Rotating Line Reveals Parity Invariant in IMO Windmill Problem — E8 Intelligence Research

FINDING: The hardest competition problems often reveal deep combinatorial or geometric invariants, not just computational complexity. The 2011 IMO windmill problem (Q2) exemplifies this: it asks for a proof that for any set of points in general position, there exists a line through a point that rotates to visit each point exactly once, revealing a parity invariant. MATH: No explicit equations or constants. Key invariant: parity of the number of points on each side of a rotating line. For \(n\) points, the line must pass through a point, and the number of points on each side changes by ±1 per step. The solution uses a combinatorial argument: choose a starting line through a point such that the number of points on one side is minimal (0 or 1), then rotate; the invariant ensures each point is visited exactly once. CONNECTION: The rotation and parity structure mirrors a discrete version of a harmonic ratio in angular spacing. The invariant (0 or 1 points on one side) corresponds to a bal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com