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Well, that's how my little sister asked it, and I couldn't provide a quick answer.

If we ignore any changes in shape from the cutting process, then my first thought was that the scaling wouldn't make any difference. But then I started thinking that more efficient packings might only become possible after we cut the apple pieces down to a certain size.

I realise this is almost unanswerable without multiple assumptions (spherical pieces of apple anyone? :) ) but at the end of the day I'm interested in what is known (if anything) about the packing of irregularly shaped solids in finite containers of similar dimension (within an order of magnitude) to the solids themselves.

Thanks

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Let $a$ be the diameter of an apple. Look at lunchboxes of height $<a$, but large other dimensions. Then you can fit $0$ apples in the box, but many if you make apple sauce. If details are provided about dimensions, more specific numerical bounds can be given. –  André Nicolas Nov 5 '11 at 7:47
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Well obviously you can. Consider the limit: if you grind your apples to a paste, you can use up the entire volume of the lunch box, while with full apples you're very unlikely to. Look up packing problems -- you can also click on the tag below your question. :-) –  ShreevatsaR Nov 5 '11 at 7:48
    
surely if the shape of the apple remains the same (so we're not just going to a paste, they remain, say, spheres) then the limit as their size goes to 0 doesn't correspond to filling 100% of the volume of the container? –  tom Nov 5 '11 at 7:59
    
@AndréNicolas if numerical bounds could indeed be established, would it make any difference for, say, a 10cm * 10cm * 10cm container whether the irregular shapes where ~3cm in size or ~1cm in size. I guess that's the gist of the question. –  tom Nov 5 '11 at 8:02
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@tom - when you cut an apple with a knife, the pieces you get are more like cubes than like spheres. So how about the following: Get the lunch box (or a handy rectangular container) and fill it with marbles, or apples. Pour water into the remaining space. Remove the marbles and measure the water. Now do the same with blocks -- pack with blocks, fill the remainder with water, remove blocks, measure amount of water. (These kinds of experiments lead to interesting places!) –  Sam Nead Nov 5 '11 at 10:31

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