Thirty-six officers problem
Encyclopedia
The thirty-six officers problem is a mathematical puzzle proposed by Leonhard Euler
in 1782.
The problem asks if it is possible to arrange 6 regiments consisting of 6 officers each of different ranks in a 6 × 6 square so that no rank or regiment will be repeated in any row or column. Such an arrangement would form a Graeco-Latin square
. Euler correctly conjectured there was no solution to this problem, and Gaston Tarry
proved this in 1901; but the problem has led to important work in combinatorics
.
Besides the 6 by 6 case the only other case where the equivalent problem has no solution is the 2 by 2 case, i.e. when there are 4 officers.
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
in 1782.
The problem asks if it is possible to arrange 6 regiments consisting of 6 officers each of different ranks in a 6 × 6 square so that no rank or regiment will be repeated in any row or column. Such an arrangement would form a Graeco-Latin square
Graeco-Latin square
In mathematics, a Graeco-Latin square or Euler square or orthogonal Latin squares of order n over two sets S and T, each consisting of n symbols, is an n×n arrangement of cells, each cell containing an ordered pair , where s is in S and t is in T, such that every row and every column contains...
. Euler correctly conjectured there was no solution to this problem, and Gaston Tarry
Gaston Tarry
Gaston Tarry was a French mathematician. Born in Villefranche de Rouergue, Aveyron, he studied mathematics at high school before joining the civil service in Algeria....
proved this in 1901; but the problem has led to important work in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
.
Besides the 6 by 6 case the only other case where the equivalent problem has no solution is the 2 by 2 case, i.e. when there are 4 officers.
External links
- Euler's Officer Problem at Convergence
- 36 cube A (solvable) puzzle that uses this problem.