We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient-based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem (Mundinger et al., 2024a). Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.

How many colors are needed to color the plane so that if you drop a matchstick both ends always land on different colors? This might sound like a puzzle, but it’s actually a famous unsolved math problem called the Hadwiger-Nelson problem.We used neural networks, the same kind of technology behind modern artificial intelligence systems, to explore possible solutions to this problem in a new way. By turning the challenge into something a computer can learn and optimize, we were able to search through countless coloring patterns and avoid the usual trial-and-error approach.This led to the discovery of two new coloring configurations that improve upon previous results for a variation of the problem. It's the first progress in over 30 years. Our approach shows how machine learning can help solve deep, abstract problems in mathematics and could inspire similar breakthroughs in other hard-to-crack areas.