Statistics on chess positions

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

Contents of the 64 squares only. This is what I call a diagram. 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 cooperate 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). Add to this fifty-move-rule information and repetition-of-position information. This information is necessary for competitive chess. For the fifty-move rule, store the number of plies since the last capture or pawn move (a number between 0 and 99). For the threefold-repetition 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. The Forsyth-Edwards Notation for chess positions is yet another definition. It can be described as definition 2, plus fifty-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.

Number of distinct chess positions uniquely realizable all ply 0 1 1 ply 1 20 20 ply 2 400 400 ply 3 1862 5362 ply 4 9825 72078 ply 5 53516 822518 ply 6 311642 9417681 ply 7 2018993 96400068 ply 8 12150635 988187354 ply 9 69284509 9183421888 ply 10 382383387 85375278064 ply 11 1994236773 726155461002

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 game problems. 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 (have a unique solution) to be considered for publication.

Number of distinct chess diagrams uniquely realizable all ply 0 1 1 ply 1 20 20 ply 2 400 400 ply 3 1862 5362 ply 4 9373 71852 ply 5 51323 815677 ply 6 298821 9260610 ply 7 1965313 94305342 ply 8 11759158 958605819 ply 9 66434263 8866424380 ply 10 365037821 81766238574 ply 11 1895313862 692390232505

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 0-10. A summary is given in the table and graph below. It may seem that we can eventually cover all the integers by increasing the ply count, but in 2005 I showed by a counting argument that there exists an n that cannot be obtained no matter the number of plies. In 2014, FIDE introduced the 75-move rule which made chess finite, leading to a simpler proof of the result.

Data on diagrams with n solutions largest n with a diagram lowest n without a diagram ply 0 1 2 ply 1 1 2 ply 2 1 2 ply 3 4 3 ply 4 16 5 ply 5 91 25 ply 6 524 93 ply 7 2899 679 ply 8 16327 3413 ply 9 135024 23993 ply 10 1351762 173609 ply 11 14538568 930853

"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 Homebase proof games for many examples. Click on a number in the table below to access a file with the diagrams.

Number of "at home" diagrams uniquely realizable with 2 solutions all ply 0 1 0 1 ply 1 0 0 0 ply 2 0 0 0 ply 3 0 0 0 ply 4 0 0 1 ply 5 0 0 0 ply 6 0 0 0 ply 7 0 0 9 ply 8 10 12 74 ply 9 41 30 255 ply 10 116 187 1350 ply 11 335 512 4719 ply 12 1111 1522 18535 ply 13 2619 3599 58489 ply 14 6067 9286 189876 ply 15 12788 21063 548129 ply 16 26692 44999 1550081

Mirror-symmetric diagrams

The symmetry considered here is horizontal symmetry with black and white interchanged. See Asymmetric play to symmetric diagrams for some examples. Click on a number in the table below to access a file with the diagrams. Note that an odd ply guarantees asymmetric solutions.

Number of mirror-symmetric diagrams with 1 symmetric solution with 2 symmetric solutions with 1 asymmetric solution with 2 asymmetric solutions with 1 of each all ply 0 1 0 0 0 0 1 ply 1 0 0 0 ply 2 20 0 0 0 0 20 ply 3 8 0 8 ply 4 85 0 0 0 0 260 ply 5 8 11 177 ply 6 372 0 0 0 6 2816 ply 7 9 8 2392 ply 8 1243 0 12 104 17 26925 ply 9 53 109 25843 ply 10 3723 27 110 467 163 232380 ply 11 685 1398 241868 ply 12 12327 134 691 1698 897 1826345 ply 13 3999 5772 2045254 ply 14 34353 442 12366 9125 3415 13226846 ply 15 15084 25330 15787105

Checkmate diagrams

The title 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 (column "all") I'm counting diagrams that are checkmate for at least one game in the specified number of plies. For diagrams that have exactly 1 checkmate solution, I check that the diagram cannot be realized in any other (non-checkmate) way. François Perruchaud showed that the test can fail 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+). Later I found more examples by computer, including 3 shorter ones in 12 plies.

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

Number of checkmate diagrams with 1 checkmate solution with 2 checkmate solutions all uniquely realizable not uniquely realizable ply 0 0 0 0 0 ply 1 0 0 0 0 ply 2 0 0 0 0 ply 3 0 0 0 0 ply 4 0 0 4 4 ply 5 3 0 38 105 ply 6 51 0 25 1251 ply 7 1106 0 1513 26542 ply 8 3813 0 5797 212907 ply 9 47300 0 82349 3555181 ply 10 216420 0 363361 25410051 ply 11 2057581 0 3735018 340122090 ply 12 10276981 3 21039750 2355723056 ply 13 74358924 43 150292121 26514174333

Diagrams with no shorter realization

Often a diagram is achievable in p plies, but in no fewer plies. In that case, a game reaching the diagram in p plies is called a "shortest proof game". In the table below, the column "uniquely realizable" therefore counts the number of dual-free shortest proof game problems. The column "all" is also interesting because each legal diagram appears exactly once in it (indexed by the length of its shortest proof game(s)). This means that the counts in that column eventually drop to zero, and their sum is equal to the number of legal diagrams. As noted on the parent page, the count is known to be non-zero for ply 366.

Number of chess diagrams that cannot be realized in fewer plies uniquely realizable all ply 0 1 1 ply 1 20 20 ply 2 400 400 ply 3 1702 5202 ply 4 8659 69731 ply 5 49401 766337 ply 6 287740 8708079 ply 7 1934794 86540204 ply 8 11569093 880526165 ply 9 65443733 7996545696 ply 10 360231372 73802185449 ply 11 1872156836 616052245142

Tempo diagrams

Diagrams that have a shorter realization can also be interesting. They are especially interesting if the diagram is uniquely realizable in p plies. Imagine the frustration when you're asked for a proof game in p plies, and you can easily do it in fewer plies but the solution in exactly p plies eludes you! One must find a way to waste moves, also called "losing tempo".

The table below counts diagrams that have a unique realization in p plies, but where the shortest possible realization is in p − t plies, where t > 0 is the tempo achieved. The table can be thought of as a breakdown of the numerical difference between the "uniquely realizable" columns from sections "Chess diagrams" and "Diagrams with no shorter realization".

Note that a tempo of 4 is impossible because any solution in p − 4 plies can be turned into 16 solutions in p plies with an initial knight dance, making the diagram not uniquely realizable in p plies. Perhaps surprisingly, some tempos larger than 4 plies are possible (but not multiples of 4). Click on a number in the table below to access a file with the diagrams.

Number of uniquely realizable chess diagrams with a given tempo tempo 1 tempo 2 tempo 3 tempo 5 total ply 0 0 0 0 0 0 ply 1 0 0 0 0 0 ply 2 0 0 0 0 0 ply 3 160 0 0 0 160 ply 4 650 64 0 0 714 ply 5 1786 136 0 0 1922 ply 6 10663 418 0 0 11081 ply 7 29731 788 0 0 30519 ply 8 186637 3426 2 0 190065 ply 9 966404 23492 634 0 990530 ply 10 4719654 86281 501 13 4806449 ply 11 22884520 264422 8060 24 23157026

Below are examples with large tempos constructed by people. As a bonus, starred (*) problems even have a unique solution in p − t plies.

More values of t (and even multiples of 4) can be obtained by changing the initial position (so-called A→B proof game problems). It is even possible to achieve every natural number starting with t=11 (so t=11,12,13,14,...) using the same pair of diagrams A & B.