Problem 633 — Visual explainer (undergrad)
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Problem statement
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Classify those triangles which can only be cut into a square number of congruent triangles.

Picture
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[Given objects + constraints]  --->  [Count / structure]  --->  [Show bound / existence]

Conjecture / Task: what to look at
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- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Benchmarks / known results (from comments)
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- It is easy to see (see for example ) that any triangle can be cut into $n^2$
  congruent triangles (for any $n\geq 1$).
- Soifer proved that there exists at least one triangle (e.g.
- one with sides $\sqrt{2},\sqrt{3},\sqrt{4}$) which can only be cut into a
  square number of congruent triangles.
- (In fact Soifer proves that any triangle for which the angles and sides are
  both integrally independent has this property.) Soifer proved that if we
  relax congruence to similarity then every triangle can be cut into $n$
  similar triangles when $n\neq 2,3,5$ and there exists a triangle that cannot
  be cut into $2$, $3$, or $5$ similar triangles.