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**irspow****Member**- Registered: 2005-11-24
- Posts: 455

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?

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

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..."

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**irspow****Member**- Registered: 2005-11-24
- Posts: 455

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 (2006-02-23 14:52:52)*

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**ryos****Member**- Registered: 2005-08-04
- Posts: 394

This looks like a chemistry problem! It's tricky. I'll have to look at it...later...

El que pega primero pega dos veces.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,529

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.

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**irspow****Member**- Registered: 2005-11-24
- Posts: 455

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.

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**irspow****Member**- Registered: 2005-11-24
- Posts: 455

Here goes the solutions, I will hide them for those who wish to solve them on their own.

The "bonus" is easier.

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