Mathematical chess problem

A 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, Thabit, Euler, Legendre and Gauss.[1] 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

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.[2] 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).

abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
16 independent kings
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
14 independent bishops
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
8 independent queens

Domination problems

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.

abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
9 dominating kings
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
5 dominating queens
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
8 dominating bishops
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
12 dominating knights

The domination problems are also sometimes formulated as to find the minimal number of pieces, which attack all squares on the board, including occupied ones.[3] The solution for rooks is to place them all on one of files or ranks. The solutions for other pieces are given below.

abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
12 kings attack all squares
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
5 queens attack all squares
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
10 bishops attacking all squares
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
14 knights attacking all squares

Domination by queens on the main diagonal of a chessboard of any size can be shown equivalent to a problem in number theory of finding a Salem–Spencer set, a set of numbers in which none of the numbers is the average of two others. The optimal placement of queens is obtained by leaving vacant a set of squares that all have the same parity (all are in even positions or all in odd positions along the diagonal) and that form a Salem–Spencer set.[4]

Piece tour problems

These kinds of problems ask to find a tour of certain chess piece, which visits all squares on a chess board. The most known problem of this kind is Knight's Tour. Besides the knight, such tours exists for king, queen and rook. Bishops are unable to reach each square on the board, so the problem for them is formulated to reach all squares of one color.[5]

Chess swap problems

In chess swap problems, the whites pieces swap with the black pieces.[6] This is done with the pieces' normal legal moves during a game, but alternating turns is not required. For example, a white knight can move twice in a row. Capturing pieces is not allowed. Two such problems are shown below. In the first one the goal is to exchange the positions of white and black knights. In the second one the positions of bishops must be exchanged with an additional limitation, that enemy pieces do not attack each other.

Knight swap puzzle
Bishop swap puzzle

See also

Notes
  1. Gik, p.11
  2. Gik, p.98
  3. Gik, p.101.
  4. Cockayne, E. J.; Hedetniemi, S. T. (1986), "On the diagonal queens domination problem", Journal of Combinatorial Theory, Series A, 42 (1): 137–139, doi:10.1016/0097-3165(86)90012-9, MR 0843468
  5. Gik, p. 87
  6. https://www.chess.com/forum/view/fun-with-chess/knight-swap-puzzle

References
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