The Invariant of Parity: Solving the 2011 IMO Windmill Problem — E8 Intelligence Research

FINDING: The "hardest" competition problems often hinge on elegant combinatorial or geometric invariants, not on complex equations or constants. The 2011 IMO windmill problem (Q2) exemplifies this: it requires proving that for any set of points in general position, there exists a line through a point that rotates to bisect the set, using parity and symmetry arguments. MATH: No explicit equations or constants. Key invariant: parity of the number of points on each side of a rotating line. The solution uses a discrete continuity argument (a "windmill" process) where the line rotates about a pivot point, swapping sides of points one at a time, maintaining a bisection property. CONNECTION: The rotating line and pivot points evoke a discrete analogue of rotational symmetry in the plane. The process implicitly relies on the symmetry of the configuration under 180° rotation (the line's orientation flips after half a turn). This mirrors the geometric harmony of cyclic order and balanced parti Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com