# Smallest sets of spheres that envelop a cube

It is trivial to find the smallest sphere that envelopes a cube. Next, 8 overlapping spheres can envelop the cube with a shorter cross distance.

What is the next smallest number of (differently sized, overlapping) spheres that will envelop the cube with a shorter cross distance?

• 3 spheres is the minimum sufficient. Jun 5 '20 at 7:16
• @Arjang: Two spheres, I would say. Jun 5 '20 at 7:47
• @TonyK : Two spheres doesn't envelope a volume but 3 spheres can Jun 5 '20 at 9:20
• @Arjang: see my answer. Jun 5 '20 at 9:46
• What is a cross distance? Jun 5 '20 at 10:18

To envelop a $$2\times 2\times 2$$ cube with a single sphere, we need a radius of $$\sqrt 3$$. But to envelop a $$2\times 2\times 1$$ cuboid with a single sphere, we only need a radius of $$\sqrt{1^2+1^2+(\frac12)^2}=\sqrt\frac94$$. And clearly two such spheres are enough to envelop a $$2\times 2\times 2$$ cube.

Edited to add: This may or may not answer the OP's question, depending on what "cross distance" means. We are currently waiting on a workable definition.

• @Thank you Tony, I see your point now. Jun 5 '20 at 10:05