Problem number 18. Three football teams participate in a sports tournament. Each team plays the other two teams exactly once. In each game, the winner gets three points and the loser doesn't get any points. If the game ends in a tie, each team gets one point. At the end of the tournament, which number of points is it impossible for any team to have? A1, B2, C4, D5, or E6. This problem sounds like it has a lot of different numbers, but actually it's a pretty simple problem. Because each team only plays two games. Because it plays each of the other two teams exactly once. So there are two games per team. Also, for each of these games, it can only score three points if they win, zero points if they lose, or one point if it's a tie. So we need to figure out which of these numbers we can or cannot get, adding up two of these numbers. Now it's possible that the team wins both games or loses both games, so the numbers can be repeated. So can we get a total of one? The only way to get a one using zeros, ones, and threes is one plus zero. So the team would have tied one game and lost another game. So that's definitely possible. For two, it's one plus one. A team can get a one if it ties, and if it ties both games, gets two points in the tournament. So that works. Four can be made by three plus one. The team had one win and one tie. That's possible. For five, we would need something like three plus one plus one. But that means it would have to be more than two games. If we obviously did one plus one, that would be too little. If we did three plus three, that would be too much. If we had a zero in there for a loss, that also would not be enough. So five is a number that we cannot get. And then just to be sure, six can be made by three plus three, meaning the team would have won both games. So that one's possible. So since the problem asks us what's impossible for a team to get at the end of this tournament, meaning having played two games, the answer is D, five.

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