# Math Is Fun Forum

Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

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## #1 2006-02-23 12:25:53

irspow
Member
Registered: 2005-11-24
Posts: 1,055

### 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?

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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## #2 2006-02-23 13:53:25

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

### Re: Number of spheres within a cube.

"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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## #3 2006-02-23 14:02:46

irspow
Member
Registered: 2005-11-24
Posts: 1,055

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

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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## #4 2006-02-23 14:11:51

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

### 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.

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## #5 2006-02-23 17:37:21

MathsIsFun
Registered: 2005-01-21
Posts: 7,711

### 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

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## #6 2006-02-24 08:25:51

irspow
Member
Registered: 2005-11-24
Posts: 1,055

### 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.

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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## #7 2006-02-25 04:10:21

irspow
Member
Registered: 2005-11-24
Posts: 1,055

### 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.

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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