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 Created by Walter Zamkauskas in Argentina in 1988 Created by Walter Zamkauskas in Argentina in 1988 First published in 1992 First published in 1992 Spread quickly on the internet, with yearly programming competitions. Spread quickly on the internet, with yearly programming competitions. First analyzed for combinatorial game theory by Berlekamp in “Sums of Nx2 Amazons” in 2000 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 Fits criteria as a combinatorial game Endgame is a sum of analyzable smaller games Endgame is a sum of analyzable smaller games Positions can be very difficult to analyze Positions can be very difficult to analyze Berlekamp calculated thermographs for 2 x n positions with one amazon per player Berlekamp calculated thermographs for 2 x n positions with one amazon per player

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. 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. 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. A total of 66,976 unique boards and 6,212,539 unique positions analyzed.

Snatzke’s Program Algorithm: Essentially just a brute force search Algorithm: Essentially just a brute force search Written in Java (JDK 1.1, later JDK 1.3) Written in Java (JDK 1.1, later JDK 1.3) Run on a 500 Mhz Pentium III with 512 MB RAM 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 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 Second try (with JDK 1.3) took one month

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

Example of a complex canonical form: Example of a complex canonical form:

Thermographs for Amazons positions are relatively simple, by Comparison: 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 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 A surprising nimber, *2 (unexpected in a partizan game) A surprising nimber, *2 (unexpected in a partizan game) The impact of one square: 7/8 vs. 1v The impact of one square: 7/8 vs. 1v

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

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

Zugzwang Positions in Amazons A simple Zugzwang position is defined as a game a|b where a,b are integers, a < b-1 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. 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. On the left (below), white will prefer that black moves first. Doesn’t matter on right.

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: 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|-2||2|0} {0|-2||2|0}

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