The Smallest Triangle
Put dots in a square. Every three of them make a triangle, and one of those triangles is the smallest. Hans Heilbronn asked, before 1951: how large can you make that smallest triangle? Try it.
1 · Keep the smallest triangle large
Drag the dots. The smallest triangle is shaded, and its area is scored as a fraction of the square. Three dots in a line make a triangle of area zero, so spread them out — but not too evenly.
Proven best values are known only for small numbers of dots: 1/8 for six, and (√13 − 1)/36 ≈ 0.0724 for eight. Proving the seven-dot value took a computer search split into 226 cases. For other sizes the page shows no target; the computer’s search is a guide, not a proof.
2 · Random is worse than careful
Scatter the same number of dots at random 400 times and take the typical smallest triangle. Then compare it with what careful placement achieves.
Random dots almost always include three that are nearly in a line. Every advance on this problem, from Erdős’s curve to the 1982 disproof to the 2026 claim, has been a better construction — a smarter way to avoid that.
3 · Infinitely many is not every
The 2026 paper claims its bound for every large number of dots. Its machine check proves it for infinitely many numbers — a sequence of sizes that never ends, but may skip sizes in between. Switch between the two.
Illustration only: the actual checked sizes come from primes in the construction and are far larger than 60. To disprove the “almost 1/n²” limit, beating it at infinitely many sizes is enough — so the disproof is fully checked. Describing every size needs one more step, which the paper takes (build a slightly bigger set, then delete dots; deleting can never make the smallest triangle smaller). The scope note says the checked construction covers the sequence only.
4 · An improvement too small to draw
As dots multiply, how fast must the smallest triangle shrink? Write the answer as n to a power. Constructions push the power up from −2; impossibility proofs push it down from −1. The truth is somewhere in the shaded gap.
Lower bounds (blue) say what careful placement achieves; upper bounds (amber) say what no placement can beat. The 2026 claim moves the lower bound from −2 to −2 + η, where the paper’s explicit η has 235 zeros after the decimal point.
5 · What is claimed, and what is checked
| Statement | Status |
|---|---|
| The “almost 1/n²” limit is false | Checked by machine, in Lean |
| The improved bound holds for an unbounded sequence of sizes | Checked by machine |
| The improved bound holds for every sufficiently large n | In the paper; “broader” than the check |
| The proof as a whole (24 pages) | No specialist review found |
Seventy-five years
- Before 1951Heilbronn, who fled Nazi Germany in 1933, guesses the smallest triangle shrinks like 1/n².
- 1951Klaus Roth publishes the first paper on the problem.
- 1964Heilbronn moves to Toronto, where he dies in 1975.
- 1982Komlós, Pintz and Szemerédi disprove his guess, by a factor of log n.
- 2023Cohen, Pohoata and Zakharov move the other side for the first time in four decades.
- Oct 6, 2026OpenAI claims a power improvement; checked for infinitely many n, claimed for every large n.