Impossible Task: Dissecting a Square into an Odd Number of Equal-Area Triangles

This article explores a deceptively simple geometric problem: can a square be dissected into any number of triangles with equal area? The answer, surprisingly, is complex. In 1970, Paul Monsky proved that it's impossible to dissect a square into an odd number of equal-area triangles. The proof cleverly combines Sperner's Lemma and 2-adic valuations. By ingeniously coloring the vertices of the triangles and analyzing the number of factors of 2 in the triangle's area using 2-adic valuation, a contradiction is reached, proving the proposition.