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Claim. Beeson, Laczkovich and Zhang [BLZ26] prove that a triangle can be cut into a positive nonsquare number of congruent triangles if and only if its angles can be labeled $(A,B,C)$ so that one of eight conditions holds: $A=B$ (the isosceles triangles); $C=\pi/2$ with legs in ratio $M/K$ for positive integers $M,K$ with $M^2+K^2$ not a square; the angles are $(\pi/6,\pi/2,\pi/3)$; $C=\pi/3$ and $\sqrt3\tan(A/2)$ is rational; $B=2A$ and $\sqrt3\tan(A/2)$ is rational; $B=2A$ and $\sin(A/2)$ is rational; $C=A/2+B$ and $2\sin(A/4)=M/K$ for positive integers $M,K$ with $2K^2-M^2$ not a square; or $C=2A+B/2$ and $\sqrt3\tan(A/2)$ is rational. The triangles Problem 633 asks for, those that can only be cut into a square number of congruent triangles, are exactly the triangles outside these eight families; by the paper's Corollary 2, outside the isosceles triangles they miss only countably many similarity classes. The statement, the paper's dependency boundary and a rewritten deduction are on Theorem 1 of the source card, whose qualifications (a misidentified reference on p. 9, an unproved remark on p. 16, and the secondary Theorem 32) do not touch the classification.

Acceptance. The site's curator, T. F. Bloom, labels the problem Solved and credits the classification to Beeson, Laczkovich and Zhang [BLZ26] (problem page last edited 2026-04-08, accessed 2026-09-05), which is the reviewed evidence. The result is a preprint, arXiv:2604.03609, first posted 2026-04-04 (v1, 21 pages) and revised to v3 on 2026-08-25 (33 pages), with no journal record found; the Creative Commons Attribution 4.0 license is from the arXiv record. The library's rewritten deductions are this corpus's own reading and award no acceptance; no independent proof review is recorded. The formal-conjectures file for the problem states the question with an unspecified answer and sorry and is not a formalization of this result.