We just crossed the halfway point of the tournament, when measured in terms of number of games played. The tournament consists of a total of 63 games, of which:
32 games have been played over the past 11 days: 4 games in each of the 8 groups;
there are 16 games remaining in the group phase: 2 games in each of the 8 groups, with 4 games per day over the next 4 days (Tues-Fri);
16 teams will advance to the knockout phase: 2 teams from each of the 8 groups;
15 games will be played in the knockout phase: 8 + 4 + 2 + 1.
So the total # of 63 can be broken down as follows:
63 = 32 + 16 + 15 = 8*4 + 8*2 + (8 + 4 + 2 + 1).
Let me know if you see anything wrong with the math here.
5 comments
Sean
June 21, 2010 at 4:45 PM
can i get a pie chart or something? Also, I sense you’re trying to unlock some sort of secret here with your fancy summation. What are you getting at, mathman?
Suman
June 21, 2010 at 5:00 PM
All will be revealed in due time. Some interesting combinatorics here. For example, the knockout phase: 15 games = 8 + 4 + 2 + 1 = \sum_{i=0}^{3} 2^i. This sum of course works for any n, where n is the number of rounds in the bracket. Here n = 3, but for the NCAA tournament with 64 teams, n = 6 (since 2^6 = 64), and there are a total of 63 games in the tournament. In general, for a tournament with 2^n teams there will be (2^n – 1) games.
Sean
June 21, 2010 at 5:55 PM
I think this was a car talk brain teaser last year.
Suman
June 22, 2010 at 1:13 AM
Could be..but sounds unlikely. Also: @johnglally pointed out that there is a 3rd place match which I didn’t take into the count..so actually the title of the post should have been “32 games done, 32 to go”..what fearful symmetry.
Sean
June 22, 2010 at 3:13 AM
Sounds unlikely? Pshaw. I say close enough! http://www.cartalk.com/content/puzzler/transcripts/201008/index.html
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