Exercise 3. Andréa, Baptiste, and Camille are playing soccer with three people. One of the players is in goal, the other two are on the field trying to score. The player who scores then becomes the goalkeeper for the next shot.

During the afternoon, Andréa was on the field 12 times, Baptiste was on the field 21 times, and Camille was in goal 8 times. Their teacher knows who scored the 6th goal. Who was it?
## Solution to Exercise 3

If we denote $\mathrm{n}$ as the number of games played, then in total there have been $\mathrm{n}$ times a goalkeeper. We now use the assumptions given in the statement. If Andrea has been on the field 12 times, then he has been $\mathrm{n}-12$ times in goal. If Baptiste has been on the field 21 times, then he has been $n-21$ times in goal. Thus, $n-12+n-21+8=n$, which gives us that $n=25$. This also means that Andrea has been in goal 13 times. But it is impossible to be in goal twice in a row. Thus, we know that Andrea was in goal at the beginning, and was in goal exactly every other time. In particular, he was in goal during the seventh game, and therefore Andrea scored the 6th goal.

Comment from the graders

The exercise was largely successful. Some proofs and write-ups are particularly efficient.