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Imagine just putting the spheres in one at a time until you can't put in any more. Why can't you put in any more? It must be that every point of [0.495,0.505]^1000000 is within 0.99 of the centre of one of the spheres. Otherwise you could put in a new sphere with that point as its centre. This means that if we imagine spheres of radius 0.99 around each of our original spheres (which only have radius 0.495) those big spheres must cover all of [0.495,0.505]^1000000. Hence their combined volume must be as large as the volume of that box. Note that I defined V to be the volume of a sphere with radius 0.99, not diameter 0.99 as in the statement of the problem.



Thanks, that's much clearer and I understand now. That's a good simple lower bound.




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