https://www.math.columbia.edu/~ums/Peg%20problem%20-%20slides.pdf
- Emch (1913) solved the problem for smooth convex curves. (Proof uses configuration spaces and homology.)
- Schnirelman (1929) solved it for any smooth Jordan curve.
Theorem. Vaughan (1977) [local-0]
Theorem. Vaughan (1977) [local-0]
Every continuous Jordan curve contains four points forming the vertices of some rectangle. This is what 3b1b did, see video here.
Theorem. Greene-Lobb (2020) [local-1]
Theorem. Greene-Lobb (2020) [local-1]
Given a smooth Jordan curve and a rectangle in the plane, then contain four points forming the vertices of a rectangle similar to R.