Kuperberg’s six-cylinder conjecture

W. Kuperberg posed the problem in 1990 and conjectured that the answer was six. Six cylinders are achievable in a regular hexagonal arrangement, so the substantive issue was proving that more than six are impossible.

Known results

• Six unit cylinders can touch the ball in a regular hexagonal arrangement with edge length $2$; each cylinder touches its two neighbors.

July 2026 computer-assisted proof

A new arXiv paper proves the upper bound of six, reducing the geometry to $2{,}954{,}984$ cases and checking rational polynomial inequalities in each case. Together with the construction, this proves the exact answer is six. The paper states that some ideas originated in work with Anthropic’s Claude, which assisted rather than independently producing the published proof.

Current status (as of July 2026): Kuperberg’s conjecture is resolved by a computer-assisted arXiv proof: exactly six cylinders are possible.