The Heilbronn Problem

Place n points in a unit-area region such that the smallest triangle determined by any three points achieves the largest possible area, A(n).

This site collects the best known configurations across three classic containers: the square, the triangle, and optimal convex regions. Each entry includes exact coordinates, symmetry and congruence analysis, references to published proofs, and an in-browser rational arithmetic verifier.

Squares Triangles Convex regions

Recent records

| date | entry | new value | gain | found by | | ---------- | -------------------------------------------------------------------- | ---------- | ------ | ---------------------------------------- | | 2026-10-01 | Triangle, n = 20 | 0.01260939 | +2.37% | Rob Gardiner | | 2026-10-01 | Square, n = 23 | 0.00975728 | +1.18% | Rob Gardiner | | 2026-09-28 | Square, n = 21 | 0.01147376 | +1.85% | Rob Gardiner | | 2026-09-28 | Square, n = 23 | 0.00964331 | +1.97% | Rob Gardiner | | 2026-09-28 | Square, n = 25 | 0.00823006 | +3.74% | Rob Gardiner | | 2026-09-28 | Square, n = 35 | 0.00443228 | +0.53% | Rob Gardiner | | 2026-09-25 | Convex, n = 31 | 0.00630967 | +3.46% | Alexandar Lackovic with help of Opus 5.5 | | 2026-09-25 | Convex, n = 33 | 0.00547013 | +3.21% | Alexandar Lackovic with help of Opus 5.5 | | 2026-09-23 | Triangle, n = 19 | 0.01348189 | +0.21% | Marc-Emmanuel Coupvent des Graviers | | 2026-09-23 | Triangle, n = 29 | 0.00612369 | +2.55% | Marc-Emmanuel Coupvent des Graviers |

The ten most recent improvements. New records are also published as an Atom feed.

Best known values

Truncated to 8 decimals; ▲ marks entries proven optimal.

The Heilbronn Problem

The Heilbronn Problem

The Heilbronn Problem

Recent records

Best known values

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