AI Claude Helps Math Expert Solve Decades-Old Problem

AI Solves Decades-Old Math Problem, Stuns Expert

A leading mathematician was recently stymied by a complex problem for weeks, only to have it solved by Anthropic’s Claude Opus 4.5 AI model. The breakthrough highlights the rapidly increasing capabilities of artificial intelligence in tackling sophisticated challenges.

The Problem and the Expert

Donald Knuth, a renowned mathematician and computer scientist, encountered the difficulty although preparing material for a future volume of his acclaimed series, The Art of Computer Programming. The problem involved researching Hamiltonian cycles on directed graphs – a complex topic within mathematical combinatorics.

Claude’s Unexpected Solution

Knuth shared his experience in a paper titled Claude’s Cycles, published in late February 2026, expressing his surprise at the AI’s ability to uncover a solution. He noted that Claude Opus 4.5 successfully addressed the issue after he had been struggling with it for an extended period.

A Shift in Perspective

The success prompted Knuth to reconsider his views on generative AI technology. Claude didn’t simply provide an answer; it reformulated the problem to analyze it more effectively, testing three different approaches before arriving at a concrete solution in Python.

Opus 4.5: A Leap Forward for AI Agents

The incident underscores the advancements made with Claude Opus 4.5, which was released in November 2025. Anthropic describes the model as state-of-the-art for coding, agents and computer use, and significantly improved for everyday tasks like working with spreadsheets and presentations. It represents a major step forward in AI capabilities and a preview of changes to how work is done.

Beyond Mathematics: Wider Implications

This isn’t an isolated incident. Claude Opus 4.5 has demonstrated exceptional performance in various areas, including coding and automation. According to Anthropic, accuracy on internal evaluations improved by 20%, and efficiency rose by 15% with the new model.

Accessibility and Pricing

Anthropic has made Opus-level capabilities more accessible by pricing the model at $5/$25 per million tokens. The company has also released updates to the Claude Developer Platform, Claude Code, and its consumer apps, including new tools for longer-running agents and improved integration with Excel, Chrome, and desktop applications.

FAQ

• What is Claude Opus 4.5? Claude Opus 4.5 is Anthropic’s newest AI model, designed for complex problem-solving, coding, and general use.
• What are Hamiltonian cycles? Hamiltonian cycles are a topic in graph theory, involving finding a path that visits each point in a network exactly once and returns to the starting point.
• Who is Donald Knuth? Donald Knuth is a highly respected mathematician and computer scientist, known for his influential series, The Art of Computer Programming.
• How is Claude Opus 4.5 priced? The model is priced at $5/$25 per million tokens, making it more accessible to a wider range of users.

Did you recognize? Claude Opus 4.5 is available on Anthropic’s apps, API, and all three major cloud platforms.

Explore more about the latest advancements in AI and their impact on various industries. Learn more about Claude.