Mathematical chess problem

Mathematical chess problem is a mathematical problem which is formulated using a chessboard and chess pieces. These problems belong to recreational mathematics. The most known problems of this kind are Eight queens puzzle or Knight's Tour problems, which have connection to graph theory and combinatorics. Many famous mathematicians studied mathematical chess problems, for example, Euler, Legendre and Gauss. Besides finding a solution to a particular problem, mathematicians are usually interested in counting the total number of possible solutions, finding solutions with certain properties, as well as generalization of the problems to N×N or rectangular boards.

Independence problems (or unguards) are a family of the following problems. Given a certain chess piece (queen, rook, bishop, knight or king) find the maximum number of such pieces, which can be placed on a chess board so that none of the pieces attack each other. It is also required that an actual arrangement for this maximum number of pieces be found. The most famous problem of this type is Eight queens puzzle. Problems are further extended by asking how many possible solutions exist. Further generalization are the same problems for NxN boards.

The maximum number of independent kings on an 8×8 chessboard is 16, queens - 8, rooks - 8, bishops - 14, knights - 32. Solutions for kings and bishops are shown below. To get 8 independent rooks is sufficient to place them on one of main diagonals. A solution for 32 independent knights is to place them all on squares of the same color (e.g. place all 32 knights on dark squares).

Another kind of mathematical chess problems is a domination problem (or covering). This is a special case of the vertex cover problem. In these problems it is requested to find a minimum number of pieces of the given kind and place them on a chess board in such a way, that all free squares of the board are attacked by at least one piece. The minimal number of dominating kings is 9, queens - 5, rooks - 8, bishops - 8, knights - 12. To get 8 dominating rooks it is sufficient to place them on any rank, one for each file. Solutions for other pieces are provided on diagrams below.

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