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.
Slide 3
Game Description Geometric 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
Slide 4
Game Description Geometric 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
Slide 5
Game Description Geometric 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
Slide 6
Game Description Geometric 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
Slide 7
Game Description Geometric 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
Slide 8
Game Description Geometric 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
Slide 9
Game Description Geometric 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 x Player 2 loses!
Slide 10
Game Description Numeric 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 2020 21212 2323 2424 3030 124816 3131 36122448 3232
9183672144 3 2754108216432 N = 2 4 * 3 3 = 432
Slide 11
Game Description Numeric 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 2020 21212 2323 2424 3030 124816 3131 36122448 3232
9183672144 3 2754108216432 N = 2 4 * 3 3 = 432 Player 1 chooses
12
Slide 12
Game Description Numeric 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 2020 21212 2323 2424 3030 124816 3131 36122448 3232
9183672144 3 2754108216432 N = 2 4 * 3 3 = 432 Player 2 chooses
9
Slide 13
Game Description Numeric 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 2020 21212 2323 2424 3030 124816 3131 36122448 3232
9183672144 3 2754108216432 N = 2 4 * 3 3 = 432 Player 1 chooses
8
Slide 14
Game Description Numeric 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 2020 21212 2323 2424 3030 124816 3131 36122448 3232
9183672144 3 2754108216432 N = 2 4 * 3 3 = 432 Player 2 chooses
2
Slide 15
Game Description Numeric 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 2020 21212 2323 2424 3030 124816 3131 36122448 3232
9183672144 3 2754108216432 N = 2 4 * 3 3 = 432 Player 1 chooses
3
Slide 16
Game Description Numeric 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 2020 21212 2323 2424 3030 124816 3131 36122448 3232
9183672144 3 2754108216432 N = 2 4 * 3 3 = 432 Player 2 loses!
Slide 17
Fair or Unfair? Strategy-Stealing Argument Suppose player one
begins by removing the bottom right-most piece
Slide 18
Fair or Unfair? 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
Slide 19
Fair or Unfair? Strategy-Stealing Argument If it is a losing
move, player two has a good countermove and player two wins
Slide 20
Fair or Unfair? 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
Slide 21
Known Special Cases m x m Chomp Player one chomps the piece
located at (2,2) x 1234 1 2 3 4
Slide 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 twos moves symmetrically Player 1 chooses (2,2)
Slide 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 twos moves symmetrically Player 2 moves
Slide 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 twos moves symmetrically Player 1 moves
symmetrically
Slide 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 twos moves symmetrically Player 2 moves
Slide 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 twos moves symmetrically Player 1 moves
symmetrically
Slide 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 twos moves symmetrically x Player 2 loses!
Slide 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
Slide 29
Three-Rowed Chomp Zeilbergers Chomp3Rows Doron Zeilberger
developed a program that computed P-positions for 3-rowed Chomp for
c 115 We will be using this notation throughout [c, b, a]
|--------c--------- | |---b---| |--------a--------|
Slide 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,b 0,a 0 ] is a P-position iff a 0 = 2
[1,1,4] i.e., [1,0,3+x] where x = 1 Move to: [1,0,2]
Slide 31
Our Research Two computers play against each other, both
eventually learn to play at their best Displays : Board 1 st
computers opening winning move P-positions and their total amount
Number of games played Adaptive Learning Program
Slide 32
Approximation of P-positions
Slide 33
Slide 34
Approximation of P-Positions
Slide 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
Slide 36
Analyzing Opening Winning Moves Computers 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
Type 1: y n = [y 0 [1]+4n,0,y 0 [3]+3n] Opening Winning Move
Conjecture for 3-Rowed Chomp Suppose x n is the set of board sizes:
3 x (1+7n), 3 x (3+7n), 3 x (4+7n), 3 x (6+7n), where n0. Then the
computers opening winning moves for x n are to the set of
P-positions y n y n has a pattern such that: y n = [y 0 [1]+4n,0,y
0 [3]+3n], where : Board Size (x 0 ): Computer 1's Opening Winning
Move: Resulting P-Positions (y 0 ): 3x13[0,0,1] 3x36[1,0,2]
3x412[2,0,2] 3x624[3,0,3] Board Size (x 1 ): Computer 1's Opening
Winning Move: Resulting P-Positions (y 1 ): 3x848[4,0,4]
3x1096[5,0,5] 3x11192[6,0,5] 3x13384[7,0,6] y 0 = { [0,0,1] }
[1,0,2] [2,0,2] [3,0,3]
No Patterns Found 4-Rows Board Size Value of N Computer 1's
Opening Winning Move Resuting P- Position 4 x1273[0,0,0,1] 4x254
[1,1,0,0] 4x310818[1,0,2,0] 4x42166[1,0,0,3] 4x543236[2,0,3,0]
4x686412[2,0,0,4] 4x7172872[3,0,3,0] 4x8345624[3,0,0,4]
4x969123456[7,2,0,0] 4x1013824288[5,0,5,0] 4x112764848[4,0,0,7]
4x125529696[5,0,0,7] 4x13110592576[6,0,7,0] 5-Rows Board SizeValue
of N Computer 1's Opening Winning Move Resuting P- Position
5x1813[0,0,0,0,1] 5x2162 [1,1,0,0,0] 5x3324108[2,0,1,0,0]
5x464836[2,0,0,0,2] 5x512966[1,0,0,0,4] 5x6259272[3,0,0,3,0]
5x7518412[2,0,0,0,5] 5x8103685184[7,1,0,0,0] 5x920736288[5,0,0,4,0]
5x104147224[3,0,0,0,7] 5x118294441472[9,2,0,0,0]
Slide 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 4
Rows: d = 2 [2,0,0,4]
[2,1,0,2][2,1,1,5][2,1,2,2][2,1,3,2][2,1,4,3][2,1,5,3][2,1,6,3]
[2,2,1,3] [2,3,0,4] [2,4,0,6] [2,5,1,5] [2,6,3,2] [2,7,4,0]
Slide 42
Pattern Found After Grouping Constant Row Value Conjecture For
n-rowed Chomp, when n3, 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 3-Rows: c = 4 [c,b,a] [4,0,4] [4,1,4] [4,2,4] [4,3,0]
4-Rows: d = 4 [d,c,b,a] [4,0,0,7] [4,0,1,5] [4,0,2,7] [4,0,3,2]
[4,0,4,4] 5-Rows: e = 1 [e,d,c,b,a] [1,0,0,0,4] [1,0,0,1,2]
[1,0,0,2,3] [1,0,0,3,3] [1,0,0,4,3] [1,0,0,5,3] [1,0,0,6,3]
[1,0,0,7,3]
Slide 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 Computers
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
Slide 44
Aknowledgements iCAMP Program Faculty Advisor: Dr. Eichhorn
Robert Campbell Game Theory fellow researchers