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 A278829 Pawn's moves in chess: possible difference between origin and destination square when the squares are numbered sequentially row by row. 5

%I #13 Dec 04 2021 22:42:10

%S -16,-9,-8,-7,7,8,9,16

%N Pawn's moves in chess: possible difference between origin and destination square when the squares are numbered sequentially row by row.

%C Let the squares of a standard (8 X 8) chessboard be numbered sequentially from 1 to 64, row by row (e.g., a1 = 1, b1 = 2, ..., a2 = 9, ..., h8 = 64). Let X be the number of a square a pawn stands on, and Y the number of a square to which it can move. Then this sequence lists all possible values of Y-X.

%C The terms are independent of the starting value used for numbering, for example, one could also use number = column + 8 * row, where row and column range from 1 to 8 (thus the numbers from 9 to 72), or from 0 to 7 (with squares numbered from 0 to 63). However, the board must be numbered row by row, i.e., moving up or down one square must yield a difference of +- 8, and moving left or right (which a pawn can't do) must give a difference of +- 1. (All directions mentioned here correspond to a standard vertical diagram with White starting at the bottom and Black starting on the top.)

%C The positive values correspond to white pawns moving upwards, the negative values to black pawns moving downwards.

%e If a white pawn moves one row up, e.g., from c2 to c3, this yields a difference Y - X = +8. If a black pawn takes a white piece to the lower right, this yields a difference Y - X = -8 +1 = -7.

%e The steps of +- 16 correspond to pawns moving two squares ahead, starting from their initial position on the 2nd row (from bottom for white, from top for black).

%Y Cf. A278824 - A278828 (analog for Knights, ..., Kings).

%Y This is a subsequence of A278827: Queen's moves.

%K sign,easy,fini,full

%O 1,1

%A _M. F. Hasler_, Nov 28 2016

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