As we all know, the centers of the 7x7 have one sticker. So I think that it's logical to conclude some may be swapped and the cube still remain solved. Is this correct? And if so, how many total "solved" positions are there?

thnx

I did a calculation of this for the 20x20x20 cube

here.

For the 7x7x7 there would be:

(4!)^36 / 2^6 = (4!)^30 * 2^12 * 3^6

That's approximately 7 * 10^47 "solved" states where you could move around centers of the same color and still appear solved.

And for the n x n x n cube it would be:

(4!)^[6*floor((n-2)^2 / 4)] / 2^floor[(n-2)^2 / 4]

or

2^[17*floor((n-2)^2 / 4)] * 3^[5*floor((n-2)^2 / 4)]

"solved" states.

Here floor(x) means the greatest integer less than or equal to x.

Chris