<t>Cool.So... that shows that the timid strategy is optimal for all p >= 1/2.The question then is, is it uniquely optimal for p=1/2. I would still expect that min(N-i,i,a) for any integer a > 0 would also be optimal.Logically, it would also make sense that if p<1/2 you should bet min(N-i,i). I am no...