Queens, Attack!
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here is a long history of mathematicians and computer scientists thinking about problems related to the game of chess. For example, there have been investigations into tours and domination, on boards of various shapes and dimensions, with unorthodox pieces such as superqueens and archbishops. Here we shall consider only traditional chess pieces—kings, queens, bishops, knights, rooks, and pawns—on an n n chessboard and explore questions of independence. We shall use the word ‘‘attack’’ to describe the square or squares that a given piece can move to and capture a piece on that square. Each piece has its unique mode of attack. A king attacks precisely one square in every direction, including rows, columns, and diagonals, while a pawn generally attacks by moving one square diagonally forward. Knights move in an ‘‘L’’ shape, able to attack any square that can be reached by sliding the piece two squares either horizontally or vertically and then moving one square in the direction perpendicular to that move. The knight’s move is not blocked by any intervening pieces. In contrast to these attacks at a fixed distance, a rook may move and attack a piece along a row or column at any distance, provided that there are no intervening pieces. Bishops do the same, except along the diagonals, while the queen combines the powers of the rook and bishop. As is well known, a regulation game of chess involves a certain setup of these pieces, with the players moving alternately, working to achieve a superior position and capture the opponent’s pieces with the ultimate goal of checkmating the opponent’s king. Many interesting questions arise, however, outside of a traditional game. Suppose, for example, that we fill the board with a uniform army consisting of only one type of piece, so that we have, say, lots of bishops attacking bishops or queens attacking queens. We let them do battle and ask when the fighting will stop, that is, when there will be an equilibrium position on the board with a maximum number of pieces such that no piece is attacking any other piece? Such questions are known as problems of independence: how many pieces of a certain type can be placed on a chessboard such that no two pieces are
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attacking each other? And we may inquire further as to the number of such arrangements.
Fixed Attacks For pieces that attack at a fixed distance—pawns, kings, and knights—finding maximal nonattacking arrangements can be relatively straightforward. Pawns attack only by moving diagonally one square forward, so we may fill every second row with pawns with the next row empty, 2 using n2 pawns altogether for n even. To see that this number is maximal, note that every 2 2 region on the chessboard may contain at most two nonattacking pawns, and so no more than half the board can be filled with nonattacking pawns. ‘‘He’s Only a Pawn in Their Game’’ Pawns are usually omitted from independence problems in recreational mathematics texts possibly because unlike every other piece, pawns can move in only one direction. Be that as it may,
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