The Windmill Problem: A Hidden Invariant and Rotational Symmetry in Finite Point Configurations — E8 Intelligence Research

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The solution to the 2011 IMO Q2 'windmill' problem reveals a hidden invariant and rotational symmetry in finite point configurations, governed by a parity and balance argument. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22075897

August 24, 2026Open Access

The Windmill Problem: A Hidden Invariant and Rotational Symmetry in Finite Point Configurations — E8 Intelligence Research

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Key Points

Investigate the dynamic combinatorial geometry and rotational invariants of the windmill process operating on finite sets of planar points.
Modeled continuous line rotation through a finite set of n points in general position (no three points collinear).
Constructed a state-balance function tracking the difference between points located to the left and right of the rotating line across successive pivot transitions.
Showed that the state-balance function changes by exactly ±2 at each pivot instant throughout a 180-degree rotation.
Demonstrated that point parity invariants ensure the windmill trajectory cycles through all points in the configuration infinitely often.

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Implication

Mathematical analysis reveals dynamic invariants in finite point configurations, demonstrating that a rotating line process visits every point infinitely often.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a8e9b58451774b83f3b4078 https://doi.org/10.5281/zenodo.22075897

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The Windmill Problem: A Hidden Invariant and Rotational Symmetry in Finite Point Configurations — E8 Intelligence Research

Key Points

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Implication

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Cite This Study

The Windmill Problem: A Hidden Invariant and Rotational Symmetry in Finite Point Configurations — E8 Intelligence Research

Key Points

No abstract yet

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Implication

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Cite This Study

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