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The question of how many smaller spheres can be fit into a larger sphere is fascinating and has been examined extensively. I was curious, though, about the scenario of packing spheres of different radii into a larger sphere. For example: What is the maximum number of rigid spheres of radius 1 and radius 2 which can be packed into a rigid sphere of radius 10, given that there must be an equal number of radius 1 and radius 2 spheres? I'm not sure what packing strategy would produce an optimal result, so any ideas would be appreciated. Thanks in advance.

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    $\begingroup$ Unequal sphere packing is fascinating and has been examined extensively. $\endgroup$ – user31373 Jul 3 '12 at 0:21
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    $\begingroup$ Another search term is "mixed sphere packing," also "multi-sized sphere packing." $\endgroup$ – Gerry Myerson Jul 3 '12 at 0:22
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A recent paper is Marshall and Hudson, Dense binary sphere packings, Beiträge Algebra Geom. 51 (2010), no. 2, 337–344, MR2682460 (2011g:52030). It deals particularly with the case where the two radii are 1 and $\gamma$, where $.444\lt\gamma\lt.482$, whereas you want $\gamma=1/2$, but it also contains an overview of the problem and an extensive list of references.

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    $\begingroup$ Thank you for the reference. I wasn't aware of how extensively the problem had been examined, but now I know where to go to learn more about it. Regards, Brian. $\endgroup$ – Brian Jul 3 '12 at 3:31

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