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# Department of Science, Math, & Technology

E = mcï¿½
Dissecting a square into triangles
Posted: Posted January 27th, 2018
Edited January 27th, 2018 by The Fly
 Take any square. If you cut it along a diagonal you get two triangles, both of equal area (half that of the square). Each of these triangles can be cut in two further triangles of half the area, so you end up with 4 triangles in all, dissected from the square and which all have the same area. When you do this you have an even number of triangles. It turns out that that there is a theorem that says that you can't dissect a square to get an odd number of triangles all of the same area. It's probably not hard to show that you can't do it with 3 triangles, but it's not so easy to prove it for general odd numbers. (I don't have a proof, by the way.) Anyone hear about this problem? Don't know its name. Have fun, The Fly ===================================
There are 2 Replies

Look's like the general solution is called Monsky's Theorem: https://en.wikipedia.org/wiki/Monsky%27s_theorem. I would not have thought of that proof. Look's like that's the only known proof right now, too!

Posted January 27th, 2018 by EN

Yes, thanks, I knew it was out there, but thought it's an interesting result to share. I heard too that it has a generalization for higher dimensions, like cubes etc, which your citation mentions as well.

Here, I found Monsky's original paper/proof:

http://ieee.scripts.mit.edu/urgewiki/images/0/00/Monsky.pdf

where he shows that in fact the areas of the dissected triangles satisfy some polynomial relation. His proof is just about 2 pages.

Posted January 28th, 2018 by The Fly
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