unit-distance problem – An OpenAI model explored how to arrange points on a grid so many pairs sit exactly one unit apart, building on a strategy tied to choosing c² = 65. The work lands in the orbit of a famous 80-year-old question associated with Paul Erdős—where a known lower boun
A simple-looking grid choice—c² = 65—turned out to be the key to making a lot of pairs of points line up at exactly one unit of distance.
In OpenAI’s setup, the diagram is based on selecting c² = 65. That number can be satisfied in more than one way: either 1² + 8² = 65 or 4² + 7² = 65. With the grid spacing set to 1/√65. every point ends up exactly one unit away from 16 other points—specifically (1. 8). (4. 7). (7. 4). and (8. 1). along with their sign-flipped versions (-1. 8). (-4. 7). and (-7. 4). (-8. 1). and so on.
The arithmetic matters because it’s not just about getting some unit-distance neighbors—it’s about getting the count to grow as the grid gets larger. The model’s reasoning also flags what can go wrong. If c² is chosen too large compared with the number of points in the grid. many of the potential one-unit-away neighbors won’t be available because they land outside the grid.
So the question becomes a balancing act: pick a c² large enough to create many valid whole-number diagonals, but not so large that the corresponding “one-unit-away” offsets push you out of range.
This is where older mathematics enters, and it arrives with a promise—but also a trap. Using insights from number theory. including Jacobi’s two-square theorem. Erdős showed that an optimally sized circle can make the number of unit-distance pairs grow faster than the number of points—though only barely. The result set up a challenge that lasted: could anyone do better?.
Erdős answered that question from both sides, even if he couldn’t fully settle it. To find an upper bound. he brought in an argument from graph theory. a different area of mathematics that tracks connections like edges between vertices. That line of reasoning showed there could only be so many unit distances overall. The problem was that his upper bound grows much. much faster than the best lower bound he was able to construct.
The gap was the point. Erdős’s conjecture leaned toward the lower-bound side of the story. He predicted. without being able to prove it. that the actual optimum would sit much closer to the lower bound than the upper one. In his view, the maximum number of unit distances would grow just barely faster than the number of points.
Erdős put the claim in precise growth terms: he conjectured that the number of unit distances would be n^(1+o(1)). That means that for sufficiently large n, the maximum number of unit distances would stay below n^(1+𝜖) for any 𝜖 > 0. In practice. it could grow a little faster than the lower-bound construction—described as n^(1 + C/(log log n)) for some constant C—but still remain in the same general neighborhood.
The model’s contribution. at least at this stage. lives inside that long-running tension: turning number-theory choices like c² = 65 into a grid that manufactures unit-distance pairs at scale. while reminding everyone that the arithmetic only works if the grid stays big enough to hold the neighbors you create.
In the end. the excitement isn’t only that a famous problem has been “solved” again in a new way—it’s that a clear mechanical choice on a grid echoes the exact structure of an 80-year-old mathematical fight: how to pack points so unit distances proliferate. and whether the true maximum follows the sharper lower bound Erdős believed in. rather than the much looser ceiling his graph-theory argument produced.
OpenAI unit distance problem Erdős conjecture graph theory Jacobi’s two-square theorem c squared equals 65 number theory grid spacing 1 over sqrt 65 math breakthrough