maximize the smallest triangle · open since 1950
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Drag the n points around the unit square. The smallest of the C(n, 3) triangles they form is your score — push it as high as you can. Heilbronn conjectured the optimum is ≈ 1/n² (a grid); Komlós–Pintz–Szemerédi 1981 disproved him with a construction that gets c·log(n)/n². The page tracks your personal best per n in your browser.
Heilbronn's conjecture: H(n) = Θ(1/n²). Disproved by Komlós–Pintz–Szemerédi 1981 — the optimum is at least c·log(n)/n². Upper bounds have been chipped down over decades: Roth (1971), Schmidt (1972), Komlós–Pintz–Szemerédi (1981), Cohen–Pohoata–Zakharov (2024). Plot below is log-log; your personal bests appear as dots when they exist.
Heilbronn asked a deceptively simple question. Pick n points in the unit square. Among the C(n, 3) triangles they form, take the smallest. How big can you make that smallest triangle, by choosing the points cleverly?
H(n) = maxP ⊂ [0,1]² min{a,b,c} ⊂ P area(abc).
Three points form a triangle of zero area only if they're collinear, so part of the game is dodging collinearity. The other part is spacing — bunching any two points together creates a long thin sliver of a triangle whenever you join them to a third point.
Heilbronn conjectured H(n) = Θ(1/n²). A square grid of √n × √n points has min triangle ≈ 1/(2n) by a careful count, and one can do slightly better with perturbations — but never, he thought, by more than a constant factor.
In 1981 Komlós, Pintz, and Szemerédi published a construction giving H(n) ≥ c · log(n) / n². The log factor was a shock. Their points are placed pseudo-randomly with a careful "removal" pass to kill collinearities; the proof that the construction works is a delicate probabilistic argument.
Even with these advances, the gap between the c·log(n)/n² lower bound and the ~1/n8/7 upper bound is wide — about a factor of n1/7. Both ends might be wrong. The right answer to H(n) is not known.
For small n, computer search has found exact (or believed-exact) optimal configurations. The current records up to n = 15 are due to Goldberg (early 1970s), Yang–Graham–Goldberg (2009), and a long tradition of careful enumeration.
Drop into the play tab and try to match them. Your browser stores your best per n in localStorage; clear-site-data resets the leaderboard.