The Big Dance

A week ago yesterday our local newspaper printed the bracket for the NCAA Division I Men's Basketball Tournament.  I kept the bracket so that I could fill it in as the games were completed.

I was reviewing my bracket today, and I was just thinking about how many games will be played in the tournament.  Well, let's see:

The First Four (new this year, 8 teams) = 4 games
Second Round (64 teams -- 60 + the First Four winners) = 32 games
Third Round = 16 games (Winners are the "Sweet Sixteen")
Fourth Round = 8 games (to get the "Elite Eight")
Fifth Round = 4 games (finally... the "Final Four")
Sixth Round = 2 games
Championship Game = well, just one game (the winner is the Champion)

That's 67 games total.

But... before you compliment me on my calculation... I showed you the "long way around" to answer this question.  There is a much more elegant solution.

The tournament started with 68 teams.  Every game has a losing team.  Because it's a single-elimination tournament, the losing team is therefore out.  The only team that doesn't lose is the Championship team.  So 67 teams lose a game.  Hey, that's 67 games!  One less game than the number of teams in the tournament!

So, how many games in a single-elimination tournament if there are:

12 teams?  (This is the case in the NFL playoffs.)
64 teams?  (The good 'ole days of March Madness.)
47 teams?  (Don't know of an example.  But see how easy it is?!)

(Postscript:  Need help?  I'll post the answers as the first comment to this post.)