Kuperberg's six-cylinder conjecture (Kuperberg 1990) asserts that at most six infinite solid right circular cylinders of radius 1 with pairwise disjoint interiors can be simultaneously tangent to a unit ball in three-dimensional Euclidean space.
Six are possible with parallel axes. In a plane through the center of the unit ball perpendicular to the axes, place the axes at the vertices of a regular hexagon of circumradius 2. The resulting cylinders touch their neighbors and the unit ball without overlapping interiors.
Matić and Radoičić (2026) announced a proof of the upper bound. Their argument reduces the problem to 2,954,984 cases, each checked using explicit polynomials whose coefficients are rational numbers. They disclose substantial assistance from Claude in developing the proof and describe a reconstruction and verification of the argument by the authors.
Matić et al. (2026a) extended the question to the standard ring of 
equal parallel cylinders tangent to a unit
ball and their two neighbors. The ring radius is 
. Allowing the cylinders
to be rearranged, no additional cylinder of radius 
can be inserted for 
,
whereas at least one can be inserted for every 
. The case 
remains open. Matić et al. (2026a) disclose
Claude assistance in exploration, certificate-search code, and verification.
Littlewood's cylinder problem instead requires pairwise contact among congruent infinite cylinders, with no common ball. Its maximum is seven (Matić et al. 2026b).
