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#1 2006-02-24 11:25:53
- irspow
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Number of spheres within a cube.
What is the maximum number of 1" spheres that will fit completely within a 24" cube?
As a bonus, what is the percentage of empty volume within this maximized arrangement?
#2 2006-02-24 12:53:25
- Ricky
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Re: Number of spheres within a cube.
1" diameter or radius?
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
#3 2006-02-24 13:02:46
- irspow
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Re: Number of spheres within a cube.
Sorry if I was not clear Ricky. The 1" is the diameter of the spheres. No tricks or gimmicks, this is just a generic geometry puzzle.
Last edited by irspow (2006-02-24 13:52:52)
#4 2006-02-24 13:11:51
- ryos
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Re: Number of spheres within a cube.
This looks like a chemistry problem! It's tricky. I'll have to look at it...later...
El que pega primero pega dos veces.
#5 2006-02-24 16:37:21
- MathsIsFun
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Re: Number of spheres within a cube.
Oh no - packing!
Intuitively a Tetrahedral packing is tightest, but because of a specific (in this case 24") limit, there may be "slack" that could be better filled with some rearrangement.
A computer program is called for (I think) and a darn good one.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#6 2006-02-25 07:25:51
- irspow
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Re: Number of spheres within a cube.
I'll give you guys a break, the slack isn't enough to change the answer for a slightly smaller cube. I just picked a 24" cube because it was a nice integer value. No computer program is necessary either. Like I said before, this is just a plain geometry problem. I'll give it another day before posting the solution if no one gets it first.
#7 2006-02-26 03:10:21
- irspow
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Re: Number of spheres within a cube.
Here goes the solutions, I will hide them for those who wish to solve them on their own.
The "bonus" is easier.