Exercise 3. Three-person football is played in successive phases: one player is the goalkeeper while the other two, called "field players," try to score a goal. As soon as a player scores, the phase ends and they become the goalkeeper for the next phase. Amandine, Bobby, and Charles are playing this game. At the end of the game, they remember that Amandine was a field player 12 times, Bobby was a field player 21 times, and Charles was the goalkeeper 8 times.

a) How many phases were there in total?

b) Who scored the sixth goal?
Solution to exercise 3 a) Let p be the number of phases: Amandine has been a goalkeeper $p-12$ times, Bobby $p-21$ times, and Charles has been a field player $p-8$ times. There is exactly one goalkeeper per phase, so the sum of the number of times each has been in this position is $p$, which is $(p-12)+(p-21)+8=p$, or $2 p-25=p$, thus $p=25$.

b) Over 25 phases, Amandine has been the goalkeeper thirteen times, which is more than half of the time. However, one cannot stay in goal two times in a row. Therefore, she was the goalkeeper during the odd phases, which are phases $1,3, \ldots, 23,25$. Thus, she was the goalkeeper during the seventh phase, so she scored the sixth goal.