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Math-882-A2 · revision r1 · on Math-882

gpt pro 5.2 attempt on Math-882

No claim recorded · LLM-generated · gpt pro 5.2 ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.

Manuscript

Contents
  1. Problem 634. Find all values $n$ such that there exists a triangle which can be cut into $n$ congruent triangles.
  2. 1) [Formal restatement]
  3. 2) [Quick literature/context check (if browsing is available)]
  4. 3) [Attack plan]
  5. 4) [Work]
  6. 4.1. Constructions giving infinite subfamilies of $\mathcal{N}$.
  7. 4.2. Obstructions.
  8. 5) [Verification]
  9. 6) [Final]
  10. 7) [Completion estimate]

Problem 634. Find all values $n$ such that there exists a triangle which can be cut into $n$ congruent triangles.

1) [Formal restatement]

Determine the set \[ \mathcal{N}:=\{n\in\mathbb{Z}_{\ge 1}: \exists\text{ a triangle }T\text{ that can be partitioned into }n\text{ congruent triangles}\}. \] Equivalently, decide for each $n$ whether at least one triangle admits an $n$-triangle congruent dissection.

2) [Quick literature/context check (if browsing is available)]

From the prompt and widely cited tiling constructions:
  • $n=k^2$ always lies in $\mathcal{N}$ (by the explicit construction proved in Problem 635).
  • Various additional infinite families are known to lie in $\mathcal{N}$ (e.g. $2k^2,3k^2,6k^2$, and $k^2+m^2$ via right-triangle similarity constructions).
  • Beeson (as cited in the prompt) shows that $7,11\notin\mathcal{N}$.
  • Recent work (e.g. Zhang's 2025/2026 arXiv preprint cited on the Erdős Problems site) studies tilings where the tile has a $2\pi/3$ angle, giving further structure and conjectures, but not a complete classification of $\mathcal{N}$.

3) [Attack plan]

  1. Separate the problem into (A) "reptile" (tile similar to the whole) cases and (B) non-similar congruent tilings.
  2. Record unconditional constructions that realize large families of $n$.
  3. Record unconditional obstructions (e.g. $n=7,11$) and known necessary conditions from angle/edge arithmetic.
  4. Explain why closing the gap to an exact description of $\mathcal{N}$ remains open.

4) [Work]

4.1. Constructions giving infinite subfamilies of $\mathcal{N}$.
  • Squares. For every $k\ge 1$, there exists a triangle cut into $k^2$ congruent triangles (Problem 635, §4.1).
  • Doubles of squares. Using the same parallelogram grid but taking the entire parallelogram rather than half gives a dissection of a parallelogram into $2k^2$ congruent triangles; choosing a triangular half of a suitable parallelogram and refining yields $2k^2\in\mathcal{N}$.
  • Sums of two squares (right-triangle similarity). There are standard "reptiling" constructions for right triangles where a larger similar right triangle is formed from a rectangular array of congruent right-triangle tiles, yielding tile counts of the form $k^2+m^2$.
  • Multiples $3k^2$ and $6k^2$. Equilateral/60--60--60 and 30--60--90 angle structures allow triangular-lattice style constructions that yield $3k^2$ and $6k^2$ families.
(Here the point is not the uniqueness of the construction, but the existence of triangles realizing these $n$.)
4.2. Obstructions.
The prompt reports that $7,11\notin\mathcal{N}$ (Beeson). Beyond these sporadic nonexistence results, general necessary conditions often come from:
  • Angle constraints: each angle of the big triangle is a sum of tile angles at boundary vertices; and the multiset of tile angles around interior vertices must sum to $2\pi$.
  • Edge constraints: each side length of the big triangle is a nonnegative integer combination of the tile side lengths, respecting which tile side can lie on which boundary side.
These are powerful enough to rule out some small $n$, but (at present) do not yield a complete characterization of $\mathcal{N}$.

5) [Verification]

  • The existence of $k^2\in\mathcal{N}$ is rigorously verified by the explicit $k^2$-tiling construction in Problem 635.
  • The additional families listed are consistent with standard tiling constructions; the key unresolved step is proving that no other $n$ occur beyond the known families and sporadic cases, or else producing new constructions for the remaining $n$.

6) [Final]

LABEL: UNRESOLVED
SUBLABEL: Partial progress (recorded constructions/obstructions) + full classification open.
I can certify large infinite families inside $\mathcal{N}$ (in particular all squares) and note known sporadic nonexistence ($7,11$), but I do not obtain a complete description of all $n$ for which some triangle admits an $n$-congruent-triangle dissection.

7) [Completion estimate]

About 25% complete: the square-family is proved constructively and the known landscape is organized, but the "if and only if" characterization of $\mathcal{N}$ is not reached.