What is the problem?
Imagine placing dots on a piece of paper. Every time two dots are exactly 1 unit apart, you draw a glowing line between them. The puzzle Hungarian mathematician Paul Erdős posed in 1946 is deceptively simple:
If you place N dots anywhere on the page, what is the maximum number of those "unit-distance" pairs you can possibly create?
For nearly 80 years, mathematicians assumed a plain square grid was the best arrangement. In May 2026, an OpenAI model disproved this — finding smarter patterns that produce more glowing connections. It is the first time AI has autonomously solved a prominent open problem central to a branch of mathematics.
- 1946 — Erdős poses the problem
- Square grids dominate for 80 years
- May 2026 — AI finds a better construction
Try it yourself — place dots, watch pairs appear
Click anywhere on the canvas to place a dot. Every pair of dots exactly 1 unit apart will light up in orange with a connecting line. Try the preset patterns to compare how many pairs each arrangement creates.
Click on the grid to add a dot · Click an existing dot to remove it
What did the AI discover?
The OpenAI model found that triangular (honeycomb) arrangements and cleverly constructed algebraic patterns from number theory can pack more unit-distance pairs than any square grid — for large numbers of dots. The AI's proof was verified as genuine mathematical research by leading mathematicians, including Cambridge's Timothy Gowers.