Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 633

claims/: The 1 claim page of Problem 633, one per claimant's result; the problem's standing derives from them.

Statement. Classify those triangles which can only be cut into a square number of congruent triangles.

Status. Solved by the classification of Beeson, Laczkovich, and Zhang, Theorem 1, arXiv:2604.03609v3. The site page accessed records the solution, and the authors' primary paper supplies the argument. The eight families in that theorem are exactly the triangles admitting a nonsquare tiling; all other triangles answer this question. The accepted claim is Beeson–Laczkovich–Zhang 2026.

Source. erdosproblems.com/633, accessed 2026-09-05. Cite as: T. F. Bloom, Erdős Problem #633, https://www.erdosproblems.com/633, accessed 2026-09-05.

References.

[BLZ26] M. Beeson, M. Laczkovich, and Y. X. Zhang, Solution of Erdős problem 633. arXiv:2604.03609v3 (2026).
[So09] Soifer, Alexander, How Does One Cut a Triangle? I. (2009), 15-23.
[So09b] Soifer, Alexander, How Does One Cut a Triangle? II. (2009), 37-39.
[So09c] Soifer, Alexander, Is there anything beyond the solution?. (2009), 47-50.

Formalization. The formal-conjectures statement, defines congruent tilings by triangles with disjoint interiors. Its erdos_633 declaration has an unspecified answer and sorry; it does not formalize the eight-family solution. The same file's historical auxiliary results also contain unproved declarations. No Lean proof of the solution is recorded.

Current assessment

The site accepts the joint paper as the solution, and its proof-claims tab lists no claim. An April 2026 comment on its thread describes a strengthening in preliminary form; the precise statement is the paper's Theorem 3 in its later version.

Beyond the site, the arXiv version history and the authors' publication list and academic page identify the same solution. Searches for later corrections or competing solutions, and for related announcements on X, found no later primary replacement. The manuscript is a preprint with no journal record found; source acceptance and independent proof review are distinct.

No independent proof-review verdict for the classification is recorded on this page.

Progress

Every triangle has a tiling into $n^2$ congruent triangles for each positive integer $n$ , obtained by drawing equally spaced parallels to its sides. An isosceles triangle has a two-tile dissection along its symmetry axis. The classification therefore concerns which additional, nonsquare counts are possible.

The site attributes to Soifer [So09b] a sufficient condition: if the three side lengths and, separately, the three angles are integrally independent, every congruent tiling has square count. It gives the triangle with sides $\sqrt2,\sqrt3,2$ as an example. It also attributes to [So09] the different statement that every triangle can be dissected into $n$ similar triangles when $n\notin\{2,3,5\}$ , and that some triangle admits none of those three counts. These historical results are given as the site states them. Similarity permits different tile sizes and is a different question from congruence.

Laczkovich's earlier angle classification and existence theorems, Beeson's counting equations, and the 2026 rationality theorem of Beeson and Zhang are the inputs to the final classification. Zhang's construction work provided additional explicit tilings and helped motivate the joint solution. The final proof establishes nonsquare counts for the whole relevant tile shapes; it does not require reproducing those particular constructions. The source explains the relationship with Problem 634, which asks about tile counts across triangles.

Known Results

Theorem 1: the exact eight-family classification, with rewritten proof and explicit external dependencies.
Corollary 2: outside the isosceles family, only countably many similarity classes admit any nonsquare tiling.
Theorem 3: a square tiling by a nonsimilar tile requires an isosceles triangle or the stated triquadratic square family.

The source's secondary tile-uniqueness claim and added exact-minimum remark have unresolved proof details, recorded on Theorem 32 and Proposition 29. Neither is used to establish the classification that solves this problem.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (25)

The Beeson–Laczkovich–Zhang classification problems/discrete_geometry/E0633/claims Problem 634 Discrete and Convex Geometry discrete_geometry/beeson_2026_solution_erdos_problem_633 Corollary 2 — Countably many non-isosceles exceptions Lemma 18 — A quartic-to-cubic transformation Lemma 24 — Rational squared tangents at rational angles Proposition 10 — Rational sides with a 60- or 120-degree angle Proposition 13 — The six necessary angle and rationality patterns Proposition 19 — Two incompatible square conditions Proposition 20 — A quartic with no rational square values Proposition 21 — A rational function excluded by curve 144.a1 Proposition 22 — A rational function excluded by curve 36.a2 Proposition 26 — The 60-degree family Proposition 27 — The doubled-angle Group 1 family Proposition 28 — The doubled-angle Group 2 family Proposition 29 — The triquadratic square criterion Proposition 30 — The final Group 2 family Proposition 31 — The second 60-degree tiling shape Proposition 9 — Rational sides in Group 1 Theorem 1 — Classification of triangles admitting nonsquare tilings Theorem 11 — Possible non-isosceles tilings Theorem 3 — Exceptions to nonsquare non-reptilings Theorem 32 — Claimed uniqueness of the nonsimilar tile

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