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A318266 Number of legal chess positions with a total of n black and white pieces, reduced for symmetry. 0
462, 368079, 125246598, 25912594054, 3787154440416, 423836835667331, 38176306877748245 (list; graph; refs; listen; history; text; internal format)



a(n) is the number of legal chess positions with n pieces on the board.

See Kirill Kryukov's webpage (in the Links section) for the explanations of these computations and of symmetries taken into account to avoid double counting, and for the definition of "legal" position. Note that some "legal" positions are not reachable in a game: e.g., white Ka1 vs. black Kc3 + Qc2 + Qb3 (black to move) is counted in a(4), although it is impossible to reach such a position.

The rapid growth of this sequence illustrates why exhaustive endgame databases at chess quickly require several hundred terabytes, even with a small number of pieces on the board.


Table of n, a(n) for n=2..8.

Kirill Kryukov, Number of Unique Legal Positions in chess endgames


There are 3612 ways to place 2 kings on a chessbard so that they are not in check, which would not be a legal position: this decomposes as 3612 = 4*(64-4) [when white king is in a corner] + 6*4*(64-6) [when white king is on a side] + (64-28)*(64-9) [when white king is not touching any side of the board]. Killing "isomorphic" positions due to symmetries reduces this number to 462 remaining different positions. Symmetries roughly lead to a division by 8, but not exactly, because they have some fixed points (e.g., the position with kings on A1 and H8 is a fixed point for the diagonal symmetry). Therefore a(2)=462.


Cf. A048987 for another enumeration of chess positions.

Sequence in context: A035847 A140905 A208619 * A295432 A213406 A294853

Adjacent sequences:  A318263 A318264 A318265 * A318267 A318268 A318269




Cyril Banderier, Aug 22 2018



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Last modified November 27 20:49 EST 2021. Contains 349395 sequences. (Running on oeis4.)