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A refrigerator is 8 inches long , 4 inches wide and 5 inches high.How many ice cubes will it hold if each cube is 2 inches on the edge.

According to me its 20 because volume of Fridge = 160 and volume of cube=8 so 160/8 = 20 However the books says the answer is 24. Am i missing something or is this a misprint ?

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A complicating factor is that you cannot fill the 5 inches of height perfectly, since you presumably don't want to slice up the cubes. – akkkk Jun 28 '12 at 12:16
It's important to develop the habit of "sanity-checking" problems like this for physical constraints you might have missed. As we see here, blindly applying the math can lead to wrong results. – rschwieb Jun 28 '12 at 12:18
So the books answer is then completely off since its assuming more cubes that i have calculated ?? – Rajeshwar Jun 28 '12 at 12:19
Smallest refrigerator I ever heard of. And if it's about ice cubes, a freezer would be a better idea. – Gerry Myerson Jun 28 '12 at 13:09
@Rajeshwar: Your reasoning is only partly right. By volume considerations, the answer is clearly $\le 20$. In fact it is less than $20$, because of the $5$ inch part. We can certainly fit in $16$. Perhaps one can fit in one more by using a strange packing. Certainly one can if we replace $5$ by $5.9$. – André Nicolas Jun 28 '12 at 17:56

You can get only $\frac{8}{2}$ times $\frac{4}{2}$ times the floor value of $\frac{5}{2}$, which is $2$ inside. This means a total of $16$ ice cubes.

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That's assuming that the obvious packing is actually optimal. That does seem pretty likely, but it might be non-trivial to prove. In any case, per the OP's argument, the answer certainly can't be more than 20. – Ilmari Karonen Jun 28 '12 at 12:20
What if some way of rotating them can pack more in? – GEdgar Jun 28 '12 at 12:26
@GEdgar:rotation doesn't decrease the volume of the cube nor does it increase the volume of the refrigerator.Rotation only helps when there is some void between filling and those voids can be reduced by rotating.Here, assuming no void there are only 20 cubes that can be put in the refrigerator. – Aang Jun 28 '12 at 12:51
I agree 16 is the right answer, but so far the reasoning is not completely convincing. The lower bound 20 as computed by the OP is correct and convincing. – GEdgar Jun 28 '12 at 14:22

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