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

462, 368079, 125246598, 25912594054, 3787154440416, 423836835667331, 38176306877748245

Offset: 2

Author

Cyril Banderier, Aug 22 2018

Keywords

• nonn
• fini
• hard

Comments

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.

Examples

			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.
		

Links

• Kirill Kryukov, Number of Unique Legal Positions in chess endgames

Crossrefs

Cf. A048987 for another enumeration of chess positions.