Thirty-six officers problem
- Thirty-six officers problem
The thirty-six officers problem is a mathematical puzzle proposed by Leonhard Euler in 1782. [Euler, L., "Recherches sur une nouvelle espece de quarres magiques" (1782).] [cite journal | url = http://books.google.com/books?id=cuudxJgEnyEC&pg=PA54&dq=euler+36-officers&as_brr=1&ei=f7m_RoiIL4LIoAKHrajwBQ | journal = Proceedings of the Royal Institution of Great Britain | title = Magic Squares and Other Problems on a Chess Board | author = P. A. MacMahon | year = 1902 | volume = XVII | pages = 50–63]
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; [cite journal | author = G. Terry | title = Le Probléme de 36 Officiers | journal = Comptes Rendu de l' Association Française pour l' Avancement de Science Naturel | volume = 1 | pages = 122–123 & 2170–2203 | year = 1900–1901 | url = http://books.google.com/books?id=qzkDAAAAIAAJ&q=36+%22Recherches+sur+une+nouvelle+espece+de+quarres+magiques%22&dq=36+%22Recherches+sur+une+nouvelle+espece+de+quarres+magiques%22&ei=Y7e_Rp2cBKHgoAKI0tXvBQ&pgis=1] but the problem has led to important work in combinatorics. [Dougherty, Steven. "36 Officer Problem." [http://academic.scranton.edu/faculty/doughertys1/euler.htm Steven Dougherty's Euler Page] . 4 Aug 2006.]
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
External links
* [http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1434&bodyId=1599 Euler's Officer Problem] at Convergence
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