Heilbronn's triangle problem: record configurations

Place n points in a square or triangle so the smallest triangle they form is as large as possible; beat a record or prove one optimal.

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1	# Heilbronn's triangle problem: record configurations
2
3	Place $n$ points in a square (or a triangle, or any convex region) so that the smallest triangle formed by three of them is as large as possible. Optimal configurations are proved only up to $n = 9$ in the square and $n = 8$ in the other containers, while records for larger $n$ change almost daily. Every candidate is verified exactly in milliseconds.
4
5	## Problem
6
7	For a container $K$ of unit area and points $p_1, \dots, p_n \in K$, let
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9	$$A(p_1, \dots, p_n) = \min_{i < j < k} \operatorname{area}(p_i p_j p_k).$$
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11	The Heilbronn number $H_n(K)$ is the maximum of $A$ over all configurations of $n$ points in $K$. The record table tracks three variants: $K$ a unit square, $K$ a triangle of unit area (the problem is affine-invariant, so all triangles are equivalent), and the "convex" variant, in which the points may lie in any convex region of unit area, chosen together with the points.
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13	Task: find configurations that beat the current records, or prove that a record is optimal. The smallest open cases are $n = 10$ in the square and $n = 9$ in the triangle and in the convex variant.
14
15	## Background
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17	Heilbronn asked how large the smallest triangle can be for $n$ points in a unit region; the asymptotic question, between the lower bound of order $(\log n)/n^2$ of Komlós, Pintz and Szemerédi and the upper bound $n^{-7/6 + o(1)}$ of Cohen, Pohoata and Zakharov (2024), is a famous problem in its own right; an OpenAI preprint (September 2026, not refereed or formalized) claims a power improvement of the lower bound, to $n^{-2+\eta}$ for a tiny fixed $\eta > 0$. For small $n$ the exact optima are hard: the configurations have many tied minimal triangles, and certified answers have needed symbolic computation or global mixed-integer optimization. Erich Friedman's Packing Center kept the record tables for decades; the current site rebuilds them from exact coordinates.
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19	## What is known
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21	From the record site (October 2026):
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23	- Square: optimal configurations are certified for $n \le 9$; Sudermann-Merx (2026) certified all of them by mixed-integer nonlinear optimization and recovered exact coordinates, after earlier proofs for $n = 7$ (Zeng and Chen, 2011) and $n = 8$ (Dehbi and Zeng, 2022). For $n = 10$ the record, $A \approx 0.0465374196$ (a root of $3456x^3 - 1764x^2 + 268x - 9$), is Comellas and Yebra's configuration from 2001–2002.
24	- Triangle: optimal for $n \le 8$ (Sudermann-Merx, 2026). The record for $n = 9$ is $43/784 \approx 0.0548469$ (Cantrell, 2006).
25	- Convex: optimal values for $n \le 8$ were formalized in Lean 4 by Stead (2026). The record for $n = 9$ is about $0.0640647523$ (Cantrell, 2007).
26	- Larger n: records are listed up to $n = 36$ in each variant and were still being improved in October 2026.
27
28	## What would count as progress
29
30	- A configuration beating a record for some $n$ and container. The site's verifier computes all $\binom{n}{3}$ triangle areas in exact rational arithmetic, so a candidate given by rational coordinates is checked in milliseconds.
31	- A proof that a record is optimal, for example the square with $n = 10$ or the triangle with $n = 9$, as a certified global optimization run whose model and certificate can be rerun, or as a Lean proof.
32	- Exact algebraic values for records known only numerically.
33
34	## References
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36	1. F. Comellas and J. L. A. Yebra, "New lower bounds for Heilbronn numbers", Electronic Journal of Combinatorics 9 (2002), R6. [doi:10.37236/1623](https://doi.org/10.37236/1623)
37	2. L. Dehbi and Z. Zeng, "Heilbronn's problem of eight points in the square", Journal of Systems Science and Complexity 35 (2022), 2452–2480. [doi:10.1007/s11424-022-1220-7](https://doi.org/10.1007/s11424-022-1220-7)
38	3. N. Sudermann-Merx, "From computational certification to exact coordinates: Heilbronn's triangle problem on the unit square using mixed-integer optimization" (2026). [arXiv:2603.11107](https://arxiv.org/abs/2603.11107)
39	4. N. Sudermann-Merx, "Heilbronn's problem in the unit triangle: certified optimal configurations for up to n ≤ 8" (2026). [arXiv:2607.15021](https://arxiv.org/abs/2607.15021)
40	5. A. Cohen, C. Pohoata and D. Zakharov, "Lower bounds for incidences" (2024), which gives the Heilbronn upper bound $n^{-7/6+o(1)}$. [arXiv:2409.07658](https://arxiv.org/abs/2409.07658)
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42	## Source
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44	Imported from [The Heilbronn Problem, record tables by Tej Stead](https://math.tejstead.com/heilbronn/), curated by Tej Stead. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.