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Topic: pack the gift (Read 684 times) |
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inexorable
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pack the gift
« on: Feb 15th, 2007, 3:46am » |
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As a Valentine's Day present, Alice got Bob a set of infinitely many cubes, having side-lengths 1, 1/2, 1/3, ..., one cube of each side. But how to gift-wrap them? Alice knows that the harmonic series diverges, so she cannot stack one on top of another in finite space. But she also knows that the total volume of the cubes is finite, since
1 + 1/(2^3) + 1/(3^3) + ... < integral of (1+ 1/x^3) dx with x limits from 1 to infinity = 1.5
So the total volume is less than 1.5 and it seems as if she could just pack them all into a gift-box of dimensions 1 × 1 × 3/2. At this point she realized that a better idea would be to give Bob the cubes and the gift box and make him do the packing.
Show how to arrange the cubes so that they fit into a 1 × 1 × 3/2 box.
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Grimbal
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Re: pack the gift
« Reply #1 on: Feb 15th, 2007, 6:11am » |
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The first cube is easy 
The second probably also.
But then... 
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towr
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Re: pack the gift
« Reply #2 on: Feb 15th, 2007, 6:16am » |
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Why should they fit?
The volume may correspond, but if I had a sphere with volume 1 I wouldn't eb able to put in an a box with volume 1, even thought he volumes are equal.
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inexorable
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Re: pack the gift
« Reply #3 on: Feb 15th, 2007, 6:29am » |
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on Feb 15th, 2007, 6:16am, towr wrote:Why should they fit?
The volume may correspond, but if I had a sphere with volume 1 I wouldn't be able to put in an a box with volume 1, even thought he volumes are equal. |
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True. But one box A does not fit in another box B only if there does not exist any combination of dimensions of A such that they are less than or equal to dimensions of B. and at each step of placing one box in box of 1X1X3/2 and considering the dimensions of remaining space it seems there is space for new box to fit.
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Grimbal
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Re: pack the gift
« Reply #4 on: Feb 15th, 2007, 9:53am » |
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OK. How to pack:
- Cube n is 1/n on a side.
- Cube 1 fills most of the box.
- The remaining space is 1x1x0.5 that can be split into 2 spaces of 1x0.5x0.5.
- Pack the even cubes 2,4,6,... in the one space by using the same method recursively, but scaled down 1/2 size. With some space left.
- Pack the remaining odd cubes 3,5,7,... in the other space as if they were as large as the even cubes 2,4,6,..., with lots of gaps everywhere.
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| « Last Edit: Feb 15th, 2007, 9:55am by Grimbal » |
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towr
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Re: pack the gift
« Reply #5 on: Feb 15th, 2007, 10:29am » |
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Not so hard, seeing it like that.. Well done.
I also hadn't noticed the '<'. About 19.9% of the box is left free.
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| « Last Edit: Feb 15th, 2007, 10:31am by towr » |
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