RETRACTED -- THIS IS WRONG because it does not consider white beads.
i can either see a simpler than the accepted answer, or the explanation is too complex. here's the algorithm in pseudocode (ignoring the case of all same color):
p = position of first bead of different color than bead 0
create array of run-lengths: r={5,15,2,9,13}
create array of sums of adjacents s={20,17,11,22,18} (wrap last)
find index, k, in s[] of max sum: here it is s[3]
position = p + r[0] + r[1] + ... + r[k]
