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Sphere Cube (Posted on 2004-02-09) Difficulty: 2 of 5
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In a cube of side 4, I pack eight spheres of unit radius.


What is the largest sphere I can place in the center (such that it doesn't overlap any of the other spheres)?

See The Solution Submitted by SilverKnight    
Rating: 3.0000 (5 votes)

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another dimension | Comment 10 of 16 |
As several people have posted, I agree that the answer is √3-1. Here's a more general problem: I have an n-dimensional 'n-cube' of side 4. I pack 2^n 'n-spheres' of radius 1 packed in it, similarly to the 3 dimensional problem. What's the largest 'n-sphere' I can place in the centre? What happens as n gets larger?

Surprisingly, the radius of the inner 'n-sphere' increases with n, and for enough dimensions it will stick out of the sides of the 'n-cube'! If I'm not mistaken, it goes to infinity with n, but I'll have to do the calculation.
  Posted by pleasance on 2004-02-13 14:06:31
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