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How many spheres of a 1-foot radius can fit in a 8x10x12

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How many spheres of a 1-foot radius can fit in a 8x10x12 [#permalink] New post 04 Nov 2006, 06:53
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How many spheres of a 1-foot radius can fit in a 8x10x12 foot container without allowing any of the spheres to touch more than six other spheres?

a. 100
b. 120
c. 144
d. 240
e. none of the above
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 [#permalink] New post 04 Nov 2006, 09:23
My answer is B - if the spheres have a 1 foot radius, then their diameters are 2 feet, making them 2 feet all 'round. So you could fit 4 along the 8 edge of the container, 5 along the 10, and 6 along the 12. 6*5*4 = 120 spheres. What's the OA?
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 [#permalink] New post 04 Nov 2006, 11:51
Sphere-packing problems are difficult and beyond the scope of this exam.
I'll bet the OA is uphillclimb's answer, but it isn't obvious. The problem is that the lattice there is a very bad packing and you can change the lattice at least along the sides to a close-packing lattice. Particularly around the corners, it is not at all clear that you can't fit more balls in than this. This problem is messed.

BTW - sphere packing is a problem that interested Gauss, Kepler, and oodles of very fine mathematicians.
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 [#permalink] New post 04 Nov 2006, 12:42
joeydvivre wrote:
Sphere-packing problems are difficult and beyond the scope of this exam.
I'll bet the OA is uphillclimb's answer, but it isn't obvious. The problem is that the lattice there is a very bad packing and you can change the lattice at least along the sides to a close-packing lattice. Particularly around the corners, it is not at all clear that you can't fit more balls in than this. This problem is messed.

BTW - sphere packing is a problem that interested Gauss, Kepler, and oodles of very fine mathematicians.


Thanks for the info on these great mathematicians....

Yes the wording should be very good even to attempt these questions
  [#permalink] 04 Nov 2006, 12:42
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