Kakeya needle problem

Let n ≥ 2. Let C^T(n) denote the minimal area |bigcup_j=1^n T_j| of a union of triangles T_j with vertices (x_j,0), (x_j + 1/n, 0), (x_j + j/n, 1) for some real numbers x_1,…,x_n, and similarly define C^P(n) denote the minimal area |bigcup_j=1^n P_j| of a union of parallelograms P_j with vertices…

From the catalogue. Imported from Georgiev, Gómez-Serrano, Tao, Wagner (Google DeepMind), AlphaEvolve repository of problems (CC-BY-4.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-kakeya-needle-problem,
  title        = {Kakeya needle problem},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/kakeya-needle-problem}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

Let $n \geq 2$ . Let $C^{T} (n)$ denote the minimal area $∣ ⋃_{j = 1}^{n} T_{j} ∣$ of a union of triangles $T_{j}$ with vertices $(x_{j}, 0)$ , $(x_{j} + 1/ n, 0)$ , $(x_{j} + j / n, 1)$ for some real numbers $x_{1}, \dots, x_{n}$ , and similarly define $C^{P} (n)$ denote the minimal area $∣ ⋃_{j = 1}^{n} P_{j} ∣$ of a union of parallelograms $P_{j}$ with vertices $(x_{j}, 0), (x_{j} + 1/ n, 0), (x_{j} + j / n, 1), (x_{j} + (j + 1) / n, 0)$ for some real numbers $x_{1}, \dots, x_{n}$ . Finally, define $S^{T} (n)$ to be the maximal "score" $\frac{\sum _{i = 1}^{n} ∣ T _{i} ∣}{( \sum _{i = 1}^{n} \sum _{j = 1}^{n} ∣ T _{i} \cap T _{j} ∣ ) ^{1/2} ∣ ⋃ _{i = 1}^{n} T _{i} ∣ ^{1/2}}$ over triangles $T_{i}$ as above, and define $S^{P} (n)$ similarly. Establish upper and lower bounds for $C^{T} (n)$ , $C^{P} (n)$ , $S^{T} (n)$ , $S^{P} (n)$ that are as strong as possible.

AlphaEvolve found a construction better than the previous record (at publication).

What counts as progress

A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
A proof that a known construction is optimal, or a better bound on the other side.
Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.

Current state

The problem

The problem

What counts as progress

Source and licence