take turns choosing a divisor of a given natural number, N • They are not allowed to choose a multiple of a previously chosen divisor • The player to choose 1 loses N = 24*33 = 432 Player 1 chooses 12

  12. GameDescriptionNumeric 2-D Chomp: • Players take turns choosing a divisor of a given natural number, N • They are not allowed to choose a multiple of a previously chosen divisor • The player to choose 1 loses N = 24*33 = 432 Player 2 chooses 9

  13. GameDescriptionNumeric 2-D Chomp: • Players take turns choosing a divisor of a given natural number, N • They are not allowed to choose a multiple of a previously chosen divisor • The player to choose 1 loses N = 24*33 = 432 Player 1 chooses 8

  14. GameDescriptionNumeric 2-D Chomp: • Players take turns choosing a divisor of a given natural number, N • They are not allowed to choose a multiple of a previously chosen divisor • The player to choose 1 loses N = 24*33 = 432 Player 2 chooses 2

  15. GameDescriptionNumeric 2-D Chomp: • Players take turns choosing a divisor of a given natural number, N • They are not allowed to choose a multiple of a previously chosen divisor • The player to choose 1 loses N = 24*33 = 432 Player 1 chooses 3

  16. GameDescriptionNumeric 2-D Chomp: • Players take turns choosing a divisor of a given natural number, N • They are not allowed to choose a multiple of a previously chosen divisor • The player to choose 1 loses N = 24*33 = 432 Player 2 loses!

  17. FairorUnfair? Strategy-Stealing Argument • Suppose player one begins by removing the bottom right-most piece

  18. FairorUnfair? Strategy-Stealing Argument • Suppose player one begins by removing the bottom right-most piece • If that move is a winning move, then player one wins

  19. FairorUnfair? Strategy-Stealing Argument • If it is a losing move, player two has a good countermove and player two wins

  20. FairorUnfair? Strategy-Stealing Argument • If it is a losing move, player two has a good countermove and player two wins • But player one could have gotten to that countermove from the very beginning • Therefore, player one has the winning move and can always win, if he/she plays perfectly

  21. Known Special Cases m x m Chomp • Player one chomps the piece located at (2,2)

  22. Known Special Cases m x m Chomp • Player one chomps the piece located at (2,2) • The board is left as an L-shape, and player one copies player two’s moves symmetrically Player 1 chooses (2,2)

  23. Known Special Cases m x m Chomp • Player one chomps the piece located at (2,2) • The board is left as an L-shape, and player one copies player two’s moves symmetrically Player 2 moves

  24. Known Special Cases m x m Chomp • Player one chomps the piece located at (2,2) • The board is left as an L-shape, and player one copies player two’s moves symmetrically Player 1 moves symmetrically

  25. Known Special Cases m x m Chomp • Player one chomps the piece located at (2,2) • The board is left as an L-shape, and player one copies player two’s moves symmetrically Player 2 moves

  26. Known Special Cases m x m Chomp • Player one chomps the piece located at (2,2) • The board is left as an L-shape, and player one copies player two’s moves symmetrically Player 1 moves symmetrically

  27. Known Special Cases m x m Chomp • Player one chomps the piece located at (2,2) • The board is left as an L-shape, and player one copies player two’s moves symmetrically Player 2 loses!

  28. Known Special Cases Two-Rowed Chomp • Proposition 0: • (a, a-1) is P-position , where a  1 • (a,b) is an N-position ONLY when a  b  0 and a≠ b+1 • Winning Moves: • (a,a-1) if a=b • (b+1,b) if a  b+2

  29. Three-Rowed Chomp Zeilberger’s “Chomp3Rows” • DoronZeilberger developed a program that computed P-positions for 3-rowed Chomp for c≤ 115 • We will be using this notation throughout |--------a--------| |---b---| |--------c---------| [c, b, a]

  30. Three-Rowed Chomp • Proposition 1: • The only P-positions [c,b,a], with c = 1, are [1,1,0] and [1,0,2] • N-positions with at least 6 pieces and with c = 1: • [1,1,1], [1,2,0], [1,0,3+x], and [1,1+y,x] • Winning Moves: • [1,1,1], [1,2,0],[1,0,3+x] to [1,0,2] • [1,1+x,y] to [1,1,0] • Proposition 2: • [2,b0,a0] is a P-position iff a0 = 2 [1,1,4] i.e., [1,0,3+x] where x = 1 Move to: [1,0,2]

  31. Adaptive Learning Program Our Research • Two computers play against each other, both eventually learn to play at their best • Displays : • Board • 1st computer’s opening winning move • P-positions and their total amount • Number of games played

  32. Approximation of P-positions

  33. Approximation of P-positions

  34. Approximation of P-Positions

  35. Initial attempt to Analyze P-Positions • Initially we decided to look at the sum of the P-positions to note obvious patterns • One obvious pattern was found (the one proposed by Zeilberger) • Was not much of a success due to the various possible arrangements of pieces

  36. Analyzing Opening Winning Moves • Computer’s opening winning moves for 3,4, and 5 rows were analyzed • One significant pattern was observed for 3-rowed Chomp, and a possible pattern was observed as well • No clear patterns were found for 4 and 5-rowed Chomp

  37. P-Positions after Computer Learned Opening Move

  38. Type 1: yn = [y0[1]+4n,0,y0[3]+3n] Opening Winning Move Conjecture for 3-Rowed Chomp • Suppose xn is the set of board sizes: 3 x (1+7n), 3 x (3+7n), 3 x (4+7n), 3 x (6+7n), where n≥0. • Then the computer’s opening winning moves for xn are to the set of P-positions yn • yn has a pattern such that: yn = [y0[1]+4n,0,y0[3]+3n], where :

  39. Type 1: Type 2: In Progress yn = [y0[1]+4n,0,y0[3]+3n]

  40. No Patterns Found

  41. Analyzing All P-positions by Grouping • 3, 4, and 5-rowed Chomp was analyzed • The P-positions within these n-rowed Chomp sets were grouped by the amount of pieces in the bottom row • The P-positions for each group were then sorted into their possible permutations

  42. Pattern Found After Grouping Constant Row Value Conjecture • For n-rowed Chomp, when n≥3, at least one subset of its total P-positions will have a pattern as follows: • n-2 columns of the data for the subset will be fixed to distinct constant values • In the following column the values will increase by a value of one • The values of the remaining columns may vary or have a constant value as well

  43. Concluding Remarks • Developed a learning program to analyze Chomp • Approximated amount of P-positions per board size • Initially analyzed sum of P-positions to find patterns • Analyzed Computer’s opening moves and resulting P-positions • Opening Winning Move Conjecture for 3-Rowed Chomp • Grouped P-positions of certain board sizes with fixed boards by amount of pieces in bottom row • Constant Row Value Conjecture

  44. Aknowledgements • iCAMP Program • Faculty Advisor: Dr. Eichhorn • Robert Campbell • Game Theory fellow researchers

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