Cube

[heman]

suppose there is a cube and we can colour the cube's faces with only two colours ..i.e.
black and white ,,how many different patterns are possible....

[Muzza]

Seems like the perfect place to apply Burnside's theorem (about orbits).

[pallidin]

Here's my take:

How many faces are on a cube? 6
How many allowable variations per face? 2
Combinations possible: 2*2*2*2*2*2 =64

Oh, but wait, this is not a strictly linear situation!
My mind is melting.

[Rach3]

all white: 1 version
1 black side: 1 version
2 black: 2 versions (adjacent/nonadjacent)
3 black: 2 versions (three in a row/three sides with common vertex)
and by symmetry...
4 black = 2 white: 2 versions
5 black: 1 version
6 black: 1 version

Total... 10 indistinguishable ways.

[heman]

Thanks Rach,,
i am clear with how that value 10 comes..but actually the theory related with Polya's theorem is n't much clear to me...i am actually not clear with how they have derived the formula.