Numberphile's Five Unsolved Math Puzzles: From Plane Coloring to Zero Division — E8 Intelligence Research

Abstract

FINDING: Five Numberphile videos cover distinct unsolved/classic problems — Hadwiger-Nelson (chromatic number of the plane), Catalan's conjecture, the 7825 problem (likely the Erdős–Straus or a specific Diophantine bound), Josephus problem, and zero-division paradoxes. | MATH: Hadwiger-Nelson: 5 ≤ χ(ℝ²) ≤ 7 (recently tightened to 5–7; the plane's unit-distance graph chromatic number). Catalan: xᵃ − yᵇ = 1 has only solution 3² − 2³ = 1 (i.e., 9−8=1). Josephus: J(n,k) = (J(n−1,k)+k) mod n, with closed form for k=2: J(n) = 2(n − 2^⌊log₂ n⌋) + 1. Zero: 0/0 undefined; 0⁰ often defined as 1 in combinatorics but indeterminate in limits. | CONNECTION: Hadwiger-Nelson relates to lattice structures — the hexagonal lattice (Eisenstein integers, root system A₂) gives the lower bound 5 via the Moser spindle (7 vertices, 11 edges) and Golomb graph; the upper bound 7 uses hexagonal tiling (crystallographic symmetry p6m). The Josephus problem's binary structure (2^⌊log₂ n⌋) echoes base-2, not base-60, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com