One move-sequence solves the Rubik's Cube!!

--> [^:::^] 

Starting from the solved state of cube, if the same move-sequence S is applied successively, then the cube will return to the solved state again

Proof:   

Since, A Rubik'sCube has finitenumber of possible states. Therefore, if we start from the solved state of a cube and apply successively the same move- sequence S, then we must return to the same state again because each time, application of S jumbles the stickers.

Let, 

m = minimum number of times S is applied to get the same arrangement k for the first time; where k<m

Hence, S^k = S^m ------------------------------------------ (i)

To prove: The cube return to the solved state again.

Proof: If we prove that k=0, then automatically it will be proved that the cube will return to the solved state again.

If k=0, then S⁰ = S^m = 1 , hence proved.

Assume: k > zero.

Multiply, both sides of (i) by S^-1 , then

S^k-1 = S^m-1 ------------------------------------------ (ii)

But, from (ii), it means that we repeated an arrangement which is contradiction to the fact that m is least number of times S is applied to repeat an arrangement.

Hence, k must be 0.

( 3x3x3 Rubik's Cube Blindfolded Tutorial)