Monsky's theorem

In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area.

The problem was posed by Fred Richman in the American Mathematical Monthly in 1965 and was proved by Paul Monsky in 1970.

[2][3][4] Monsky's proof combines combinatorial and algebraic techniques and in outline is as follows: By Monsky's theorem, it is necessary to have triangles with different areas to dissect a square into an odd number of triangles.

Lower bounds for the area differences that must occur to dissect a square into an odd numbers of triangles and the optimal dissections have been studied.

[6][7][8] The theorem can be generalized to higher dimensions: an n-dimensional hypercube can only be divided into simplices of equal volume if the number of simplices is a multiple of n!.

A square can be divided into an even number of triangles of equal area (left), but into an odd number of only approximately equal area triangles (right).