10. Let $n \ge 5$ be an odd number and let $r$ be an integer such that $1\le r \le (n-1)/2$. IN a sports tournament, $n$ players take part in a series of contests. In each contest, $2r+1$ players participate, and the scores obtained by the players are the numbers $$-r, -(r-1),\cdots, -1, 0, 1 \cdots, r-1, r$$in some order. Each possible subset of $2r+1$ players takes part together in exactly one contest. let the final score of player $i$ be $S_i$, for each $i=1, 2,\cdots,n$. Define $N$ to be the smallest difference between the final scores of two players, i.e., $$N = \min_{i<j}|S_i - S_j|.$$Determine, with proof, the maximum possible value of $N$.