Every Jordan curve in the plane contains four points forming a square.

Statement

Every simple closed curve in $\mathbb{R}^2$ inscribes a square (Toeplitz 1911). Known for smooth curves (and rectangles of all aspect ratios for smooth Jordan curves, Greene-Lobb 2020).

Assessment

Renown 4/5

A century-old classic revitalized by Greene-Lobb's 2020 symplectic breakthrough.

Attackability 1/5

A counterexample is a pathological continuous curve, not a finite object; the smooth case is proved, so any counterexample lives in fractal territory beyond finite certificates.

finite witness

1/5

oracle cost

1/5

freshness

1/5

seedability

1/5

Claims

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