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
Amazons
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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
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
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Exhaustive Search in Amazons
(Snatzke)
Snatzke’s 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.
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Snatzke’s 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
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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.
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Example of a complex canonical form:
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Thermographs for Amazons positions are
relatively simple, by Comparison:
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Complexity of canonical data grows
exponentially with the size of the board, but
complexity of thermographs remains
constant above board size 15
Some Interesting Special Cases
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A surprising nimber, *2 (unexpected in a
partizan game)
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The impact of one square: 7/8 vs. 1v
Experiments in Computer
Amazons (Mueller and Tegos)
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Line Segment Graphs for positions
Defective Territories
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A k-defective territory provides k less moves than
the number of empty squares.
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Determining if a territory proves a certain number
of moves is an NP-complete problem.
Zugzwang Positions in Amazons
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A simple Zugzwang position is defined as a game
a|b where a,b are integers, a < 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. Doesn’t matter on right.
More Complex Zugzwangs
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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:
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{0|-2||2|0}
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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