Monsky's theorem

Monsky's theorem states that a square cannot be partitioned into an odd number of triangles of equal area.

Connect

2-adic valuation: The exponent of 2 in a nonzero rational number, extended to a valuation on the 2-adic numbers. Its parity divides coordinates into classes used to color points in Monsky's proof.
Triangulation: A subdivision of a geometric region into simplices whose intersections are shared faces. The theorem concerns a triangulation of the square using triangles.
Paul Monsky: An American mathematician known for proving that a square cannot be divided into an odd number of equal-area triangles. He established the theorem in 1970 using a valuation-based coloring argument.
Monsky's theorem for rectangles: The extension that a rectangle cannot be partitioned into an odd number of equal-area triangles. It shows the obstruction is not limited to squares.
2-adic numbers: A completion of the rational numbers under the absolute value determined by powers of 2. Their arithmetic supplies the valuation that drives the proof's coloring.
Triangle: A polygon with three sides and three vertices. The pieces in the partition are triangles, measured by their areas.
Dissections of polygons: Problems and methods concerning partitions of polygons into smaller polygonal pieces. Monsky's result is a striking restriction within polygon dissection theory.
Equal-area triangulation of a square: A triangulation of a square whose triangles all have the same area. Such triangulations exist for every positive even number of triangles, unlike odd counts.
Sperner's lemma: A combinatorial lemma guaranteeing a small simplex with every color under specified boundary-coloring conditions. The proof uses a colored-triangle principle closely related to Sperner's lemma.
Area: A measure of the size of a two-dimensional region. Equal areas imply that each triangle occupies the square's area divided by the number of pieces.