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Topic: Perfect Cubes (Read 595 times) |
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Whiskey Tango Foxtrot
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Perfect Cubes
« on: Jun 6th, 2006, 12:11pm » |
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Given a perfect cube and the ability to cut a perfectly straight line, it would take 3 cuts to make 8 perfect cubes, each of exactly the same size.
How many cuts would it take to make 27 cubes?
How many to make 64 cubes?
How many to make n cubes?
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"I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use." - Galileo Galilei
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JohanC
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Re: Perfect Cubes
« Reply #1 on: Jun 6th, 2006, 1:49pm » |
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It depends whether or not you are allowed to rearrange the pieces before each subsequent cut.
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rmsgrey
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Re: Perfect Cubes
« Reply #2 on: Jun 6th, 2006, 2:30pm » |
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I'd want to be able to make a perfect plane cut - cutting along a line would offer no guarantees about the other end of the knife.
For making 27 cubes, it requires at least 6 cuts since the central cube needs to be cut on each face, and each cut can expose at most one face of the central cube
For 64, JohanC's point is significant with rearrangement, you can get it down to 6 cuts - bisect and then stack the two pieces and bisect parallel to the largest faces. Restore to the original arrangement and repeat for the other directions. Without rearrangement, you obviously need 3 in each direction to form the 4*4*4 array
For larger cube numbers, k3, I have partial results: without rearrangement, you need 3(k-1) cuts. With rearrangement, it looks like 3(ceiling(log2k)), but I don't have a full proof
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Whiskey Tango Foxtrot
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Re: Perfect Cubes
« Reply #3 on: Jun 18th, 2006, 3:24pm » |
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Yes, I was more interested in the case where the cubes can be moved. If they can't, this becomes very simple.
I have yet to prove the answer as well, by the way. I thought this up one morning and was interested if anyone had any techniques for finding a solution. My efforts have been fruitless thus far.
This might be too difficult for the easy section, though. If so, my mistake administrators.
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Grimbal
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Re: Perfect Cubes
« Reply #4 on: Jun 19th, 2006, 7:17am » |
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It is not that difficult.
Just consider the size of the largest remaining piece so far. The best you can do in a cut is to divide one dimension by 2, rounded up. This gives a minimum number of cuts of [log2(size x)] + [log2(size y)] + [log2(size z)], where [] is rounding up. And that number can be reached with the simple method of halving and stacking.
PS: the above is actually how to cut a block of a given size into unit cubes. For the problem at hand, the unit size would be 1/k of the size of the original cube, where k3=n, making the number of cuts 3·ceiling(log2(k)).
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| « Last Edit: Jun 19th, 2006, 8:16am by Grimbal » |
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Whiskey Tango Foxtrot
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Re: Perfect Cubes
« Reply #5 on: Jun 19th, 2006, 2:23pm » |
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Thanks guys. That was banging around in my head for a while.
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"I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use." - Galileo Galilei
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