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Topic: number of cube configurations (Read 1052 times) |
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sportsdude
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number of cube configurations
« on: Jun 23rd, 2013, 6:03pm » |
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A 3x3x3 cube is comprised of 27 unit cubes. By removing one of these cubes, we can create one of 4 different possible shapes: one with a corner missing, one with the center of an edge missing, one with the center of a face missing, and one with the center of the cube missing.
How many configurations are possible if we remove 2 units?
Is the solution generalizable to n units from a kxkxk cube?
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jollytall
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Re: number of cube configurations
« Reply #1 on: Jun 24th, 2013, 12:52pm » |
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Let me give it an ugly try:
1. One of the missing is a corner
1.1. The other is a corner too
1.1.1. They are on the same edge (1 case/3 symmetrical)
1.1.2. They are on the same side, diagonal (1/3)
1.1.3. They are opposite on the cube (1/1)
1.2. The other is on an edge.
1.2.1. The other is on the edge of the missing corner (1/3)
1.2.2. The other is on the edge of the opposite corner (1/3)
1.2.3. The other is on the "belt" in between (12/6) - as corrected below, here also the mirror can be counted twice.
1.3. The other is a side center
1.3.1. The other is on the same side as the missing corner (1/3)
1.3.2. The other is on an opposite side (1/3)
1.4. The other is in the center (1/1)
2. One of the missing is on an edge
2.1. The other is also on an edge
2.1.1. The other is on the same side, neighbour (1/4)
2.1.2. The other is on the same side, opposite (1/2)
2.1.3. The other is totall the opposite (1/1)
2.1.4. The other is one of the remaining (2/4). A bit of explanation: The 4 cases are 2 pairs. The pairs can be moved into each other by rotation, but the two in the pair are mirror symmetrical (optical isomers as in chemistry would be called).
2.2. The other is a side center.
2.2.1. Neighbour to the missing edge (1/2)
2.2.2. Neighbour to the opposite edge (1/2)
2.2.3. The other two sides (1/2)
2.3. The other is the middle piece (1/1)
3. One of the missing ones is a side center
3.1. The other is also a side center
3.1.1. Neighbouring side (1/4)
3.1.2. Opposite side (1/1)
3.2. Middle piece (1/1)
If I can add up, it is 20 or 2122 depending whether optical isomers are the same or different.
I would not try the generalisation.
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| « Last Edit: Jun 25th, 2013, 3:24am by jollytall » |
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sportsdude
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Re: number of cube configurations
« Reply #2 on: Jun 24th, 2013, 6:03pm » |
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Mirror images are considered distinct, so I think you missed a few.
For instance, 1.2.3 would add two to the total (that is, if we name the members of one face 1,2,3,4...9 from left to right and top to bottom, "1 and 6" and "1 and 8" are separate configurations).
Then again, I'm not entirely sure of my own answer, so I might be wrong 
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jollytall
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Re: number of cube configurations
« Reply #3 on: Jun 25th, 2013, 3:23am » |
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I think you are right. I change the above. Then the total is 20 or 22, counting or not mirrored ones.
You mention a "few". Which else is more than one?
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| « Last Edit: Jun 25th, 2013, 3:25am by jollytall » |
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rmsgrey
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Re: number of cube configurations
« Reply #4 on: Jun 25th, 2013, 3:42am » |
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For 2.1.4., since both cubes are missing from edges, you only get enantiomers if you distinguish the "first" and "second" holes somehow. Otherwise, you can get the mirror image by rotating so that the "second" occupies the place of the "first" and the "first" occupies the place of the reflection of the "second".
If you are distinguishing between first and second holes, then you're also missing cases where the first is an edge and the second a corner, the first is a face and the second edge or corner, and where the first is the central cube.
(see below)
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| « Last Edit: Jun 26th, 2013, 3:45am by rmsgrey » |
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jollytall
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Re: number of cube configurations
« Reply #5 on: Jun 25th, 2013, 5:24am » |
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I am not convinced.
Let's take the cube in front of us. The "first" missing one is on the top side closest to us. The second ones are on the back side, middle row, on the two sides.
To get from the first to the second, we define a move sequence. First go from the first parallel to the edges across one of the sides, so to reach the side where the second is missing. There take a 45 degree turn left or right to get to the second.
If above, we take e.g. on the back side the right one, as the second, then the move is as follows: Go straight on the top side and then turn 45 right on the back side to get there. If you want to come back from the second to the first, then the first step is to come straight front on the right sige and then 45 right to go up to the first on the front side.
I.e. A "TURN-RIGHT" pair will not become a TURN-LEFT pair even if you swap first and second.
(To your other suggestion, I would not label first and second, because then a lot of new pairs would born.)
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rmsgrey
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Re: number of cube configurations
« Reply #6 on: Jun 26th, 2013, 3:44am » |
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on Jun 25th, 2013, 5:24am, jollytall wrote:
Yeah, I missed that I'd given my physical cube a 90-degree turn in rotating one edge to the other's position, which put the other edge in the location of the first's reflection, but with the cube oriented illegally.
Oh well...
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