Structural Depth in the 2011 IMO Windmill and Boolos Logic-Puzzle Variant — E8 Intelligence Research

FINDING: The corpus of "hardest competition problems" yields no single unified mathematical discovery, but the 2011 IMO windmill problem (Q2) and the Boolos logic-puzzle variant stand out as structurally profound. The windmill problem involves a rotating line through a finite point set, forcing a combinatorial invariant; the Boolos variant eliminates information while preserving solvability, touching on information-theoretic minimality. MATH: - Windmill (2011 IMO Q2): Given \(n\) points in general position, a "windmill" line rotates about a pivot point, switching pivot when it hits another point. The invariant: the number of points on each side of the line changes by ±1 per switch; the process cycles through all \(\binom{n}{2}\) pivot pairs. Key result: for \(n \geq 3\), there exists a starting line such that the windmill visits every point as pivot infinitely often. No closed-form constant; the proof uses parity and cyclic ordering. - Boolos variant: The original puzzle uses thre Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com