Statistics on chess positions

Chess positions are more tricky to define than chess games. I can see at least 4 possible definitions:

1. Contents of the 64 squares only. This is what I call a diagram.
2. Add whose turn it is, castling rights, and any en passant square. This is what I call a position. This information is sufficient for chess enumeration and games in which the two players contribute to achieve a goal ("help" stipulations). This definition is fundamental to chess, and is used to decide whether two chess positions are the same (FIDE Laws of Chess, Article 9.2). To avoid unnecessary duplication of positions, en passant availability is noted only if the en passant capture is possible (by a legal move).
3. Add to this 50-move-rule information and repetition-of-position information. This information is necessary for competitive chess. For the "50 move" rule, store the number of plies since the last capture or pawn move (a number between 0 and 99). For the "draw by repetition of position" rule, store every move since the last "irreversible" move (like a capture or a pawn move). Because of this, the concept isn't really interesting, it's almost like representing a position with the list of every move since the beginning.
4. The Forsyth-Edwards Notation for chess positions is yet another definition. It can be described as definition 2, plus 50-move-rule information (but no repetition-of-position information). In addition it stores the move number of the game. An en passant square is noted even if no en passant capture is possible.

As you can see many definitions are possible. This page discusses definitions 2 and 1 (in this order).

Note that neither "position" nor "diagram" has a standard definition in chess literature, so when you read something outside of this page you should focus on the intent and not on the actual word used. For example, chess problems are implictly problems about what I call diagrams because castling and en passant information is missing. Recovering this information is sometimes even the problem!

Chess positions

A chess position is "uniquely realizable" if there is only one chess game that leads to the position in the specified number of plies.

Chess diagrams

A chess diagram is "uniquely realizable" if there is only one chess game that leads to the diagram in the specified number of plies. In the language of chess problems, these are called "dual-free proof games". A "proof game" is a legal (though possibly weird) chess game reaching a given diagram, thereby proving that the diagram is legal (reference). A chess problem must usually be dual-free to be considered for publication.

Diagrams with n solutions

Sometimes a diagram with multiple solutions can be fun:

François Labelle & computer

Retros mailing list, January 19, 2004

Proof game in 3.5 moves (2004 solutions)

Which values of n can be obtained in this way? I know the answer for plies 0-8. A summary is given in the table and graph below:

"At home" diagrams

A chess diagram is called "at home" if all the surviving pieces are apparently on their start squares (aka "deletion", "chez soi"). See "At Home" proof games for many examples. Click on a number in the table below to access a file with the diagrams.

Mirror-symmetric diagrams

The symmetry considered here is horizontal symmetry with black and white interchanged. Mirror-symmetric diagrams are interesting when the ply count is odd. See Asymmetric play to symmetric diagrams for some examples. Click on a number in the table below to access a file with the diagrams.

Checkmate diagrams

The subject says it all: the diagram shows a checkmate. Actually it's more tricky than it looks: checkmate is a property of "position", not diagram, and it is possible for the same diagram to be checkmate or not checkmate depending on what the last move was. For example:

So technically in the table below I'm counting diagrams that are checkmate for at least one game (in the specified number of plies). For diagrams that are "uniquely realizable" or "with 2 solutions" I make sure that the diagram cannot be realized in any other (non-checkmate) way. I didn't find any example where this mattered in 11 plies or less, but François Perruchaud showed that it matters at ply 13: there is only one way to reach the diagram above in 6.5 moves with checkmate (1.e3 h5 2.Bc4 h4 3.Bxf7+ Kxf7 4.Qf3+ Kg6 5.Qf4 Kh5 6.g3 g6 7.g4#), but the diagram is not uniquely realizable because there are 4 other ways to reach the diagram in 6.5 moves without checkmating (for example 1.e3 h5 2.Bc4 h4 3.Bxf7+ Kxf7 4.Qf3+ Kg6 5.Qe4+ Kh5 6.Qf4 g6 7.g4+).

Click on a number in the table below to access a file with the diagrams.

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