Problem 1604

Solution:

The answer is $\mathbf{(B)}$. Let us build a table with one row for each play played and one column for each player of the Geodesics. We write +1 in a cell if the player corresponding to the row was on the field during the play corresponding to the column and the play was won by the Geodesics. Similarly, we write -1 for the pairs (player, play) given by the plays lost by the Geodesics with that player on the field. Finally, we write 0 in the cells corresponding to pairs (player, play) for which the player was not on the field in the corresponding play. Let us now consider the sum of all the numbers in the table.

By hypothesis we know that Alice's column sums to +7, Barbara's column sums to -2, and all the other columns sum to 0. The sum of all the numbers in the table is therefore 5. On the other hand, each row of the table has sum $+k$ (if the play was won by the Geodesics) or $-k$ (if it was lost): indeed, in each row there are exactly $k$ nonzero numbers (corresponding to the $k$ players on the field), and they are either all equal to +1, if the play was won by the Geodesics, or all equal to -1, otherwise. Denoting then by $V$ and $P$ the number of plays won and the number of plays lost by the Geodesics, we have obtained the equation $5 = k(V-P)$. From this it follows that $k$ divides 5, and since $k > 1$ we have $k = 5$.