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I have a point in space, K, and I am trying to draw the highest possible number of overlapping spheres, of radius R, that contain said point. So far, the answer is "an infinite amount". The difficulty arises when I try to enforce the clause that each sphere is only allowed to overlap with up to 7 centre-points of other spheres.

How many spheres can be drawn?

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I don't know the answer for 7 spheres, but if you are not allowed to overlap with any sphere's, you can get a lower bound as well as an upper bound around $2^{cd}$, from the Kissing problem.

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