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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?

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

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  • $\begingroup$ @Thank you Tony, I see your point now. $\endgroup$
    – jimjim
    Jun 5 '20 at 10:05

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