[Bozeman]

Let 10 say 1 if there are an odd number of white hats he sees, 0 if even.

Now each successor can correctly name his hat by seeing if he sees the same evenness as 10, counting the correctly named hats 9 through n+1.

So you should get 9 + 1/2 on average.

Craig

PS for the 3 hat problem a similar trick should work modulo 3.