Question 113
In the $O$-$E$ game, a round starts with player $A$ paying $c$ cents to player $B$. Then $A$ secretly arranges the numbers $1, 3, 5, 7, 9, 11, 13$ in some order as a sequence $a_{1}, a_{2}, \ldots, a_{7}$, and $B$ secretly arranges $2, 4, 6, 8, 10, 12, 14$ as a sequence $b_{1}, b_{2}, \ldots, b_{7}$. Finally, the players show their sequences and $B$ pays $A$ one cent for each $i$ in $X=\{1,2,3,4,5,6,7\}$ such that $a_{i}\lt b_{i}$. This finishes the round. What number $c$ would make the game fair? (The game is fair if the total payments by $A$ to $B$ equals the total payments by $B$ to $A$ after all possible distinct rounds are played exactly once.)