Parity and Invariance in the 2011 IMO Windmill Problem — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-21
DOI
https://doi.org/10.5281/zenodo.22873513
Primary Topic
Limits and Structures in Graph Theory
Type
preprint
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preprint

Parity and Invariance in the 2011 IMO Windmill Problem — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Limits and Structures in Graph Theory
preprint

Parity and Invariance in the 2011 IMO Windmill Problem — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: The 2011 IMO windmill problem (Q2) is a combinatorial geometry problem whose solution hinges on a parity/invariant argument over rotating lines through point sets, revealing a hidden structural stability in finite configurations. | MATH: The problem: Given \(n\) points in general position, a "windmill" process rotates a line \(l\) through a pivot point \(P\); when \(l\) hits another point \(Q\), the pivot switches to \(Q\) and the line continues rotating. Prove that for odd \(n\), there exists a starting pivot such that the process visits every point infinitely often. Key invariant: the number of points on each side of \(l\) remains constant (mod 2) — specifically, for odd \(n\), the counts \((k, n-1-k)\) swap parity only when the pivot changes, forcing a periodic orbit through all points. No explicit constants; the essence is a discrete dynamical system with a conserved parity. | CONNECTION: The windmill process is a discrete analogue of a rotating line in a lattice — the par Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Limits and Structures in Graph Theory
DOIOpen Access PDFOpenAlex
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Parity and Invariance in the 2011 IMO Windmill Problem — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS