The Hidden Projective Symmetry Behind IMO's Windmill Problem — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The corpus reveals no single "hardest problem" but a cluster of structurally distinct challenges — Putnam's elegant-combinatorics, IMO's windmill (dynamic geometry), Collatz (iterative arithmetic), and the MATH dataset (benchmarking) — with the deepest mathematical content in the windmill problem's hidden projective symmetry. MATH: - **Windmill (IMO 2011 P2):** For odd n, a finite set S of points in general position. A "windmill" is a line through a pivot point p ∈ S, rotating continuously; when it hits another point q ∈ S, the pivot switches to q (line passes through q and continues rotating). **Key result:** There exists a starting line such that every point of S is a pivot infinitely often. The proof uses a parity/involution argument: the number of points on each side of the rotating line changes by ± 1 at each pivot switch, and for odd n, the line must pass through exactly one point at all times (since n−1 is even, sides ba
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026hidden,
title = {{The Hidden Projective Symmetry Behind IMO's Windmill Problem — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23587065},
url = {https://zenodo.org/records/23587065},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Hidden Projective Symmetry Behind IMO's Windmill Problem — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23587065
UR - https://zenodo.org/records/23587065
AB - FINDING: The corpus reveals no single "hardest problem" but a cluster of structurally distinct challenges — Putnam's elegant-combinatorics, IMO's windmill (dynamic geometry), Collatz (iterative arithmetic), and the MATH dataset (benchmarking) — with the deepest mathematical content in the windmill problem's hidden projective symmetry. MATH: - **Windmill (IMO 2011 P2):** For odd n, a finite set S of points in general position. A "windmill" is a line through a pivot point p ∈ S, rotating continuously; when it hits another point q ∈ S, the pivot switches to q (line passes through q and continues rotating). **Key result:** There exists a starting line such that every point of S is a pivot infinitely often. The proof uses a parity/involution argument: the number of points on each side of the rotating line changes by ± 1 at each pivot switch, and for odd n, the line must pass through exactly one point at all times (since n−1 is even, sides ba Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -