combinatorics

The article explores the number of unique 4x4 Sudoku solutions, finding only twelve distinct structures, and presents a computational trick for identifying minimal subsets.

EULER is a multi-agent system that explores cross-domain transfers to automatically prove or refute mathematical conjectures, validated with stress tests and evaluated on 120 recent conjectures, producing proofs, refutations, and partial results.

This article improves the lower bound for the smallest square that can pack 17 unit squares, achieving a new value of 4.5058, building on previous work and incorporating AI-assisted methods.

The paper introduces FAR, a human-AI discovery paradigm that automates the search for mathematical problems from literature, with a pilot in combinatorics demonstrating its effectiveness in identifying conjectures and resolutions.

The paper introduces a new paradigm for AI-assisted mathematical discovery, where experts define research directions and an AI system automates problem discovery and triage, demonstrated through a combinatorics case study.

A paper reports an AI-agent-driven attack on Conway's 99-graph problem, providing partial-credit bounds, a forced-structure reduction, and a verifiable framework, without solving the open question.

A web app that lets you scroll through all 43 quintillion possible Rubik's Cube states, visualizing the combinatorial space.

The author claims to have solved six open problems posed by the mathematician Paul Erdős within a span of five days.

This paper from the Axiom team investigates the rarity of lattice triangles, presenting a mathematical result on the distribution of convex lattice polygons.

ComBench is an Olympiad-level combinatorics benchmark with 100 problems designed to evaluate rigorous proof reasoning and constructive realization in large language models, revealing that frontier models like GPT-5.5 achieve only 65.4% overall average and that these two capabilities are distinct.

Thomas Bloom provides an expository blog post on recent counterexamples to the Erdős unit distance conjecture and sum-product conjecture over the reals, including an OpenAI-assisted disproof of the unit distance conjecture and a collaborative disproof of the sum-product conjecture, sketching the constructions and intuition behind them.

This paper trains a small one-layer encoder-decoder transformer on the zeta map bijection for Dyck paths and uses mechanistic interpretability to extract a new explicit algorithm called the scaffolding map, demonstrating an AI-assisted approach to mathematical discovery.