amazons n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Amazons PowerPoint Presentation
Download Presentation
Amazons

Loading in 2 Seconds...

play fullscreen
1 / 16

Amazons - PowerPoint PPT Presentation


  • 225 Views
  • Uploaded on

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 First published in 1992

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

Amazons


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
amazons

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

amazons1
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
amazons as a combinatorial game
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
exhaustive search in amazons snatzke
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.
snatzke s program
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
results
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.
slide10
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
some interesting special cases
Some Interesting Special Cases
  • A surprising nimber, *2 (unexpected in a partizan game)
  • The impact of one square: 7/8 vs. 1v
defective territories
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.
zugzwang positions in amazons
Zugzwang Positions in Amazons
  • 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
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|-2||2|0}
open questions future work
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?