How to determine the number of a cube within a bigger cube?

Hi all,

I have a cube, sized 39 x 13 x 8. I need to find out how many of them can fit in a cube of 100 x 100 x 100. I need to find the highest number possible.

Do you know of a way to do that without having to draw them, with trial and error?

Many thanks in advance

Septronic

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I do not have a definite answer, and I am surely no expert on this topic. But I can say that similar problems are hard: en.wikipedia.org/wiki/Bin_packing_problem In your case, a computer program might be the easiest way out. – jmc May 18 '13 at 14:09
Your latter cube is indeed a cube, but the first-mentioned "cube" is not. Such a rectangular body is called a cuboid. – John Bentin May 18 '13 at 17:42
Do you think trial and error will find the optimal packing? I feel like I have to think "outside the box" to pack 5 unit squares in a 2.8x2.8 square. Hopefully this sort of packing is not necessary for your problem. – Dustin G. Mixon May 18 '13 at 18:59
@John : $\:$ I suspect that what the OP has is a rectangular prism, since a cuboid can be more general. – Ricky Demer May 18 '13 at 20:02
FYI - I asked a follow-up question, and got an answer that might interest you: mathoverflow.net/questions/131084/is-this-cube-packing-possible – Dustin G. Mixon May 19 '13 at 0:25

1 Answer

It might be hard to get an optimal packing, but it looks like this paper can get you close. Page 24 illustrates different instances of their algorithm's solution. For example, this is how they pack 255 rectangles of size 137x95 into a 2530x1320 rectangle:

For more information, google "manufacturer's pallet loading problem."

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