Full Breakdown
ChatGPT Solves Erdos Distance Problem
6/25/2026, 10:28:06 PM
Historical Roots
Erdos asked how to place N points on a plane to maximize pairs at identical distances. Three points can form an equilateral triangle; four points can make a diamond, while a dense square grid yields many equal distances via Pythagorean triples. For eight decades only marginal improvements were recorded.
Principal Figures
OpenAI delivered the solution, vetted by Daniel Litt (U. of Toronto) and Will Sawin (Princeton). Noga Alon highlighted the problem’s ubiquity among combinatorial geometers. Misha Rudnev (U. of Bristol) expressed surprise at timing. Google DeepMind pursues a “carpet-bombing” strategy on open problems.
Official Perspectives
Noga Alon noted the problem’s attention among combinatorial geometers and other mathematicians. Misha Rudnev said he never expected it solved in his lifetime. Daniel Litt observed that a group of mathematicians could have found the same solution given enough time. Will Sawin refined the construction modestly.
Criticisms and Limits
Fluency can hide errors; the ORCA benchmark shows models scoring only 45–63 % on 500 tasks, with rounding and calculation mistakes. Grok 4.20 reached 70.4 % accuracy, while ChatGPT 5.3 scored 48.4 % in free tier. Experts note Riemann Hypothesis remains beyond current AI, underscoring token-budget limits.
Conflicting Views
Some view the AI construction as genuinely creative; others argue a group of mathematicians could have found the same result. No consensus exists on whether the solution exceeds human imagination or merely reflects exhaustive search.
Quantitative Context
ChatGPT solved the Erdos problem after a single prompt, while DeepMind’s “carpet-bombing” targets hundreds of open problems with token budgets. Benchmarks show Grok 4.20 at 70.4 % accuracy, Claude Sonnet 4.6 at 53.2 %, and ChatGPT 5.3 at 48.4 % on math tasks.
Verbatim Quotes
"This is a problem that I didn’t expect to see solved in my lifetime."—M. Rudnev, Bristol
"If enough mathematicians were locked in a room for long enough, they would have found the same solution."—D. Litt, Toronto
"It would be fair to say that every mathematician working in Combinatorial Geometry thought about this problem, and lots of mathematicians working in other areas spent at least some time thinking about it."—N. Alon, Princeton
"The next AI race will not be won by the model that sounds smartest in a demo. It will be won by the model that survives contact with the spreadsheet."—Slate
Future Directions
OpenAI and DeepMind will keep applying token budgets to many open problems, aiming to separate “easy-hard” questions from intractable ones. The approach should highlight challenges such as the Riemann Hypothesis, while mathematicians focus on new concepts. Verification pipelines and hybrid human-AI workflows are advocated to guard against errors.
