Parity and Rotation: The Hidden Invariant in the IMO Windmill Problem — E8 Intelligence Research

FINDING: The 2011 IMO windmill problem (Q2) is a deceptively simple combinatorial geometry problem whose solution hinges on a parity/invariant argument about rotating lines through point sets, revealing deep structure in finite point configurations. | MATH: The problem: Given n points in general position (no three collinear), a "windmill" is a process where a line rotates about a pivot point, switching pivots when it hits another point. Key invariant: the number of points on each side of the line changes by exactly ±1 at each switch; the process cycles through all points as pivots infinitely. The proof uses the fact that for any line through a point, the count of points on each side is fixed mod 2 (parity invariant), and the total "winding" over a full cycle returns to the initial state — a topological/parity argument, not metric. No explicit constants arise; the essence is combinatorial symmetry. | CONNECTION: The windmill's pivot-switching rule mirrors the structure of root systems ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com