Parity Invariant and Pivot Existence in the IMO Windmill Problem — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-19
DOI
https://doi.org/10.5281/zenodo.22841559
Primary Topic
Computability, Logic, AI Algorithms
Type
preprint
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preprint

Parity Invariant and Pivot Existence in the IMO Windmill Problem — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Computability, Logic, AI Algorithms
preprint

Parity Invariant and Pivot Existence in the IMO Windmill Problem — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: The 2011 IMO windmill problem reduces to a parity-invariant on the number of points on each side of a rotating line, with the invariant being the total count of "switches" modulo 2, and the solution hinges on the existence of a point that is always on the rotating line (the pivot), which is guaranteed by a combinatorial pigeonhole argument on the sequence of pivots. | MATH: Let \(n\) points in general position (no three collinear). A windmill process: a line \(L\) rotates about a pivot point \(p_i\), and when \(L\) hits another point \(p_j\), the pivot switches to \(p_j\), and the line continues rotating. Key invariant: the number of points on each side of \(L\) changes by exactly \(\pm 1\) at each switch, but the *difference* between left and right counts is invariant modulo 2. Specifically, if \(n\) is odd, there exists a pivot \(p\) such that the line always has \((n-1)/2\) points on each side — this is achieved by choosing the initial line to pass through two points and ro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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