Reference request: Euler's 36 officer problem

Question

I am recently reading MacNeish's 1922 paper "Euler Squares," in which he mentions that Euler's 36 officer problem has been proven first by Tarry in 1901 via analyzing all possible cases. However, MacNeish then wrote on page 1 that

A geometric proof by methods of Analysis Situs has been given by J. Petersen (Annuaire des Mathématiciens, 1901-02, pp. 413-426).

so I wonder whether there are any English publications that explain Peterson's solution of Euler's 36 officer problem.

Answer 1

I saw the original paper of Petersen in http://prosopomaths.ahp-numerique.fr/annuaire-laisant, but it's in French.

Knuth says https://www.math.uci.edu/~brusso/[14]BosShrParCJM1960.pdf this is the first rigorous proof. It's too long for a comment. I updated it as an answer. I don't know if Bose, etc's proof is brute-force approach or not. But I'am wondering what's the "flaw" in Petersen's proof.

And the following references is quoted from Knuth's The art of computer programming Volume 4, Chapter 1, page 7:

Euler’s conjecture about the remaining cases n = 10, n = 14, ... was “proved” three times, by J. Petersen [Annuaire des mathématiciens (Paris: 1902), 413–427], by P. Wernicke [Jahresbericht der Deutschen Math.-Vereinigung 19 (1910), 264–267], and by H. F. MacNeish [Annals of Math. (2) 23 (1922), 221– 227]. Flaws in all three arguments became known, however; and the question was still unsettled when computers became available many years later. One of the very first combinatorial problems to be tackled by machine was therefore the enigma of 10 × 10 Græco-Latin squares: Do they exist or not?

In 1957, L. J. Paige and C. B. Tompkins programmed the SWAC computer to search for a counterexample to Euler’s prediction. They selected one particular 10×10 latin square “almost at random,” and their program tried to find another square that would be orthogonal to it. But the results were discouraging, and they decided to shut the machine off after five hours. Already the program had generated enough data for them to predict that at least 4.8 × 10 11 hours of computer time would be needed to finish the run!

Shortly afterwards, three mathematicians made a breakthrough that put latin squares onto page one of major world newspapers: R. C. Bose, S. S. Shri- khande, and E. T. Parker found a remarkable series of constructions that yield orthogonal n×n squares for all n > 6 [Proc. Nat. Acad. Sci. 45 (1959), 734–737, 859–862; Canadian J. Math. 12 (1960), 189–203]. Thus, after resisting attacks for 180 years, Euler’s conjecture turned out to be almost entirely wrong.

Their discovery was made without computer help. But Parker worked for UNIVAC , and he soon brought programming skills into the picture by solving the problem of Paige and Tompkins in less than an hour, on a UNIVAC 1206 Military Computer. [See Proc. Symp. Applied Math. 10 (1960), 71–83; 15 (1963), 73–81.]