I was just wondering, bearing in mind a 3x3 cube has rotatable centres, how many states actually count as "solved"?

At first I thought it was 4^6, then I thought it might be half that since they always rotate in pairs of 90 degrees...

Anyone better at maths who can explain it?

The first 5 faces can be in any 4 orientations. The last one has only 2 options. I don't understand the math behind it but I know that these are the restrictions.
So it's 4^5 * 2 = 2048

It also depend on rather you consider global rotations of the entire puzzle as different states. If you do then you can multiply this by 24. I remember that figure as 6*4. You can pick 1 of the 6 faces to be up. You then can pick only 1 of 4 faces to be forward.

Other then that I agree with GuiltyBystander.

Carl

Surely there are an infinite number of solved states: solved puzzle on a table, on a chair, under this book, or that book, 2mm this way, 4cm that way, rotated by 12 degrees to the left, raised by 3 meters, 8 degrees warmer, moving at 12m/s this way, accelerating 5mm/hour that way, spinning at 6 rpm clockwise about this axis, or 5.7625548 million rpm anticlockwise about that axis, etc.

Come on... do you really think Monk would consider those solved stated? To be solved it must be at rest in the dead center of a perfectly white square room. Preferably a class 1 clean room that is temperature and humidity controlled. With all the side of the cube parallel to the walls of the room. Arg!!!! 24 is not a multiple of 10... take back everything I just said. Monk would clearly just take all the stickers off and replace them with all white stickers. There... just 1 solved state. Monk is happy now.

Carl

*laughs* Thanks for my first laugh in the morning hehehe