Invariant Parity and Rational Rotation in the IMO Windmill Problem — E8 Intelligence Research

Abstract

FINDING: The 2011 IMO windmill problem (Problem 2) uses a rotating line through a finite set of points; the parity of the number of points on each side of the line is invariant under the windmill process, and the line rotates through rational angle multiples relative to the point set. MATH: - Invariant: For any line through a point \(P\) in a set of \(n\) points (no three collinear), the number of points on each side of the line has fixed parity (odd/even) throughout the process. - The line rotates by steps that are rational multiples of \(\pi\) (specifically, the angle between consecutive point-pair lines is determined by the discrete geometry of the set). - No explicit constants (0.382, 0.618, etc.) appear; the key is combinatorial parity and angular discreteness. CONNECTION: - The rational angle multiples hint at a discrete rotational symmetry, reminiscent of crystallographic restrictions (e.g., only rotations by \(2\pi/k\) for \(k=1,2,3,4,6\) in periodic lattices). Howeve Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com