A gambler is presented with a sequence of $n \geqslant 6$ random numbers, $N_{1}, N_{2}, \ldots, N_{n}$, one at a time. The distribution of $N_{k}$ is

$$P\left(N_{k}=k\right)=1-P\left(N_{k}=-k\right)=p,$$

where $1 /(n-2)<p \leq 1 / 3$. The gambler must choose exactly one of the numbers, just after it has been presented and before any further numbers are presented, but must wait until all the numbers are presented before his payback can be decided. It costs $£ 1$ to play the game. The gambler receives payback as follows: nothing if he chooses the smallest of all the numbers, $£ 2$ if he chooses the largest of all the numbers, and $£ 1$ otherwise.

Show that there is an optimal strategy of the form "Choose the first number $k$ such that either (i) $N_{k}>0$ and $k \geq n-r_{0}$, or (ii) $k=n-1$ ", where you should determine the constant $r_{0}$ as explicitly as you can.