Salvador Badillo-Rios and Verenice Mojica
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.
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
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
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
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
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
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
Game Description
Geometric 2-D Chomp:
 Two-player game
 Players take turns
x
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!
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
20
21
22
23
24
30
1
2
4
8
16
31
3
6
12
24
48
32
9
18
36
72
144
33
27
54
108
216
432
N = 24 * 33 = 432
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
20
21
22
23
24
30
1
2
4
8
16
31
3
6
12
24
48
32
9
18
36
72
144
33
27
54
108
216
432
N = 24 * 33 = 432
Player 1 chooses 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
20
21
22
23
24
30
1
2
4
8
16
31
3
6
12
24
48
32
9
18
36
72
144
33
27
54
108
216
432
N = 24 * 33 = 432
Player 2 chooses 9
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
20
21
22
23
24
30
1
2
4
8
16
31
3
6
12
24
48
32
9
18
36
72
144
33
27
54
108
216
432
N = 24 * 33 = 432
Player 1 chooses 8
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
20
21
22
23
24
30
1
2
4
8
16
31
3
6
12
24
48
32
9
18
36
72
144
33
27
54
108
216
432
N = 24 * 33 = 432
Player 2 chooses 2
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
20
21
22
23
24
30
1
2
4
8
16
31
3
6
12
24
48
32
9
18
36
72
144
33
27
54
108
216
432
N = 24 * 33 = 432
Player 1 chooses 3
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
20
21
22
23
24
30
1
2
4
8
16
31
3
6
12
24
48
32
9
18
36
72
144
33
27
54
108
216
432
N = 24 * 33 = 432
Player 2 loses!
Fair or Unfair?
Strategy-Stealing Argument
 Suppose player one begins
by removing the bottom
right-most piece
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
Fair or Unfair?
Strategy-Stealing Argument
 If it is a losing move, player
two has a good countermove
and player two wins
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
Known Special Cases
m x m Chomp
1
2
 Player one chomps the piece 1
located at (2,2)
2
3
4
x
3
4
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)
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
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
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
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
Known Special Cases
m x m Chomp
 Player one chomps the piece
located at (2,2)
x
 The board is left as an L-
shape, and player one
copies player two’s moves
symmetrically
Player 2 loses!
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
Three-Rowed Chomp
Zeilberger’s “Chomp3Rows”
|--------a--------|
 Doron Zeilberger developed a
program that computed
P-positions for 3-rowed
Chomp for c≤ 115
 We will be using this
notation throughout
|---b---|
|--------c--------|
[c, b, a]
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]
[1,1,4] i.e., [1,0,3+x] where x = 1
 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
Move to: [1,0,2]
Our Research
Adaptive Learning Program
 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
Approximation of P-positions
Amount of P-Positions as a Function of N
(3 Rows)
60
y = 5.6519ln(x) - 17.226
Total Number of P-Positions
50
40
30
20
10
0
0
10000
20000
30000
40000
"N" Value
50000
60000
70000
80000
Approximation of P-positions
Amount of P-Positions as a Function of N
(4 Rows)
180
y = 0.906x0.4431
Total Number of P-Positions
160
140
120
100
80
60
40
20
0
0
20000
40000
60000
"N" Value
80000
100000
120000
Approximation of P-Positions
Amount of P-Positions as a Function of N
(5 Rows)
250
Total Number of P-Positions
y = 0.3951x0.5494
200
150
100
50
0
0
10000
20000
30000
40000
50000
"N" Value
60000
70000
80000
90000
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
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
P-Positions after Computer
Learned Opening Move
3-Rows
Board Size:
Value of N
Computer 1's Opening Winning Move
Resuting P-Position
3x1
9
3
[0,0,1]
3x2
18
18
[1,1,0]
3x3
36
6
[1,0,2]
3x4
72
12
[2,0,2]
3x5
144
72
[3,2,0]
3x6
288
24
[3,0,3]
3x7
576
144
[4,3,0]
3x8
1152
48
[4,0,4]
3x9
2304
576
[6,3,0]
3x10
4608
96
[5,0,5]
3x11
9216
192
[6,0,5]
3x12
18432
2304
[8,4,0]
3x13
36864
384
[7,0,6]
3x14
73728
9216
[10,4,0]
3x15
147456
768
[8,0,7]
Type 1: y = [y [1]+4n,0,y [3]+3n]
n
0
0
Opening Winning Move
Conjecture for 3-Rowed Chomp
 Suppose xn is the set of board sizes:
Board Size
(x0):
Computer 1's Opening
Winning Move:
Resulting P-Positions
(y0):
3x1
3
[0,0,1]
3x3
6
[1,0,2]
3x4
12
[2,0,2]
3x6
24
[3,0,3]
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 :
Board Size
(x1):
Computer 1's Opening
Winning Move:
Resulting P-Positions
(y1):
[0,0,1]
3x8
48
[4,0,4]
3x10
96
[5,0,5]
3x11
192
[6,0,5]
3x13
384
[7,0,6]
y0 = {
[1,0,2]
[2,0,2]
[3,0,3]
}
Type 1: y = [y [1]+4n,0,y [3]+3n]
n
0
Type 2: In Progress
0
Board Size
(x0):
Computer 1's Opening
Winning Move:
Resulting P-Positions
(y0):
3x1
3
[0,0,1]
3x3
6
[1,0,2]
3x4
12
[2,0,2]
3x6
24
[3,0,3]
Board Size
(x1):
Computer 1's Opening
Winning Move:
Resulting P-Positions
(y1):
3x8
48
[4,0,4]
3x10
96
[5,0,5]
3x11
192
[6,0,5]
3x13
384
[7,0,6]
Board Size:
Computer 1's Opening Winning
Resuting P-Position
Move
3x2
18
[1,1,0]
3x5
72
[3,2,0]
3x7
144
[4,3,0]
3x9
576
[6,3,0]
3x12
2304
[8,4,0]
3x14
9216
[10,4,0]
No Patterns Found
4-Rows
5-Rows
Board
Size
Value of N
Computer 1's Opening
Winning Move
Resuting PPosition
4 x1
27
3
[0,0,0,1]
4x2
54
54
[1,1,0,0]
4x3
108
18
[1,0,2,0]
4x4
216
6
[1,0,0,3]
4x5
432
36
4x6
864
4x7
Board Size Value of N
Computer 1's Opening
Winning Move
Resuting PPosition
5x1
81
3
[0,0,0,0,1]
5x2
162
162
[1,1,0,0,0]
5x3
324
108
[2,0,1,0,0]
[2,0,3,0]
5x4
648
36
[2,0,0,0,2]
12
[2,0,0,4]
5x5
1296
6
[1,0,0,0,4]
1728
72
[3,0,3,0]
5x6
2592
72
[3,0,0,3,0]
4x8
3456
24
[3,0,0,4]
5x7
5184
12
[2,0,0,0,5]
4x9
6912
3456
[7,2,0,0]
4x10
13824
288
[5,0,5,0]
5x8
10368
5184
[7,1,0,0,0]
4x11
27648
48
[4,0,0,7]
5x9
20736
288
[5,0,0,4,0]
4x12
55296
96
[5,0,0,7]
5x10
41472
24
[3,0,0,0,7]
4x13
110592
576
[6,0,7,0]
5x11
82944
41472
[9,2,0,0,0]
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
4 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]
 The P-positions for each
group were then sorted into
their possible permutations
[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
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
5-Rows: e = 1
3-Rows: c = 4
4-Rows: d = 4
[e,d,c,b,a]
[d,c,b,a]
[c,b,a]
[1,0,0,0,4]
[4,0,0,7]
[1,0,0,1,2]
[4,0,4]
[1,0,0,2,3]
[4,0,1,5]
[1,0,0,3,3]
[4,1,4]
[4,0,2,7]
[4,2,4]
[1,0,0,4,3]
[1,0,0,5,3]
[4,0,3,2]
[1,0,0,6,3]
[4,3,0]
[4,0,4,4]
[1,0,0,7,3]
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
Aknowledgements
 iCAMP Program
 Faculty Advisor: Dr. Eichhorn
 Robert Campbell
 Game Theory fellow researchers