Windmill Problem: Rotating Line Invariant via Parity and Pivot — E8 Intelligence Research

Q: What does this research mean for the field?

The solution to the 2011 IMO Q2 windmill problem relies on a dynamic, rotating line that maintains an invariant half-plane balance through parity and a fixed median pivot point. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22075549

August 24, 2026Open Access

Windmill Problem: Rotating Line Invariant via Parity and Pivot — E8 Intelligence Research

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Key Points

Analyze the combinatorial geometry and invariant properties of the rotating sweep-line windmill process across finite planar point configurations.
Modeled a continuous line rotating through an n-point planar configuration with no three collinear points.
Tracked directional angle progression, pivot transitions upon point intersections, and half-plane point distributions.
Demonstrated that continuous monotonic rotation maintains exactly ⌊n/2⌋ points on each side of the sweep line.
Identified that each pivot transition alters side point counts by ±1, preserving half-plane balance via median pivot selection.

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Implication

Theoretical analysis demonstrates invariant properties of the windmill process in planar point sets, indicating that dynamic pivot switches preserve balanced half-plane partitions.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a8e9bcc451774b83f3b4af6 https://doi.org/10.5281/zenodo.22075549

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Windmill Problem: Rotating Line Invariant via Parity and Pivot — E8 Intelligence Research

Key Points

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Implication

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Cite This Study

Windmill Problem: Rotating Line Invariant via Parity and Pivot — E8 Intelligence Research

Key Points

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Implication

Discussion

Cite This Study

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