Thirty-six officers problem

The thirty-six officers problem is a mathematical puzzle proposed by Leonhard Euler in 1782.[1][2]

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;[3][4] but the problem has led to important work in combinatorics.[5]

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.

References

  1. ^ Euler, L., Recherches sur une nouvelle espece de quarres magiques (1782).
  2. ^ P. A. MacMahon (1902). "Magic Squares and Other Problems on a Chess Board". Proceedings of the Royal Institution of Great Britain XVII: 50–63. http://books.google.com/?id=cuudxJgEnyEC&pg=PA54&dq=euler+36-officers. 
  3. ^ Tarry, Gaston (1900). "Le Probléme de 36 Officiers". Compte Rendu de l'Association Française pour l'Avancement de Science Naturel (Secrétariat de l'Association) 1: 122–123. 
  4. ^ Tarry, Gaston (1901). "Le Probléme de 36 Officiers". Compte Rendu de l'Association Française pour l'Avancement de Science Naturel (Secrétariat de l'Association) 2: 170–203. 
  5. ^ Dougherty, Steven. "36 Officer Problem." Steven Dougherty's Euler Page. 4 Aug 2006.

External links