49 directions in three-dimensional space with integer coordinates. Among them are 36 triples of mutually perpendicular directions. Try to mark some directions as true so that every triple contains exactly one true direction: it cannot be done.
Drag to turn the sphere. Click a point to mark that ray true, click again to clear it. Faint points are on the far side.
Each point is a ray, a line through the origin, drawn where it meets the upper half of the unit sphere. Three mutually perpendicular rays form a triple, drawn as a triangle of three quarter circles. The bigger a point, the more triples it belongs to: 33 of the rays lie in two or more triples, and the other 16 lie in exactly one, where they complete a triple.
A Kochen–Specker colouring in basis form assigns 1 (true) or 0 (false) to every ray so that each triple contains exactly one 1. No such assignment exists for these 49 rays; the best attempt satisfies 35 triples and breaks one. The set is also critical: remove any one of the 49 rays and a colouring becomes possible. Both facts were checked by a SAT solver.
The set was found by a computer search over rays with small coordinates. It is the smallest set that search has produced in about 165,647 attempts (last updated 23 October 2026); whether smaller sets of this kind exist is open as far as the search goes.
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