Proof of an olympiad

Question

3 friends Ronny, Steven and Tom play soccer one afternoon. Because they're three of them, it's pretty hard to play soccer so they decide the following: there will be 2 field players and they try to score at the third player who's the goal keeper.

If there has been made a point then a new game will start: the goal keeper will become field player and the one who made the point will become the goal keeper.

That afternoon Ronny has been field player 12 times, Steven 21 tiled. Tom stood 8 games as 1 goal keeper.

Who scored the 6th goal point?

My trial: Assume there are only 21 games. That means Steven never has been a goal keeper. Ronny has been goal keeper 9 times and Tom 8 times as mentioned above. If Steven never has been goal keeper and Ronny starts by making a goal then Tom will be the one who scores the 6th point. Since Steven never scores and Ronny only scores the uneven goals. And the vice versa for when Tom starts making a goal.

Is this the right way about going about this question?

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Answer 1

Say, there are $n$ games.

We know Tom has been a goalie $8$ times. Steven will be a goalie $(n-21)$ times and Ronny will be a goalie $(n-12)$ times.

So number of games, $n = 8 + n - 21 + n - 12 \,$ i.e $n = 25$

Now that tells us that Ronny has been a goalie $13$ times and has been a field player $12$ times. That is only possible if he started as a goalie in the first game and was goalie every alternate games - 1, 3, 5, 7, ...25. So he must have made the 6th goal to be a goalie in the 7th game.