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Nine Men's Morris. Martin Boyd Christopher Hirunthanakorn. Game Overview. Two player game RULES Players alternate turns placing pieces on the board If a mill is formed, player may remove an opponent's piece mill - three pieces formed along a line

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Nine men s morris

Nine Men's Morris

Martin Boyd

Christopher Hirunthanakorn

Game overview
Game Overview

  • Two player game


    • Players alternate turns placing pieces on the board

    • If a mill is formed, player may remove an opponent's piece

      • mill - three pieces formed along a line

    • After both players place nine pieces, players move their pieces to any free adjacent spot

    • The game is over when a player has less than 3 pieces or no legal moves remain

Example continued
Example (continued)

Game classification
Game Classification

  • Determinate

  • Zero-sum

  • Symmetric

  • Perfect Information

  • Sequential

  • Normal


  • One of the oldest games played to date

    • 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
Research Goals/Questions

  • 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
Analyzing the Game

  • 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
Previous Publications

  • Ralph Gasser (Swiss computer scientist)

    • 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
Five Men's Morris

  • Players have 5 pieces instead of 9

  • 16 spots instead of 24

Game states and combinatorics
Game States and Combinatorics

  • 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

    • About 1.74 * 10^11 states based on possible combinations of placement (16*15*14*13*12*11*10*9*8*7*6)

    • 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
Basic Program Structure













Basic program structure1
Basic Program Structure

  • Data of the Game Board is stored in 3 arrays

    • Basic Array


    • Mill Array


    • Connection Array


Adaptive program
Adaptive Program

  • Runs the Game MANY times

  • Contains Matchboxes that punish a player if that player loses thus not repeating the same mistake twice.






















Adaptive program1
Adaptive Program

  • 2 different Adaptive Programs written for Five Men's Morris

  • Opening Stage Adaptive

    • Contains a Matchbox for each player to select spots

  • Second Stage Adaptive

    • Contains two Matchboxes for each player to move pieces and the other for removing pieces

Adaptive results
Adaptive Results

  • Opening Stage Adaptive

    • 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

    • After 10 million runs still dead even

    • The program requires more runs to draw a conclusion.

Ai logic minimax and negamax
AI Logic (Minimax and Negamax)

  • 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.

    • Negamax differs in how it eliminates certain states that can not be achieved to increase search speeds

Current State

Next State




Next Next State





Ai logic scoring
AI Logic (Scoring)

  • 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

    • next to open connection or own piece = +1

    • next to opponent's piece = -1

    • sets up 2/3 parts of a mill = +2

    • blocks opponent's mill = +2

    • makes a mill = +3

Ai results
AI Results

  • Player 1 using AI, Player 2 playing randomly

    • After 1000 runs multiple times, Player 1 wins roughly about 70% of the time

  • Both players using AI

    • 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
General Strategy

  • 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

    • 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
Future Work

  • Improve AI and adaptive learning programs to be more efficient

    • 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