1 / 44

Mathematical and Computational Analysis of Chomp

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

elu
Download Presentation

Mathematical and Computational Analysis of Chomp

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related