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Thedutchcuber

Post subject: Rubiks cube positions? Posted: Tue May 21, 2013 9:24 am 

Joined: Mon Feb 11, 2013 8:33 am

I've a question about the number of positions for a Rubik's cube. Is there a formula for the number of positions on a Rubik's cube. If there is a formula like that than my question is if its possible to calculate it for higher order cubes ( 4x4, 5x5 etc.)
_________________ PB: 3x3: 21.09 2x2: 3.00 My Shapeways shop My YouTube channel!!


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cisco

Post subject: Re: Rubiks cube positions? Posted: Tue May 21, 2013 9:46 am 

Joined: Sun Feb 17, 2008 2:32 pm

It's all about combinations There are 8 corners that can be in 8, 7, 6, ..., 1 positions, so 8! There are 12 edges that can be en 12, 11, 10, ..., 1 positions, so 12! Corners can be in 3 different orientations, so 3^8 Edges can be in 2 different orientations, so 2^12 But not really... there are permutation restrictions, so you have to divide by 2. And there are orientation restrictions, so you have to divide by 2*3. result: 8! * 12! * 3^7 * 2^11 / 2 = 43.252.003.274.489.856.000 different positions http://en.wikipedia.org/wiki/Rubik's_Cube#Permutations


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Brandon Enright

Post subject: Re: Rubiks cube positions? Posted: Tue May 21, 2013 10:37 am 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

We had a thread about counting the permutations of the Dioctipoid here: viewtopic.php?f=1&t=22891 Some of the content there could be useful. There was a discussion about the NPyraminx and NFTO permutation calculations here: viewtopic.php?p=287293#p287293And I posted a formula for the Nminx here: viewtopic.php?p=285637#p285637Have you tried to calculate the positions for an NxNxN cube yet? If not I can walk you through it.
_________________ Prior to using my real name I posted under the account named bmenrigh.


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Thedutchcuber

Post subject: Re: Rubiks cube positions? Posted: Wed May 22, 2013 6:41 am 

Joined: Mon Feb 11, 2013 8:33 am

bmenrigh wrote: Have you tried to calculate the positions for an NxNxN cube yet? If not I can walk you through it. First thank you!! But I haven't calculated the positions for an nxnxn cube? Max Knoop
_________________ PB: 3x3: 21.09 2x2: 3.00 My Shapeways shop My YouTube channel!!


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Paradox

Post subject: Re: Rubiks cube positions? Posted: Wed May 22, 2013 7:03 am 

Joined: Sun Feb 20, 2011 5:14 pm Location: Hertfordshire, UK

cisco wrote: result: 8! * 12! * 3^7 * 2^11 / 2 = 43.252.003.274.489.856.000 different positions
Out of curiosity, does this number exclude identical but 'rotated' positions? In other words the same position of cubes in relation to eachother, but with the cube upside down for example.
_________________ Q: How many puzzles does a collector need in their collection? A: Just one more.


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Konrad

Post subject: Re: Rubiks cube positions? Posted: Wed May 22, 2013 9:21 am 

Joined: Thu Sep 17, 2009 6:07 am Location: Germany, Bavaria

Paradox wrote: cisco wrote: result: 8! * 12! * 3^7 * 2^11 / 2 = 43.252.003.274.489.856.000 different positions
Out of curiosity, does this number exclude identical but 'rotated' positions? In other words the same position of cubes in relation to eachother, but with the cube upside down for example. If you fix the 6 centres in space, you can permute edges and corners and reorientate them relative to this position in space in the ~43*10^18 different configurations. The 24 different orientations in space are not counted as different positions. BTW, on a 3x3x3 Supercube (showing the orientations of all 6 centres) you have to multiply the calculated figure 43252003274489856000 by 2048 (4^6/2 ; divided by 2 because the last centre can be reorientated by 180° degrees only, independent on the other 5. (btw, never trust Excel doing large integer calculations. In this case you will get 43.252.003.274.489. 900.000. Gus pointed me to http://www.wolframalpha.com for correct calculations )
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Brandon Enright

Post subject: Re: Rubiks cube positions? Posted: Wed May 22, 2013 11:44 am 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

Paradox wrote: cisco wrote: result: 8! * 12! * 3^7 * 2^11 / 2 = 43.252.003.274.489.856.000 different positions
Out of curiosity, does this number exclude identical but 'rotated' positions? In other words the same position of cubes in relation to eachother, but with the cube upside down for example. To echo what Konrad said, the answer is "yes" under some assumptions. For any of the 4.3 * 10^19 positions. If you pick up that cube, flip it around at random, and set it down, the resulting position is identical and only counted once. The reason is that the ((8! * 12!) / 2) * (2^12 / 2) * (3^8 / 3) calculation doesn't take into account centers. The center core can be in 24 orientations but because the core shows the orientation of the puzzle we can ignore it. If you want to treat the centers as identical (like the Void Cube) you need to divide the 4.3 * 10^19 value by 12. The reason this is 12 and not 24 has to do with the parity of the edges and corners.
_________________ Prior to using my real name I posted under the account named bmenrigh.


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Brandon Enright

Post subject: Re: Rubiks cube positions? Posted: Wed May 22, 2013 11:50 am 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

Thedutchcuber wrote: bmenrigh wrote: Have you tried to calculate the positions for an NxNxN cube yet? If not I can walk you through it. First thank you!! But I haven't calculated the positions for an nxnxn cube? Max Knoop Have you tried? Are you able to calculate the positions for the 3x3x3? If so, can you calculate the number of positions for the 4x4x4 and 5x5x5? If you can detail what you know and can do it'll give us a better idea of where to start.
_________________ Prior to using my real name I posted under the account named bmenrigh.


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katsmom

Post subject: Re: Rubiks cube positions? Posted: Wed May 22, 2013 3:30 pm 

Joined: Sun Apr 16, 2000 3:17 pm Location: Hong Kong


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Brandon Enright

Post subject: Re: Rubiks cube positions? Posted: Wed May 22, 2013 3:51 pm 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

katsmom wrote: After an adult beverage... I read the title of this and thought.... katsmom bringing new meaning to terms like "superflip" and "finger tricks" since 2000
_________________ Prior to using my real name I posted under the account named bmenrigh.


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