Monsky’s Theorem: Why a Square Cannot Be Split into an Odd Number of Equal-Area Triangles

Imagine an ordinary square. How do you cut it into an even number of triangles of equal area?

Easy. Cut the square along its diagonal and you get two triangles of equal area. Cutting into 4, 6, 8, or any even number of equal-area triangles is just as straightforward.

The Question at the Heart of Monsky’s Theorem

But here is a different question: can you cut a square into an odd number of triangles of equal area — say, 3, 5, or 7?

The answer will surprise you: no, you cannot.

But what is even more surprising than the answer itself is the proof.

Why Geometry Alone Cannot Crack Monsky’s Theorem

The only known proof does not proceed through geometry, as you might initially expect, but through number theory — specifically, through 2-adic numbers.

2-adic numbers are a way of measuring how divisible a number is by 2. The vertices of the triangulation are assigned one of three colors according to a 2-adic rule. It turns out that there will always be a triangle with all three colors whose 2-adic area is “large” — greater than 1.

The Contradiction That Makes Monsky’s Theorem Work

But if you cut the square into an odd number n of equal-area triangles, each triangle has area 1/n — and for odd n, the 2-adic size of this fraction is exactly 1. “Greater than 1” and “exactly 1” cannot hold simultaneously. Contradiction. Therefore, an odd triangulation is impossible.

The Unexpected Beauty of Monsky’s Theorem

A problem that looks purely geometric at first glance turns out to hinge on the structure of points in the plane through the property of divisibility by 2.

That is the kind of unexpected mathematics Monsky’s theorem gives us.