Monsky's Theorem

\( ewcommand{\RR}{\Bbb R} ewcommand{\QQ}{\Bbb Q} ewcommand{\ZZ}{\Bbb Z}\) For which \(n\) can you cut a square into \(n\) triangles of equal area? This question appears quite simple; it could have been posed to the Ancient Greeks. But like many good puzzles, it is a remarkably stubborn one. It was first solved in 1970, by Paul Monsky. Despite the completely geometric nature of the question, his proof relies primarily on number theory and combinatorics! There is a small amount of algebraic machinery involved, but his proof is quite accessible, and we will describe it below. If you have a napkin on hand, it should be straightforward to come up with a solution for \(n = 2\) and \(4\). A little more thought should yield solutions for any even \(n\). One such scheme is depicted below: But when \(n\) is odd, you will have considerably more trouble. Monsky’s theorem states that such a task is, in fact, impossible. Monsky's Theorem The unit square cannot be dissected into an odd numbe ... Read full article.