A067875 One player's total legal chess moves by piece type on standard chessboard.

668, 1964, 2434, 3248, 5152, 8400

Offset: 1

Author

Rick L. Shepherd, Feb 25 2002

Keywords

• fini
• full
• nonn

Comments

The terms are given in order of increasing numbers of total moves for the six piece types; that is, a(1) Pawn, a(2) Knight, a(3) King, a(4) Bishop, a(5) Rook and a(6) Queen.

The sum of these six terms is 21866, the total number of moves available to White or to Black. Hence 43732 moves, the answer to a question raised in the link below, are available to both players.

Notes: (1) Capturing, for example, a Knight on a particular square counts as a different move from capturing, for example, a Rook on the same square.

(2) Moving a piece to a square without capturing is counted separately from captures by the piece on that square.

(3) The two castling moves are counted as King moves only.

(4) The 14 en passant captures are included in the Pawn moves.

(5) Two-square initial Pawn moves are included.

(6) Pawn promotion on a particular square to a Bishop, for example, counts as a different move from promotion to a Queen on the same square.

(7) Pieces of the same type and color are considered indistinguishable.

(8) Moves causing check, discovered check, double check, checkmate, or stalemate are not distinguished from other moves.

Valid boards and moves require (these two somewhat subtle realizations):

(9) A King cannot capture Pawns on their original squares when they would be attacking the King; this would require the King to have made an illegal move earlier by walking into check.

(10) A King cannot diagonally capture Bishops on their home corner squares - again this would require the King to have made an illegal move earlier. However, a King can diagonally capture Bishops on the other two corners as the Bishop can be in position *after* the King is - via Pawn promotion in this case.

References

• Inspired by a question posed by Tim Krabbé.

Links

• T. Krabbé, Open Chess Diary, Item 105, Mar 29 2001: Never in a trillion

Crossrefs

Cf. A035005 - A035008, A033586 (count the moves per piece type differently).

Formula

a(6) = a(4) + a(5) (Queen moves equal sum of Bishop and Rook moves). Generalizing all terms for n X n chessboards other than 8 x 8 requires defining how many pieces and how many types of pieces are originally on the board and/or can be promoted to, especially because of the way captures are counted.