Tiling a triangle into a prime number of congruent triangles

2026/07/26 by Michael Beeson
Mathematics · #math.MG #math.NT

paper · pdf

Abstract

We show that, apart from a few known exceptions, if a triangle is cut into N congruent triangles, then N is not a prime number. The exceptions are cutting an isosceles triangle in half, cutting an equilateral triangle in three, a 3-tiling of a 30-60-90 triangle, and the long-known biquadratic tilings of a certain right triangle, when N is a sum of two squares.

Citations

• Tiling an Equilateral Triangle
• No triangle can be cut into seven congruent triangles
• Tilings of an Isosceles Triangle

Related

• No Prime Tiling of an Isosceles Triangle
• Tiling an Equilateral Triangle
• No triangle can be cut into seven congruent triangles
• Triangle Tiling II: Nonexistence theorems
• Tilings of an Isosceles Triangle