# Orthogonal Latin Square 6*6

I need to make remarks about Tarry's Proof for the nonexistence of 6x6 Latin Squares as part of my final exam for a class I'm in. Problem is, I can't find it ANYWHERE on the internet. I can only find minor comments about it that don't explain what he did. Does anyone know where I can find it online? Or does anyone know where I can find a detailed explanation of his proof? if you have other proof for this question please share that... i need to proof of this question

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If you have any more Latin squares questions, please make sure to tag them [latin-square]. –  Douglas S. Stones Apr 13 '13 at 19:29

Tarry, G. Le probleme des 36 officiers. Comptes Rendus Assoc. France Av. Sci. 29, part 2: 170–203, 1900

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It's quite clear that the OP knows what the reference is, and is having trouble accessing it. –  Douglas S. Stones Apr 13 '13 at 19:27
See my answer for a link on the original paper. A physical copy of the journal of the French "Association for l'Avancement des Sciences" is not so easy to find outside France (where most University libraries have it). –  ogerard Apr 14 '13 at 9:08

Tarry's paper is available in digitized form at the online French National Library, http://gallica.bnf.fr/ as well as many volumes of 19th and early 20th century scientific journals.

Here is a link for the first page:

http://gallica.bnf.fr/ark:/12148/bpt6k2011936/f175.image

It is quite a long paper (34 pages) investing potential cases sorted according to various conditions. He starts from general 6x6 latin squares and recognizes that the 9408 individual cases ("permutation carrées" = square permutations = latin squares) can be sorted into 17 types. He coins several names such as "écusson" (small shield, with an heraldic meaning) as a way to present the type of permutations (in term of cycles) of the rows and columns of latin squares. He finds 16 different possible "écussons" and he sorts the 17 types into 3 classes, depending on presence of cycles of order 2.

Hope it will get you started.

If you have trouble with the french, or with using Gallica to download the whole article do not hesitate to ask specific questions.

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Since your after remarks, here's some.

• It's easy enough these days to prove the non-existence on a computer. There are only $12$ main classes to exclude (ref.).

• There's a short proof of this result:

Stinson, A short proof of the nonexistence of a pair of orthogonal latin squares of order six, J. Comb. Theory A, 36 (1984) 373-376

• There's a recent graph theory proof of the non-existence of self-orthogonal Latin squares (Latin squares that are orthogonal to their transpose) of order $6$; see:

Burger, Kidd, and van Vuuren A graph-theoretic proof of the non-existence of self-orthogonal Latin squares of order 6, Disc. Math. (2011) 311, pp. 1223-1228.

PS. If you're desperately after the paper, I suggest you email either Brendan McKay and/or Ian Wanless. They're the most likely people to have obscure papers on Latin squares.

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