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Amazons

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Spread quickly on the internet, with yearly programming competitions. ... Zugzwang Positions in Amazons. A simple Zugzwang position is defined as a game a|b ... – PowerPoint PPT presentation

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Title: Amazons


1
Amazons
  • Experiments in Computer Amazons,
  • Martin Mueller and Theodore Tegos, 2002
  • Exhaustive Search in the Game Amazons
  • Raymond Georg Snatzke, 2002
  • Presented by Joel N Paulson

2
Amazons
  • Created by Walter Zamkauskas in Argentina in 1988
  • First published in 1992
  • Spread quickly on the internet, with yearly
    programming competitions.
  • First analyzed for combinatorial game theory by
    Berlekamp in Sums of Nx2 Amazons in 2000

3
Amazons as a Combinatorial Game
  • Fits criteria as a combinatorial game
  • Endgame is a sum of analyzable smaller games
  • Positions can be very difficult to analyze
  • Berlekamp calculated thermographs for 2 x n
    positions with one amazon per player

4
Exhaustive Search in Amazons (Snatzke)
  • Snatzkes Goal Evaluate canonical forms of all
    games with 0 or 1 amazon per player that fit into
    an 11 x 2 board.
  • Approach Program written to analyze all such
    games, ignoring identical positions, starting
    with the smallest.
  • A total of 66,976 unique boards and 6,212,539
    unique positions analyzed.

5
Snatzkes Program
  • Algorithm Essentially just a brute force search
  • Written in Java (JDK 1.1, later JDK 1.3)
  • Run on a 500 Mhz Pentium III with 512 MB RAM
  • Took four months to run the first time, with JDK
    1.1 and some code errors
  • Second try (with JDK 1.3) took one month

6
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7
Results
  • Very complex Canonical Forms for larger positions
  • Berlekamp Proved that depth of the canonical
    subgame tree for an Amazons position can be up to
    ¾ the size of the game board.

8
  • Example of a complex canonical form

9
(No Transcript)
10
  • Thermographs for Amazons positions are relatively
    simple, by Comparison
  • Complexity of canonical data grows exponentially
    with the size of the board, but complexity of
    thermographs remains constant above board size 15

11
Some Interesting Special Cases
  • A surprising nimber, 2 (unexpected in a partizan
    game)
  • The impact of one square 7/8 vs. 1v

12
Experiments in Computer Amazons (Mueller and
Tegos)
  • Line Segment Graphs for positions

13
Defective Territories
  • A k-defective territory provides k less moves
    than the number of empty squares.
  • Determining if a territory proves a certain
    number of moves is an NP-complete problem.

14
Zugzwang Positions in Amazons
  • A simple Zugzwang position is defined as a game
    ab where a,b are integers, a lt b-1
  • Trivial in most games, but will have to be played
    out in Amazons, since it matters who moves first.
  • On the left (below), white will prefer that black
    moves first. Doesnt matter on right.

15
More Complex Zugzwangs
  • Player who moves first must either take their own
    region and give region C to the opponent, or take
    region C and block off their own region
  • 0-220

16
Open Questions/Future Work
  • Do nimber positions greater than 2 exist on a
    single board?
  • 4x4 Amazons has been solved as a win for the
    second player. 5x5 is a first player win. What
    about 6x6?
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