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Inspiration for Artists
National Collegiate Athletic Association (NCAA) men's basketball tournament,
which is taking place now, also known as "March Madness," can
be a wonderful source of mathematical exploration. (Indeed there are many
other college basketball tournaments in March aside from the men's Division
I, including the women's Division I, the men's and women's National Invitation
Tournament, and men's Division II and III for schools with smaller enrollments
or athletic budgets. Any of these tournaments would be equally appropriate
to follow. However the men's Division I gets more publicity than any of
The layout of the entire tournament, including what schools are selected, what seeds they get, and when and where they play, is determined by a committee. The bracket, or draw sheet, is a graphic representation of the tournament. The myriad of patterns which can be found in the bracket is remarkable. To get a copy go to www.finalfour.net.
.THE COLLEGES AND UNIVERSITIES
State universities predominate now (37 out of the 64 teams chosen this year have a state in their names); most, perhaps all, of them are public. Still others, while private are large schools. But there are always some smaller or "unknown" colleges which produce upsets and evoke references to "Cinderella."
Every year some of the players and coaches have fascinating stories to tell, from someone's life- or career-threatening injury or medical condition to the brilliant tactician who toiled in obscurity for decades before being thrust "overnight" into the limelight.
the tournament committee makes an attempt to accommodate teams so that
inconvenience is minimized. Yet some geographical anomalies always occur.
(One reason for this is a good one: a rule instituted for reasons of fairness,
that no team gets to play in its own arena, at least until the Final Four.)
Here are some mathematical questions to ponder:
Why 64 teams? What is special about
this number? How many teams qualify for the playoffs in the National Basketball
Association, the National Hockey League, or Major League Baseball? How
many players are admitted into the US Open Tennis Tournament? Do you notice
any patterns in these numbers?
Jim's Responses to Questions Posed in March Madness
1. Some numbers, when you keep halving them, end up at 1. 64 is such a number. If you continue to halve 64, see where you end up: 64, 32, ...This property of 64 does not hold for most numbers. Satisfy yourself that this is so. Try halving other numbers, e.g., 60, 100, 42. Do you reach 1? (Odd numbers are not mentioned, because you can't halve them even once.)
The numbers which, when halved, end up at 1, are also called powers of 2. 64 can be expressed as a product of a string of 2s: 64 = 2 x 2 x 2 x 2 x 2 x 2 = 2 to the 6th power. In both the NBA and the NHL 16 teams qualify for the playoffs; 8 qualify in MLB. 128 players qualify for the U.S. Open. These numbers are also powers of 2.
2. Twelve teams qualify for the NFL playoffs. As of March 5, 101 players had qualified for the 2001 Masters, based on 17 different criteria. (See www.themasters.com) 12 and 101 are not powers of 2. The events in Question 1 are elimination tournaments with no byes. In other words, every qualifying team (or player) plays in the first round and advances to the next round only if it wins its game (or series of games).
The designs of the NFL playoffs and The Masters are different. In the NFL playoffs 4 teams do not play in the first round, i.e., they earn byes by virtue of their regular season play; that brings the number of teams in the second round to 8, a power of 2.
The Masters, and most golf tournaments, are not elimination tournaments at all. Why? Golf, unlike, tennis and basketball, is not one person or team against another. After a round there are not an equal number of winners and losers. In golf players play against the entire field and, after each round, they are ranked in a continuum from the top to the bottom. (The field in a golf tournament is usually "cut" in the middle of the tournament, but that is for purposes of economy and focus and not a necessity, as is elimination in an elimination tournament.)
3. When planting seeds in a row, a gardener will spread them out so that each can get sufficient nutrients to flourish. It is the same in a tournament. Similarly rated teams are spread out so they don't play each other in early rounds. The rationale is equity. It is not that all teams are treated as equals, in which case a random draw would be appropriate. No, in an equitable setting, all teams are accorded the status they have earned by virtue of their play during the regular season. Thus the seeds. The higher a team is seeded, the lower is the seed of its opponent .
The tournament is divided into 4 regions, with 16 teams in each region, seeded from 1 to 16. Each region can be thought of as a separate sub-tournament which starts with 16 teams and ends up with 1. The four "winners" playoff in the Final Four. In the first round, 1 plays 16, 2 plays 15, etc. (Notice how the total of the seeds in any game is 17, e.g., 1 + 16 = 17.) In the second round, assuming the higher-seeded team wins, 1 plays 8, 2 plays 7, etc. (The total is 9.) In each succeeding round the same pattern obtains.
Roughly speaking the top 4 teams in the polls are seeded #1, teams rated 5-8 are seeded #2, etc. A rule of thumb for converting from ranking to seeding would be to divide by 4 and, if you don't have a whole number, round up. (Thus, a ranking of 17 would translate to a #5 seed, i.e., 17 divided by 4 = 4.25, then round up to 5.)
4. Sixty three games are played. You could figure that, with 64 teams at the start, 32 games are played in the first round, 16 games in the second round, etc. To calculate the total you simply need to follow the same halving pattern to its end: 32 + 16 + 8 + 4 + 2 + 1 = 63.
But there is a more elegant way of thinking about the question: You start with 64 teams. Teams get eliminated as they lose, at the rate of one per game. How many teams have to lose in order to determine a winner? 63. Thus you need 63 games to eliminate 63 teams.
If you look at the bracket, the symmetry of the tournament leaps out at you. All the patterns that happen in the East and West get reflected by events in the Midwest and South. Thus there is a bilateral symmetry --- and thus an even number of games --- until the final game. There is no counterpart, or mirror, to the final game. Thus the total is an odd number of games.
Dr. James Neuberger
holds a doctorate in Mathematics Education from Rutgers University and
is an avid sports fan. He lives in New York City.
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