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The ACC basketball tournament, one of the great cultural events of the year in North Carolina, changed to a new format several years ago. Some people didn't like the old format (which is why it was changed) and some people don't like the new one. Let's take a look at the mathematical problem of designing the tournament, and see what constraints the designers had to face. Could it have been done other ways?
The tournament is a single elimination tournament, meaning that a team is out of the tournament as soon as it loses one game. Our first question is, how many games will there be in the tournament?
Theorem: If there are N teams in the tournament, there must be exactly N - 1 games.
It's easy to see why! Each game eliminates exactly one team, and there is exactly one team, the champion, which is never eliminated. So we need N - 1 games to eliminate everyone except the champion.
Now for most of its history there were 8 teams in the Atlantic Coast Conference. The conference was founded with 8 teams in 1954. South Carolina withdrew in 1971, leaving only 7 teams, but Georgia Tech joined in 1978, restoring the number to 8. For the 8-team tournament there was a standard format, which looks like this:
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(#1 or #8 vs. #4 or #5) |
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(#2 or #7 vs. #3 or #6) |
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The numbers are the seedings of the teams: the teams are "seeded" according to their order of finish in the regular season (with some tie-breaker rules we don't need to consider). Contrary to the numbering, the #1 team is said to have the highest seeding and # 8 the lowest. Notice that for the quarterfinals the sum of the seedings for each game is 9. In the semifinals, if all the higher seeded teams win in the quarterfinals the sum of the seedings for each game is 5 (#1 against #4 and #2 against #3). This is the traditional way to arrange tournaments, so we take it as RULE 1: in each round, assuming the higher seeded teams win all previous games, the sum of the seedings for each game should be a constant.
In 1991, Florida State joined the conference, making a total of 9 teams. Now the old format needed to be expanded.
Theorem 2: In a 9-team single elimination tournament, there must be a first-round game in which two of the teams play for the right to enter the quarterfinals.
Again it's easy to see why: there will be 2 teams in the finals; they must be the winners of 2 semifinal games; and the teams playing in the semifinals must be the winners of 4 quarterfinals. Thus exactly 8 teams play in the quarterfinals, so we must have exactly one game prior to the quarterfinals to eliminate the 9th team.
The new tournament format looked like this:
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The first round game, held the day before the rest of the tournament, was called the play-in game, because, in effect, the two bottom teams in the standings were playing each other for the right to enter the "real" tournament. No wonder they felt like second-class citizens. Furthermore, attendance at the play-in game was poor. There were calls for improvement in the design, and these calls were "Do away with the play-in game!"
By Theorem 2, it can't be done. There must be a play-in game, or, at least, there must be a first-round game! However, Theorem 2 doesn't tell us how to assign the teams to the different games. It isn't necessary for #8 and #9 to play the play-in game. This assignment was based on RULE 2A: The worst thing is to be in the play-in game, so this fate must fall to the two lowest-seeded teams.
What actually happened is that Rule 2A has been replaced by RULE 2B: The worst thing is to be forced to play #1, so that fate should fall to the lowest seeded team. The next-worst thing is to be in the play-in game, so that fate should fall to teams #8 and #7. With this rule, the tournament format changes to the following:
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Next, somebody noticed that the mathematics also doesn't tell us what day the games must be played on, as long as they are played in the right order. Part of the problem with the old design was that only one game was played on the first day. One or more of the quarterfinals games can be played on the first day. So the new design was tweaked just a little so that the new tournament looks like this:
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The play-in game has not been eliminated, but this design disguises it fairly effectively.
Could it have been done otherwise? Only by applying the same principles in slightly different ways. The mathematics says there are really only two decisions to make: which two teams have to face each other in the first round game and which of the quarterfinal games should be played on the first day. If one wanted to play three games on the first day, for example, the tournament could look like this:
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Can you find any other possibilities? It might be an interesting problem for students to find all possible arrangements of the tournament -- subject to rule 1 on seedings (without this rule, the number of possibilities explodes, since any team can be paired with any other team).
Internet Sources
FEEDBACK: We'd be glad to have your comments and suggestions!
Originally posted February 2, 1998. Revised March 6, 2000, and February 22, 2001.
Copyright © 2001, Center for Mathematics and Science Education. Teachers have permission to duplicate this page for use in teaching their own classes. All other rights reserved. You are welcome to link to this page, but do not copy its contents.
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http://www.unc.edu/depts/cmse/math/tourneys.html Center for Mathematics and Science Education PHONE: voice (919) 966-5922; fax (919) 962-0588 |