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This is a simple question that I have done on a test paper, but the given answer is confusing me.

The question states that a person has a bunch of cubes of volume 3 cubic centimetres. The person also has a container of size 6cm x 6cm x 3cm. What is the maximum number of small cubes that can be fit in the large container? No matter how I try I get 36 cubes, but the answer given is 32. How is this so?

numbermaniac
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  • Hint: if a cube has volume $3$ cubic centimeters, what is the sidelength of the cube? – Peter Woolfitt Jun 09 '14 at 06:40
  • The container has a volume of 108 cubic centimeters but you won't be able to fill it up completely. As Peter Woolfitt says, how long is the side of each small cube? – JRN Jun 09 '14 at 06:44

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Each small cube has a side length of $\sqrt[3]{3}\approx 1.442$ cm. Two cubes would thus fit in the height ($2\sqrt[3]{3}<3$). Four cubes would fit in the length ($4\sqrt[3]{3}<6$) and four cubes would fit in the width. You could thus fit $2\times 4\times 4=32$ small cubes in the container.

JRN
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  • It is assumed that you cannot chop up the small cubes into smaller pieces. – JRN Jun 09 '14 at 06:53
  • Now, is it possible to prove that we can't fit more cubes in by allowing the cubes to not be parallel to the sides of the box? – JimmyK4542 Jun 09 '14 at 06:53
  • @JimmyK4542, very good observation! I don't have time to think about it right now (I have to leave soon) but perhaps someone else can think of a proof. – JRN Jun 09 '14 at 06:54
  • @JimmyK4542. This is a very good point since $32$ small cubes occupy $96$ $cm^3$ and so $12$ $cm^3$ are left. I am almost sure that shaking the container will allow to put at least one extra small cube. Now, the problem is to prove it !! Cheers. – Claude Leibovici Jun 09 '14 at 07:09
  • Wow, you guys have blown my mind. I did not realise this method of calculation. Thanks guys! (I'm interested to see @JimmyK4542 's idea in action.) – numbermaniac Jun 09 '14 at 08:10