When two people play a game, usually one loses and one wins, and the winner is the champion.
If there are three players, and we want to have a single competition between every pair, then we need to set up three one-on-one matches. But the results of such a tournament might still not produce an obvious single champion. For instance, if A beat B, B beat C, and C beat A, there seems to be no way to choose one of the three players as superior to the others.
Let's consider the general case in which we have N players. (To avoid silliness, we assume that N is at least 2!). We set up a round robin competition, that is, each player plays one game against all other players. From the results of this competition, we need to declare the championship.
We would like to do this in a way that does minimum violence to our simplistic idea of a champion, while always guaranteeing that we are able to a champion.
Our approach is to allow multiple champions. We declare a player A to be a champion if it is true that, for each other player B, either A beat B or else A beat a player that beat B.
This would mean that in our little three player example, all three players would be declared "champions", for instance.
It is obvious that if someone is a champion in the usual sense (by beating everyone), then that person will be declared a champion in this new sense.
But some questions remain to be answered, of course, and who better to answer them than you?
Puzzle #1: Must there always be at least one champion?
Puzzle #2: Can there be exactly two champions?
Puzzle #3: In a game with N players, how many champions can there be?
I give up, let me see the solution.
Blame this one on Scott Hansen!