Windmill Invariants vs. Standard Olympiad Exercises: A Comparative Study — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: The 2011 IMO windmill problem (Q2) and the 1988 IMO Problem 6 (legendary hardest) reveal deep combinatorial-geometric invariants; the 2026 Problem 1 is a standard olympiad exercise with no novel structure. | MATH: Windmill problem: Given \(n\) points in general position, a "windmill" process rotates a line through a pivot point, switching pivot to the next point hit; the invariant is that the line always passes through a fixed set of points, and the process cycles through all \(\binom{n}{2}\) pairs exactly once — equivalent to a "halving line" sequence. 1988 P6: If \(a,b\) are positive integers and \(\frac{a^2+b^2}{ab+1}\) is an integer, then it is a perfect square — the invariant is \(k = \frac{a^2+b^2}{ab+1} \in \{1,4,9,\dots\}\), proven via Vieta jumping (descent on \(a+b\)). | CONNECTION: The windmill problem's pivot-switching rule generates a **root-system-like** structure: the sequence of halving lines corresponds to a **Coxeter element** in the affine Weyl group of type Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204907244
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