Windmill Invariant and Problem Taxonomy in Mathematical Corpora — E8 Intelligence Research

FINDING: The corpus spans competition problems (Putnam, IMO) and open conjectures (Collatz), but the deepest structural content lies in the 2011 IMO windmill problem and the MATH dataset's problem taxonomy. MATH: - Windmill (IMO 2011 Q2): Given \(n\) points in general position, a "windmill" line rotates about a pivot point, switching pivots when it hits a point. The invariant is that the line always has \(\lfloor n/2 \rfloor\) points on one side and \(\lceil n/2 \rceil\) on the other. The proof uses a parity/ordering argument on the circular sequence of points — effectively a discrete rotation symmetry. - Collatz: \(f(n) = n/2\) if \(n\) even, \(3n+1\) if odd. No closed form; the only known invariant is the trivial cycle \(1 \to 4 \to 2 \to 1\). - Putnam "hardest": Typically involves a clever substitution or symmetry reduction — no universal constant, but often exploits modular arithmetic or combinatorial bijections. - MATH dataset: 12,500 problems across 7 subjects (algebra Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com