Euler and the problem of the 36 officers – 02/14/2023 – Marcelo Viana

In 1782, the Swiss mathematician Leonhard Euler (1707–1783) formulated a puzzle that is somewhat reminiscent of the sudoku pastime. Six army regiments each have six officers of six different ranks. How can these 36 officers be arranged in a 6-by-6 square so that in each row of the square are all the regiments and all the ranks and the same goes for each column?

If we change the number N of regiments and ranks to 3 (9 officers) or 4 (16 officers), solutions are quite easy to find (try it!), and Euler also figured out how to solve the problem when N is 5 (25 officers) or 7 (49 officers). But the case of the 36 officers withstood all his efforts. “After all the work to solve this problem, we were forced to recognize that such an arrangement is absolutely impossible, although we cannot prove this fact”, he lamented.

In fact, the proof took 129 years: it was found by the French mathematician Gaston Tarry in 1901. Another demonstration that the 36 officers problem is impossible was given in 1934 by the British statisticians Ronald Fischer and Frank Yates, whose interest in the question was very curious. : they wanted to statistically study the effect of six different fertilizers on six types of agricultural crops.

To do this, they designed an experiment carried out on a square plot divided into 36 smaller identical squares: in each square a single fertilizer would be used in a single harvest. To minimize the risk of bias, it was desirable that each row and each column contain all crops and all fertilizers. This means that to implement the experiment it would be necessary to solve Euler’s problem!

Despite not having solved it, Euler advanced a lot in the problem, showing that there is always a solution when the number N of regiments and patents is of the form 4n, 4n+1 or 4n+3, where n is an integer number.

N=4n+2 is missing, which includes the N=6 case. For a long time, specialists believed that in these cases there would never be a solution. But this “Euler conjecture” was not correct: in 1959, the Americans RC Bose, SS Shrikhande and ET Parker showed, with the help of computers, that there is always a solution except, curiously, in the case N=6.

But even this case has a quantum solution: in work published a year ago in the scientific journal Physical Review Letters, a group of researchers showed that for N=6 it is possible to organize the officials in the way desired by Euler if we assume that they are in a state of overlapping different charters and different ranks.

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