# Question about positions of the Rubik's cube



## Robert-Y (Mar 20, 2009)

How many possible positions can be attained by the Rubik's cube, assuming that you can pop out the corners and edges and place them where you like *but not the centres*?


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## AvGalen (Mar 20, 2009)

Robert-Y said:


> How many possible positions can be attained by the Rubik's cube, assuming that you *can *pop out the corners and edges and place them where you like *but not the centres*?



12 times the normal amount

Or are you allowing positions where not all corners and/or edges are place back after the poppingout


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## Robert-Y (Mar 20, 2009)

Yes, all the corners and edges must be placed back into the cube. Is the answer really that simple? I thought maybe I was being simple minded when I thought it was just 12 x 43 quintillion


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## qqwref (Mar 20, 2009)

Yeah, it's just that. The more mathematical way to calculate it is
(corner possibilities)*(edge possitibilities)
= (8! 3^8)*(12! 2^12) = 5.19024039 x 10^20.


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## Lucas Garron (Mar 20, 2009)

Since someone's gonna ask what happens when you're allowed to move centers:

Fix a corner.
(7!*3^7)*(12!*2^12)*(6!)
=15570721178816348160000

And because math is so beautiful: Fix an edge.
(8!*3^8)*(11!*2^11)*(6!)
=15570721178816348160000

(Extra factor of 360.)

EDIT:
While we're at it, fix a center.
(8!*3^8)*(12!*2^12)*(5!)/4
=15570721178816348160000


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## qqwref (Mar 20, 2009)

You mean an extra factor of 30. When you fix a corner or edge there are 24 ways to fix centers if you're not allowed to move the center caps; if you can move the center caps there are 720 ways.


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