Blog

Notes on the geometry behind the puzzles — why the math works, where intuition breaks, and what’s still open.

• 2026-04-20 The Inscribed Square Problem: 110 years unsolved Otto Toeplitz asked in 1911 whether every simple closed curve contains four points that form a square. More than a century later, it’s still open for arbitrary curves.
• 2026-04-20 Cutting shapes in half: why a perfect bisector always exists The Intermediate Value Theorem guarantees that every bounded planar region has a 50/50 area bisector in every direction. A short tour of why.
• 2026-04-20 Courant–Robbins: any region splits into four equal pieces with two perpendicular cuts A classic theorem from What Is Mathematics? — any planar region can be divided into four equal quarters by two perpendicular straight lines.
• 2026-04-20 Where is the center of mass of a shape with a hole? The centroid of an annulus sits at its empty center — outside the shape itself. A look at the qualitative rules of planar centroids.
• 2026-04-20 Inscribed equilateral triangles in any curve (Nielsen–Wright) Unlike the still-open Toeplitz square, the equilateral-triangle version is fully solved: every Jordan curve contains infinitely many inscribed equilateral triangles.