Heuristics for an Artificial Intelligence problem
All white tiles must be moved all the way to the left.  Call the left most position 0.  Then, in the final position, the sum of white distances from the axis is 0+1+...(N - 1).

Calculate this value, the "solution distance" of the tiles up front.  As a heuristic, add the zero-based positions of white tiles up, and subtract from this sum the "solution distance" calculated a priori.

This heuristic gives you a minimal cost of finding a solution (eg. it will cost at least that much to get all white tiles in the correct place).