The Hidden Symmetry of the Windmill Problem — E8 Intelligence Research

FINDING: The "windmill" problem (2011 IMO Q2) is the most mathematically rich of the listed items — it involves a dynamic geometric process with a finite configuration of points, and its solution hinges on an invariant (the "pivot" point) that cycles through the set, revealing a hidden symmetry in the ordering of points. | MATH: The problem: Given a finite set \(S\) of \(n\) points in the plane, no three collinear, and a line \(\ell\) through one point \(P \in S\), rotating \(\ell\) continuously about \(P\) until it hits another point \(Q \in S\), then pivoting about \(Q\), etc. The invariant: the number of points on each side of the line remains constant (specifically, \(\lfloor n/2 \rfloor\) on one side, \(\lceil n/2 \rceil\) on the other) — this is preserved because each pivot swaps exactly one point across the line. The solution shows that for any initial configuration, the process cycles through all \(n\) points as pivots, returning to the start after \(n\) pivots. This is a discr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com