# How many games should be played to avoid tying?

There are $n\ge3$ players playing a game. In this game, one person will come out in first place, one in second, and so on. It's impossible to tie. The person in first place gets $n$ points, the person in second place gets $n-1$, and so on, so that the person in last place gets $1$ point. After playing this game some fixed number of times, the scores are tallied up and the winner is whoever has the highest score.

Can we fix a number of games (other than one game) in advance, so that a tie for first place is impossible? With two players this is easy - just play an odd number of games. But otherwise?

• Yup, this is basically the solution I used as well, although I expressed it as "if you can tie with $n$ and $m$ games, you can tie with $n+m$ games" and the fact that you can tie for 2 and 3 games. By the way, it might be a good idea to spoiler at least the first line of your answer, since knowing the answer might influence how a person thinks about the problem. Mar 14, 2015 at 1:49