Nine Men's Morris
Martin Boyd
Christopher Hirunthanakorn
Game Overview
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Two player game
RULES
o Players alternate turns placing pieces on the
board
o If a mill is formed, player may remove an
opponent's piece
 mill - three pieces formed along a line
o After both players place nine pieces, players
move their pieces to any free adjacent spot
o The game is over when a player has less than 3
pieces or no legal moves remain
Example
Example
(continued)
Game Classification
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Determinate
Zero-sum
Symmetric
Perfect Information
Sequential
Normal
Background
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One of the oldest games played to date
o Game board carving from 1400 BCE found in
Egypt
Also known as Mill, Merelles, or Cowboy Checkers
Popular variants of the game include Three Men's,
Six Men's, and Twelve Men's Morris
Research Goals/Questions
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Look for an optimal strategy for piece
placement
Find an optimal strategy for gameplay
Is there a winning strategy for either player?
Is the game fair?
Analyzing the Game
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Searched for previous work on the game
Game States and Combinatorics
Created program with a GUI
Analyzed Five Men's Morris
Created an Adaptive Program
Created an AI
Used python as the programming language
Previous Publications
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Ralph Gasser (Swiss computer scientist)
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Proved that perfect play in Nine Men's Morris
results in a draw and is impossible for humans to
achieve
Analyzed the midgame and endgame by going
through all possible game states and labeling them
a win or lose position
Did not provide any advice on the optimal strategy
or fairness of the game
Five Men's Morris
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Players have 5 pieces instead of 9
16 spots instead of 24
Game States and Combinatorics
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A game state is defined as the game board and all
relevant information defining it such as Last player to
move and position of last move
Board to the right is the game state where player 1
just went but could have placed it on either side
Used combinatorics to estimate the number of game
states possible
o About 1.74 * 10^11 states based on possible
combinations of placement
(16*15*14*13*12*11*10*9*8*7*6)
o Can be reduced using symmetry of game states to
about 7.26 * 10^8 (31+14*13*12*11*10*9*8*7*6)
Basic Program Structure
A
B
C
D
E
1
2
B2
3
4
5
E5
Basic Program Structure
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Data of the Game Board is stored in 3 arrays
o Basic Array
[A1,A3,A5,B2,B3,B4,C1,C2,C4,C5,D2,D3,D4,E1,E3,E5]
o Mill Array
[[0,A1,A3,A5],[0,B2,B3,B4],...,[0,E1,E3,E5]]
o Connection Array
[[A1,A3,C1,0],[A3,A1,A5,B3],...,[E5,C5,E3,0]]
Adaptive Program
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Runs the Game MANY times
Contains Matchboxes that punish a player if
that player loses thus not repeating the
same mistake twice.
Matchboxes
A
1
2
3
4
5
B
C
D
E
MOVES
A5
B4
C2
C4
C5
D2
D3
D4
E5
Adaptive Program
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2 different Adaptive Programs written for Five Men's
Morris
Opening Stage Adaptive
o Contains a Matchbox for each player to select spots
Second Stage Adaptive
o Contains two Matchboxes for each player to move
pieces and the other for removing pieces
Adaptive Results
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Opening Stage Adaptive
o After 60 million runs (On the last 10 million)
 72544 won by Player 1
 44059 won by Player 2
 9883397 end in a draw
 Player 1 has 20% advantage on win/loss
 However most opening stages end in draw
Second Stage Adaptive
o After 10 million runs still dead even
o The program requires more runs to draw a
conclusion.
AI Logic (Minimax and Negamax)
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AI is based on the game theory decision rule of
Minimax and Negamax
Both determine the worth of a game state using a set
of conditions
Efficiently searches through possible states and
presents the best one.
o Negamax differs in how it eliminates certain states
that can not be achieved to increase search speeds
Current State
Next State
Next Next State
1
2
0
6
2
5
9
AI Logic (Scoring)
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Plays the game more intelligently by choosing
the best move from all possible moves for that
game board
Moves are scored based on the resultant game
board
o next to open connection or own piece = +1
o next to opponent's piece = -1
o sets up 2/3 parts of a mill = +2
o blocks opponent's mill = +2
o makes a mill = +3
AI Results
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Player 1 using AI, Player 2 playing randomly
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Both players using AI
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After 1000 runs multiple times, Player 1 wins
roughly about 70% of the time
After 1000 runs multiple times, neither player has
an advantage over the other (around 50% each)
AI will require more improvements and test
runs to get solid results
General Strategy
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Take spots on both rings
Take spots with the most connections
Block your opponent's move in a way that you
don't trap yourself
Try to force your opponent to allow you to
make a mill
o Ex) player 1 takes outside corners and
player 2 tries to block
If possible, set up two potential mills next to
each other so that a mill can be made by
moving back and forth
Future Work
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Improve AI and adaptive learning programs to
be more efficient
o Currently the Adaptive takes too long to run
through the required number of games
Confirm the patterns found apply to Nine Men's
Morris by running the programs on it
Come up with a more detailed strategy that
will handle every situation