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.

Without any prior background I looked at an order-6 latin square and replaced the integers with integers from the set {x,y: 1,1;1,2 ... 1,6; 2,2; ...... 6,5; 6,6}.

I found a neat way to present the array.  I had not found a "google' reference until "36 officers" was mentioned.

Even reading one of those I couldn't see what was wrong with my spreadsheet representation for sometime.  Sums of columns of tens and columns of units digits were 21, and seemingly were those for the rows (well they were for the tens but I hadn't verified the units).  Everything appeared to be working fine but on reexamination the  the last 6 integers were duplicates of some already present.

In this case I fit within the group to which broll mentions, those who might be considered "beginners" but I don't mind, I enjoyed the task, even to point of deciding that it was nigh on a hopeless task to try to write a program - the permutations that would have required, and the runtime!

Posted by brianjn on 2012-09-10 04:03:38