Question 7
Players numbered $1$ to $1024$ play in a knock-out tournament. There are no draws, the winner of a match goes through to the next round and the loser is knocked-out, so that there are $512$ matches in the first round, $256$ in the second and so on. If $m$ plays $n$ and $m \lt n - 2$ then $m$ always wins. What is the largest possible number for the winner?