---

Claim. S. L. Snover, C. Waiveris and J. K. Williams, Rep-tiling for triangles, Discrete Math. 91 (1991), no. 2, 193–200, classify the cuttings of a triangle into $N$ congruent triangles similar to itself. $N$ is $n^2$ (any triangle), $n^2+m^2$ (for non-square $N$, only the right triangle with legs in ratio $n:m$) or $3n^2$ (only the triangle with angles $\pi/6$, $\pi/3$ and $\pi/2$), and every value in these families occurs. The altitude to the hypotenuse splits the right triangle with legs in ratio $n:m$ into two copies similar to it, which are cut into $n^2$ and $m^2$ congruent pieces; the triangle with angles $\pi/6$, $\pi/3$ and $\pi/2$ is cut into three copies similar to it. Harries's manuscripts ([[problems/discrete_geometry/E0634/claims/2026_07_24_harries|first claim page]], [[problems/discrete_geometry/E0634/claims/2026_07_27_harries|second claim page]]) cite the paper for these counts and this construction. Such a cutting is a cutting into congruent triangles, so these values occur for Problem 634.

Covers. The values $n^2+m^2$ and $3n^2$ ($n,m\ge1$) occur. Nothing is claimed about cuttings whose pieces are not similar to the whole triangle.

Depends on. No page of this wiki.

Standing. Accepted on its refereed publication in Discrete Mathematics. The site labels the problem OPEN, so its remark crediting the paper is commentary and reviewed is not listed.