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A318266 Number of legal chess positions with a total of n black and white pieces, reduced for symmetry. 0

%I #12 Sep 01 2018 11:48:17

%S 462,368079,125246598,25912594054,3787154440416,423836835667331,

%T 38176306877748245

%N Number of legal chess positions with a total of n black and white pieces, reduced for symmetry.

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

%C 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.

%C 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.

%H Kirill Kryukov, <a href="http://kirill-kryukov.com/chess/nulp/method.html">Number of Unique Legal Positions in chess endgames</a>

%e 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.

%Y Cf. A048987 for another enumeration of chess positions.

%K nonn,fini,hard

%O 2,1

%A _Cyril Banderier_, Aug 22 2018

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