Symmetry in Rotation: The Windmill Problem's Hidden Invariant — E8 Intelligence Research

FINDING: The 2011 IMO "windmill" problem (Q2) is a combinatorial geometry problem whose solution hinges on a parity/invariant argument over rotating lines, revealing a hidden symmetry in point configurations. The 2026 IMO Problem 1 is a new, as-yet-unpublished challenge, but the search results indicate it is being presented as accessible yet deep. MATH: The windmill problem: Given \(n \geq 3\) points in general position, a "windmill" process rotates a line \(L\) through a pivot point \(P\) (one of the given points). When \(L\) passes through another point \(Q\), the pivot switches to \(Q\), and the direction of rotation reverses. The theorem: For any such configuration, there exists a starting line such that every point is a pivot infinitely often. The invariant: the number of points on each side of \(L\) changes by exactly \(\pm 1\) at each switch, and the total "winding" is periodic. No explicit constants; the proof uses a parity argument on the cyclic order of points. CONNECTION: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com