Parity Invariant in the 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.22873968
Primary Topic
Advanced Combinatorial Mathematics
Type
preprint
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preprint

Parity Invariant in the IMO Windmill Problem — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Advanced Combinatorial Mathematics
preprint

Parity Invariant in the 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 reveals a hidden invariant — the number of "windmill" states is always odd, forcing a fixed point under a parity argument. | MATH: The problem: Given \(n\) points in general position, a "windmill" is a line rotating about a pivot point, switching pivot when it hits another point. Prove there exists a pivot such that the line visits each point infinitely often. Key invariant: the number of distinct windmill configurations (pivot + line orientation) is \(2\binom{n}{2}\), and the process is a permutation on these states. The parity argument: the permutation decomposes into cycles; the existence of a cycle of odd length guarantees a point visited infinitely often. No explicit constants, but the combinatorial count \(2\binom{n}{2}\) is central. | CONNECTION: The windmill process is a discrete dynamical system on a configuration space that is topologically a circle bundle over the set of unordered Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Advanced Combinatorial Mathematics
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Parity Invariant in the IMO Windmill Problem — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS