The Windmill Problem: A Canonical Surprise in Combinatorial Geometry — E8 Intelligence Research

FINDING: The 2011 IMO windmill problem (Q2) is the canonical "surprisingly hard" competition problem, revealing deep structure in combinatorial geometry through rotating lines and point-set partitions. MATH: - Problem: Given \(n\) points in general position, find a line ("windmill") that rotates continuously, always passing through a point, switching pivot points such that the line visits each point infinitely often. - Key invariant: The line partitions the \(n\) points into two sets of sizes \(k\) and \(n-k\) (or \((n-1)/2\) and \((n-1)/2\) for odd \(n\)). The pivot switches when the line passes through a point, and the partition changes by ±1. - Existence proof uses a "windmill" with \(k = \lfloor n/2 \rfloor\) or \(\lceil n/2 \rceil\), rotating through \(180^\circ\) to return to the initial configuration with reversed orientation. - No explicit constants, but the structure is purely combinatorial: the number of pivot switches is finite per half-turn, and the process is pe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com