# Question about positions of the Rubik's cube

#### Robert-Y

##### Member
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?

#### AvGalen

##### Premium Member
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

#### Robert-Y

##### Member
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

#### qqwref

##### Member
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.

#### Lucas Garron

##### Member
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

Last edited:

#### qqwref

##### Member
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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