Mathematical and Computational Analysis of Chomp. Salvador Badillo -Rios and Verenice Mojica. Goal. - PowerPoint PPT Presentation
Computational Analysis of Chomp
Mathematical and Computational Analysis of ChompSalvador
Badillo-Rios and Verenice MojicaGoalThe 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.Game DescriptionGeometric 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 turnGame DescriptionGeometric 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 turnPlayer 1 makes a moveGame DescriptionGeometric 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 turnPlayer 2 makes a moveGame DescriptionGeometric 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 turnPlayer 1 makes a moveGame DescriptionGeometric 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 turnPlayer 2 makes a moveGame DescriptionGeometric 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 turnPlayer 1 makes a moveGame DescriptionGeometric 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 turnxPlayer 2 loses!Game DescriptionNumeric 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
loses2021222324301248163136122448329183672144332754108216432N = 24
* 33 = 432 Game DescriptionNumeric 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
loses2021222324301248163136122448329183672144332754108216432N = 24
* 33 = 432 Player 1 chooses 12Game DescriptionNumeric 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
loses2021222324301248163136122448329183672144332754108216432N = 24
* 33 = 432 Player 2 chooses 9Game DescriptionNumeric 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
loses2021222324301248163136122448329183672144332754108216432N = 24
* 33 = 432 Player 1 chooses 8Game DescriptionNumeric 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
loses2021222324301248163136122448329183672144332754108216432N = 24
* 33 = 432 Player 2 chooses 2Game DescriptionNumeric 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
loses2021222324301248163136122448329183672144332754108216432N = 24
* 33 = 432 Player 1 chooses 3Game DescriptionNumeric 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
loses2021222324301248163136122448329183672144332754108216432N = 24
* 33 = 432 Player 2 loses!Fair or Unfair?Strategy-Stealing
ArgumentSuppose player one begins by removing the bottom right-most
piece
Fair or Unfair?Strategy-Stealing ArgumentSuppose player one
begins by removing the bottom right-most piece
If that move is a winning move, then player one wins
Fair or Unfair?Strategy-Stealing ArgumentIf it is a losing move,
player two has a good countermove and player two wins
Fair or Unfair?Strategy-Stealing ArgumentIf it is a losing move,
player two has a good countermove and player two winsBut player one
could have gotten to that countermove from the very
beginningTherefore, player one has the winning move and can always
win, if he/she plays perfectly
Known Special Casesm x m ChompPlayer one chomps the piece
located at (2,2)
x12341234Known Special Casesm x m ChompPlayer one chomps the
piece located at (2,2)
The board is left as an L-shape, and player one copies player
twos moves symmetricallyPlayer 1 chooses (2,2)Known Special Casesm
x m ChompPlayer one chomps the piece located at (2,2)
The board is left as an L-shape, and player one copies player
twos moves symmetricallyPlayer 2 movesKnown Special Casesm x m
ChompPlayer one chomps the piece located at (2,2)
The board is left as an L-shape, and player one copies player
twos moves symmetricallyPlayer 1 moves symmetricallyKnown Special
Casesm x m ChompPlayer one chomps the piece located at (2,2)
The board is left as an L-shape, and player one copies player
twos moves symmetricallyPlayer 2 movesKnown Special Casesm x m
ChompPlayer one chomps the piece located at (2,2)
The board is left as an L-shape, and player one copies player
twos moves symmetricallyPlayer 1 moves symmetricallyKnown Special
Casesm x m ChompPlayer one chomps the piece located at (2,2)
The board is left as an L-shape, and player one copies player
twos moves symmetricallyxPlayer 2 loses!Known Special
CasesTwo-Rowed ChompProposition 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
Three-Rowed ChompZeilbergers Chomp3RowsDoron Zeilberger
developed a program that computed P-positions for 3-rowed Chomp for
c 115We will be using this notation throughout[c, b,
a]|--------c---------||---b---||--------a--------|Three-Rowed
ChompProposition 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 = 1Move to: [1,0,2]Our ResearchTwo computers play against each
other, both eventually learn to play at their best
Displays :Board1st computers opening winning move P-positions
and their total amount Number of games playedAdaptive Learning
ProgramApproximation of P-positionsApproximation of
P-positionsApproximation of P-PositionsInitial attempt to Analyze
P-PositionsInitially we decided to look at the sum of the
P-positions to note obvious patternsOne obvious pattern was found
(the one proposed by Zeilberger)Was not much of a success due to
the various possible arrangements of pieces
Analyzing Opening Winning MovesComputers 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 ChompP-Positions
after Computer Learned Opening Move3-RowsBoard Size:Value of
NComputer 1's Opening Winning MoveResuting
P-Position3x193[0,0,1]3x21818[1,1,0]3x3366[1,0,2]3x47212[2,0,2]3x514472[3,2,0]3x628824[3,0,3]3x7576144[4,3,0]3x8115248[4,0,4]3x92304576[6,3,0]3x10460896[5,0,5]3x119216192[6,0,5]3x12184322304[8,4,0]3x1336864384[7,0,6]3x14737289216[10,4,0]3x15147456768[8,0,7]Type
1: yn = [y0[1]+4n,0,y0[3]+3n]Opening Winning Move Conjecture for
3-Rowed ChompSuppose xn 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 xn are to the set of P-positions ynyn has a
pattern such that: yn = [y0[1]+4n,0,y0[3]+3n], where :
Board Size (x0):Computer 1's Opening Winning Move:Resulting
P-Positions
(y0):3x13[0,0,1]3x36[1,0,2]3x412[2,0,2]3x624[3,0,3]Board Size
(x1):Computer 1's Opening Winning Move:Resulting P-Positions
(y1):3x848[4,0,4]3x1096[5,0,5]3x11192[6,0,5]3x13384[7,0,6]y0 = {
[0,0,1]}[1,0,2][2,0,2][3,0,3]Type 1: yn =
[y0[1]+4n,0,y0[3]+3n]Board Size (x0):Computer 1's Opening Winning
Move:Resulting P-Positions
(y0):3x13[0,0,1]3x36[1,0,2]3x412[2,0,2]3x624[3,0,3]Board Size
(x1):Computer 1's Opening Winning Move:Resulting P-Positions
(y1):3x848[4,0,4]3x1096[5,0,5]3x11192[6,0,5]3x13384[7,0,6]Board
Size:Computer 1's Opening Winning MoveResuting
P-Position3x218[1,1,0]3x572[3,2,0]3x7144[4,3,0]3x9576[6,3,0]3x122304[8,4,0]3x149216[10,4,0]Type
2: In ProgressNo Patterns Found4-RowsBoard SizeValue of NComputer
1's Opening Winning MoveResuting P-Position4
x1273[0,0,0,1]4x25454[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-RowsBoard
SizeValue of NComputer 1's Opening Winning MoveResuting
P-Position5x1813[0,0,0,0,1]5x2162162[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]Analyzing
All P-positions by Grouping3, 4, and 5-rowed Chomp was analyzedThe
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 permutations4 Rows: d =
2[2,0,0,4][2,0,0,4][2,0,0,4][2,0,0,4][2,0,0,4][2,0,0,4][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][2,2,1,3][2,2,1,3][2,2,1,3][2,2,1,3][2,2,1,3][2,2,1,3][2,3,0,4][2,3,0,4][2,3,0,4][2,3,0,4][2,3,0,4][2,3,0,4][2,3,0,4][2,4,0,6][2,4,0,6][2,4,0,6][2,4,0,6][2,4,0,6][2,4,0,6][2,4,0,6][2,5,1,5][2,5,1,5][2,5,1,5][2,5,1,5][2,5,1,5][2,5,1,5][2,5,1,5][2,6,3,2][2,6,3,2][2,6,3,2][2,6,3,2][2,6,3,2][2,6,3,2][2,6,3,2][2,7,4,0][2,7,4,0][2,7,4,0][2,7,4,0][2,7,4,0][2,7,4,0][2,7,4,0]Pattern
Found After Grouping Constant Row Value ConjectureFor 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 valuesIn the following column
the values will increase by a value of oneThe 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]Concluding
RemarksDeveloped a learning program to analyze ChompApproximated
amount of P-positions per board sizeInitially analyzed sum of
P-positions to find patternsAnalyzed Computers opening moves and
resulting P-positionsOpening Winning Move Conjecture for 3-Rowed
ChompGrouped P-positions of certain board sizes with fixed boards
by amount of pieces in bottom rowConstant Row Value
ConjectureAknowledgementsiCAMP ProgramFaculty Advisor: Dr.
EichhornRobert CampbellGame Theory fellow
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