Orthogonal Latin Square 6*6

Question

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

Answer 1

A modern account of the problem (including an interesting way of solving it) can be found in this paper by Steven Dougherty:

http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=ALLF&pg7=ALLF&pg8=ET&review_format=html&s4=dougherty&s5=coding&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=3&mx-pid=1268565

Another very simple and ellegant proof from Stinson is the following:

http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=ALLF&pg7=ALLF&pg8=ET&review_format=html&s4=stinson&s5=short&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=4&mx-pid=744085

Hope you find them useful.

Answer 2

If I remember correctly, Tarry's original proof used brute force. So there isn't much to mention about the 1900 proof except the fact that it settled Euler's 36 Officer Problem (see Edit2 at the end of this post).

The simplest nonbrute-force proof I know is due to Stinson:

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

He proves the nonexistence in terms of transversal design. The paper is only 4 pages long and accessible to everyone. The other proof Chema mentioned uses coding theory, and it might take a bit longer to digest.

Edit: By "accessible," I mean it's easy to follow. It's behind a paywall, so you might have to resort to library loan or shoot an email to the author if your university doesn't have access to the journal.

Edit2: I couldn't find the original articles by Tarry online or in my university's library. But Tarry's original proof is briefly mentioned in textbook "Design Theory" by C. C. Lindner and C. A. Rodger from CRC Press (on pages 96 and 97 in the first edition and on pages 120 and 123 in the second edition). You can preview the exact lines where the authors say Tarry's proof was an exhaustive search if you search for "Tarry" in the textbook on googlebook:

http://books.google.com/books/about/Design_theory.html?id=KZOTkih8RcIC

If you only need a line or two that briefly explain what the original proof is like on the internet, this may be good enough. The googlebook link is for the first edition of the textbook, by the way.

Answer 3

It looks like you can click through to a copy of Tarry's paper from his French Wikipedia page, http://fr.wikipedia.org/wiki/Gaston_Tarry

Added 4/24/13: Based on the OP's stated reason for asking the question, it's unclear if an answer is still of any use (it's also unclear if the OP has ever returned to look at any of the answers), but for anyone interested, there's a nice history of the problem in a paper by Klyve and Stemkoski which appeared in the College Mathematics Journal, January, 2006, and according to which, "The first actual proof of the 36-officer problem was probably given by Thomas Clausen [Sh], an assistant to Heinrich Schumacher, a nineteenth-century astronomer in Altona." The reference [Sh] is:

[Sh] Letter from Shumacher to Gauss, regarding Thomas Clausen, August 10, 1842. Gauss, Werke Bd. 12, p.16. G¨ottingen, Dieterich, pub. 1863.

They add that "Sadly, although Clausen published over 150 papers during his scientific career, few of them remain, and no record of his alleged proof can be found."