Chess Problems by Computer

In December 2003, I wrote a program to search for chess games that are completely determined by the last move. Among other chess problems, the search uncovered this brain-teaser:

Black mates White on Black's 5th move by playing a rook, but the rook itself does not attack White's king. How did the game go? (the solution is unique)

Since then I improved my program and used it to search for other types of chess problems.

Chess problems

• Basic chess problems
• Moves that determine all the previous moves
• Pawn promotion moves that determine the position
• Massacre proof games
• Some published proof games

Chess statistics

As a byproduct of the searches, I also get some statistics on the first few plies.

• Statistics on chess games
• Statistics on chess positions
• Statistics on one-sided chess

Plot of chess statistics

The curves "all games", "distinct positions", and "uniquely realizable diagrams" all start with the sequence 1,20,400, but asymptotically they are very different:

• The number of chess games is unbounded and grows exponentially. Remember that the "draw by repetition of position" rule, and the "fifty-move" rules are not automatic, players can play forever if neither chooses to claim those draws.
• The number of distinct positions is bounded and eventually cycles between the same four integers, each about half the total number of legal chess positions which is estimated to be between 10^43 and 10^50. The longest known dualistic proof game by L. Ceriani and K. Fabel (1947), shows that at least one position is visited for the first time at ply 366.
• The number of uniquely realizable diagrams eventually drops to zero. The longest known dual-free proof game by D. Pronkin & A. Frolkin (1989) shows that it's non-zero at ply 115.