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March 16

Dividing a square into triangles

Maybe I am missing something obvious, but here is an apparently simple problem that is baffling me. The problem is: divide a square into an odd number of triangles, each with the same area (the triangles do not need to be congruent). The problem is easily solved if we replace "odd" with "even", or replace "triangles" with "rectangles" or even "irregular pentagons". But I can't find a solution for an odd number of triangles, nor can I find a proof that this is impossible. Any thoughts ? Gandalf61 (talk) 08:59, 16 March 2012 (UTC)[reply]

Sperner's lemma and 2-adic valuation proves that it cannot be done. A search for those should give you the answer. 84.197.178.75 (talk) 09:58, 16 March 2012 (UTC)[reply]

Thank you ! Following your hints I found that this is known as Monsky's theorem, it was proved in 1970 by Paul Monsky, there is an explanation of the proof here, and it also appears in Chapter 20 of the latest edition of Aigner and Ziegler's Proofs from THE BOOK (although not, alas, in my 1999 edition). ~~And we don't seem to have a Wikipedia article on this !~~ Anyway, I am somewhat relieved that the proof is not trivial. Gandalf61 (talk) 11:08, 16 March 2012 (UTC)[reply]

Done Thank you

There's also a 171 page pdf presentation/slide show that gives every step. www.math.osu.edu/~fowler.291/teaching/talks/cutting.pdf 84.197.178.75 (talk) 15:00, 17 March 2012 (UTC)[reply]