Geometry vs Binary: Unsolved Mathematics — unsolved problems in combinatorial geometry 2024 — E8 Intelligence Research

## Binary Failure → Geometry Solution **Analysis of Findings:** The search results highlight three classic unsolved geometry problems: **Moser's worm problem**, **the perfect cuboid problem**, and **the Toeplitz conjecture** (inscribed square problem). These are deceptively simple-sounding problems that have resisted brute-force computation and statistical sampling for decades. **Where binary/statistical approaches FAIL:** - **Moser's worm problem** (find the smallest convex shape that can cover any curve of length 1): Binary search over shape spaces fails because the space of all convex shapes is infinite-dimensional and non-parametric. Statistical sampling of "worm" curves cannot guarantee coverage of all possible curves—there are uncountably many. - **Perfect cuboid problem** (integer edges, face diagonals, and space diagonal): Brute-force search over integer triples has been pushed to enormous bounds (10^15+) with zero solutions found. But a statistical "it probably doesn't exi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com