Is it possible to arrange 6 regiments each consisting of 6 officers of different ranks in a 6 by 6 square, so that no rank or regiment will be repeated in any row or column?


Attributed to Leonhard Euler.

L.Euler was a leading Swiss mathematician, who spent most of his adult life in St. Petersburg, Russia.
The following military ranks existed inter alia in Russia's infantry circa 1760:
polkovnik, pod-polkovnik, premier major, second major, poruchik, lieutenant.
The above lines, although redundant, were added to avoid ambiguity
and to ascertain that 6 specific , well-defined and identical sets of 6 ranks are addressed in L.E.'s problem.

(Found in various references on the web,) the answer is No.
In 1901, the French mathematician Gaston Tarry proved
the non-existence of the order six square. Later, with the use of a computer, mathematicians Parker, Bose and Shrikhande proved that the only impossible Euler squares were of order two and six.

Posted by Dej Mar on 2012-09-08 13:13:56