One great problem, three kinds of immersion
From a story to your own conjecture
Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.
- 1 · DiscoverMeet the questionLearn when it appeared and why it still holds people’s attention.
- 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
- 3 · DevelopYou are hereBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
Inscribed Square Problem — Does Every Closed Curve Inscribe a Square?
Problem
For any continuous simple closed curve in the plane, must there be a square with all four vertices on ? Posed by Toeplitz (1911), open for ~115 years for general curves.
Why it matters
Circles, triangles, polygons all clearly admit inscribed squares. But for highly irregular (near-fractal) curves, do squares always hide inside? The 4-point simultaneous constraint creates topological obstructions that Jordan-curve methods cannot resolve.
Progress so far
Stromquist (1989) proved inscribed squares for sufficiently smooth curves. Greene & Lobb (2020, Annals) proved every aspect ratio of rectangle inscribes in any smooth Jordan curve — a major advance. But the general continuous case remains open. Tao called it "the most natural-sounding open problem in elementary geometry."
Further reading
💡 Explore together, one line at a time(0 contributions)
Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.
What did you notice?
You do not need a complete proof. A small observation can open the next path.
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