A mathematical chess problem is a mathematical problem which is formulated using a chessboard and

chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...

pieces. These problems belong to

recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...

. The most well-known problems of this kind are the

eight queens puzzle The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; thus, a solution requires that no two queens share the same row, column, or diagonal. There are 92 solutions. ...

and the

knight's tour A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square exactly once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again im ...

problem, which have connection to

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

and combinatorics. Many famous mathematicians studied mathematical chess problems, such as,

Thabit Thabit ( ar, ) is an Arabic name for males that means "the imperturbable one". It is sometimes spelled Thabet. People with the patronymic * Ibn Thabit, Libyan hip-hop musician * Asim ibn Thabit, companion of Muhammad * Hassan ibn Sabit (died 67 ...

, Euler, Legendre and

Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

. 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 M×N boards.

Independence problem

An ''independence problem'' (or ''unguard'') is a problem in which, given a certain type of chess piece (queen, rook, bishop, knight or king), one must find the maximum number that can be placed on a chessboard 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 the

. Problems are further extended by asking how many possible solutions exist. Further generalizations apply the problem to NxN boards. An 8×8 chessboard can have 16 independent kings, 8 independent queens, 8 independent rooks, 14 independent bishops, or 32 independent knights. Solutions for kings, bishops, queens and knights are shown below. To get 8 independent rooks is sufficient to place them on one of main diagonals.

Domination problems

A ''domination'' (or ''covering'') ''problem'' involves finding the minimum number of pieces of the given kind to place on a chessboard such that all vacant squares are attacked at least once. It is a special case of the

vertex cover In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. In computer science, the problem of finding a minimum vertex cover is a classical optimiza ...

problem. The minimum number of dominating kings is 9, queens is 5, rooks is 8, bishops is 8, and knights is 12. To get 8 dominating rooks, it is sufficient to place one on each file. Solutions for other pieces are provided on diagrams below. The domination problems are also sometimes formulated as requiring one to find the minimal number of pieces needed to attack all squares on the board, including occupied ones. For rooks, eight are required; the solution is to place them all on one file or rank. The solutions for other pieces are given below. Domination by queens on the main diagonal of a chessboard of any size can be shown equivalent to a problem in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

of finding a

Salem–Spencer set In mathematics, and in particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression. Salem–Spencer sets are also called 3-AP-free sequences or progression-free sets. They have a ...

, 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.

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 A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square exactly once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again im ...

. 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.

Chess swap problems

In chess swap problems, the whites pieces swap with the black pieces. 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.

Notes

References

* (in German). Some chapters of the book are available online
Евгений Гик "Шахматы и математика"
and a

(in Russian).

External links

by Weisstein, Eric W. from

MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...

.
Chess-Piece Arrangement Problems
by George Jelliss (from The Games and Puzzles Journal).

by Ed Pegg Jr. {{DEFAULTSORT:Mathematical Chess Problem *