January Question:

There are 213 teams prepared to compete in a regional hockey tournament. The tournament is a single elimination tournament: one loss and your team is out of the tournament. How many games must be played in order to determine the winner of the Regional Hockey trophy?

January Answer:

This problem requires us to adjust our thinking. Initially, it may seem very difficult. Let's consider two different approaches. First, we could set up a simpler problem showing the schedule for various teams.

It seems that for these 4 teams, 3 games are required. We could continue to build on this bracket schedule until all 213 teams have been scheduled. This becomes tricky, though. Depending on the number of teams scheduled, byes may be necessary!

The second approach is much easier. We can simply consider the losers of the tournament. Each time a game is played, one team is eliminated -- the losing team. In order for the trophy to be awarded, there must be enough games to decide only one winner. In other words, 212 teams must be losers. The fastest approach is to have one team the final winner and the other 212 teams lose their first game. Thus, a minimum of 212 games must be played.