the big loser

THE BIG LOSERAuthor(s): Daniel MarksSource: The Mathematics Teacher, Vol. 92, No. 3 (MARCH 1999), pp. 208-213Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27970916 .

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Daniel Marks

IH E BIG

March, 1997, sixty-four teams competed for the national championship of men's college basketball in the National Collegiate Athletic Association Division I Men's Basketball Tournament. As is often pointed out, only one team can win such a tournament. In each round the teams play each other in pairs, the winners of that round playing in another round, until after six rounds of play (26 =

64), only one team remains, earning the title of national champion. This particular type of tourna ment is called a single-elimination tournament because a single loss eliminates a team from fur ther competition.

In the 1997 men's college basketball tournament, University of Arizona won the national champion ship, defeating all six opponents it faced. From one

point of view, the other sixty-three teams had an

equal degree of disappointment in the competition because all of them went home with one loss. For one team, though, the tournament was especially disappointing. Butler University lost in the first round to University of Cincinnati, which then lost to Iowa State University, which lost to University of California at Los Angeles (UCLA), which lost to

University of Minnesota, which lost to University of

Kentucky, which then lost the championship game of the tournament to Arizona. No team that beat Butler, or beat a team that beat it, or beat a team that beat a team that beat it, and so on, could defeat any subsequent opponent. With all due respect to the athletic program at Butler, noting that only sixty-four of the three hundred Division I schools are invited to compete in the tournament, that team was what could be called the tourna ment's big loser.

In every single-elimination tournament, just as

only one team can be the winner, one, and only one, team can be the big loser; the team whose defeat

begins a string of losses that stretches from begin ning to end of the tournament. Mathematics con tains many unexpected facts, though, and if Butler

still wishes to find a silver lining in its performance, here it is: the team most likely to be the big loser in a single-elimination tournament is not the worst team in the tournament. If the tournament goes according to plan, with every team beating the teams that it is expected to beat, then the big loser is, oddly enough, a team about two-thirds of the

way down the rankings. The identity of the team in

greatest jeopardy of becoming the big loser is the

subject of this article. This article explores several facts about the big loser, offering them in a hierar

chy that may be appropriate for creating various short- and long-term projects for a high school mathematics class.

SETTING THE STAGE Nearly every single-elimination tournament is

organized to guarantee that the best teams in the tournament will not face each other in the early rounds. No one wants the two best teams to meet each other in the very first game: one of them would suffer a humiliating and unfair exit without winning any games at all, and spectators during the remain der of the tournament would be deprived of the chance to see that team play. So single-elimination tournaments use a seeding system. The team iden tified by tournament organizers as the best team in the field, before the tournament, is designated the number-one seed; the second best team is the number-two seed; and so on. The seeding may include every team in the tournament, as in the NCAA men's and women's basketball tournaments;

Daniel Marks, [email protected], teaches mathe matics at Auburn University at Montgomery, Montgomery, AL 36124-4023. His degree is in mathematics education, and his main interest is in passing his love for mathemat ics on to his students.

208 THE MATHEMATICS TEACHER



Assume that the higher-seeded team wins every game. In round 1, seed #1 plays seed #16, seed #9 plays seed #8, and so on. In the next round, the winner of the game between #1 and #16? that is, #1?plays the winner of the game between #9 and #8?namely, #8?and so on. The winner of every game moves on to the next round. Who is the big loser?

Fig. 1 A sixteen-team tournament

or best sixteen, as in most major professional tennis

tournaments; or any power of 2. Once the seedings are established, matching teams for competition in the tournament generally follows this basic rule:

Assuming that the higher seeded team wins

every game, then the number-one seed will

always face the worst team still remaining, the number-two seed will face the second-worst team

remaining, and so on.

Tournaments occasionally follow a slightly re

laxed version of this rule. The NCAA men's basket ball tournament, however, in each of its four sixteen team regional competitions, does match up the teams

according to the rule. In a four-team tournament, for example, the

competition usually begins with the number-one seed playing the number-four seed, and number two

playing number three, with the winners playing against each other for the championship. If all goes

according to plan, that is, the "better" team wins each game, the number-one and number-two seeds will face each other in the final game; and the big loser will be the number-three seed, not number

four, since number three will lose to number two, who in turn loses to number one.

This article assumes that the tournament under

study uses the basic rule. The article poses a series of questions, identifies the knowledge needed to answer the question, describes how the question might be assigned in a classroom, and, of course,

gives the answer.

LEVEL 1 TASK: IDENTIFYING THE BIG LOSER IN A SIXTEEN-TEAM TOURNAMENT

Question. In a sixteen-team tournament, if every team defeats the teams that it is expected to defeat, that is, the higher-seeded team wins every game, which team will be the big loser?

Necessary skills or knowledge. Little or none, if the tournament chart shown in figure 1 is used. Teachers will need to explain the chart, showing students how to interpret it. This activity is for stu dents at or above the junior high level.

Answer. The number-eleven seed will be the big loser. That team will lose in the first round to the sixth-seeded team, which will lose in the second round to number three, which in turn will lose to number two, which will lose in the championship game to number one.

Method of solution. Students should work in

groups of three or four. The teacher should point out that the seemingly obvious answer, that is, the number-sixteen seed, the worst team in the field, is

in truth not the answer, since it will lose to a team that will win the rest of its games. Students should use figure 1 to obtain the solution by working back ward: tracing a string of defeats from the champion ship game back to the opening round. The number two seed loses in the final game; it had beaten the number-three seed in the previous game; that team had previously defeated the number-six seed, which had beaten the number-eleven seed in the first round of the tournament. The number-eleven seed is therefore the big loser.

LEVEL 2 TASK: THE LIKELIHOOD OF A GIVEN TEAM'S BEING THE BIG LOSER In reality, of course, the higher seed does not win

every game. In nearly every tournament, upsets, that is, games with unexpected outcomes, happen, the sixteenth seed may even beat the number-one seed in the opening game of a sixteen-team compe tition. The eleventh may not only avoid being the

Vol. 92, No. 3 ? March 1999



big loser, it may even win the tournament. The actual probability that a given team will be

the winner of the tournament, or the big loser in it, depends on how likely each team is to beat each other team in the tournament; that is, no single answer exists to the question of how likely a team is to win or lose the tournament. The answer instead depends on assumptions that are made about all the teams' relative strengths. One can make any assumptions, but the assumptions should

obey certain rules. An example of a reasonable set of rules would be the following:

1. The probability that a higher-seeded team beats a lower one should be more than 0.5.

2. The farther apart two teams are in the seedings, the higher the probability should be that the team that is seeded higher will win.

Question. In a four-team tournament?where the number-one seed plays the fourth seed, the number two seed plays the third seed, with the winners

playing for the championship?suppose that the

probability that the number-one seed will beat the fourth seed is 0.8; that the probability of the number two seed's beating the fourth seed or the number one seed's beating the number-three seed is 0.7; and that the probability that the number-one seed will beat the number-two seed, the number-two seed will beat the number-three seed, or the number three seed will beat the number-four seed is 0.6. These probabilities follow rules (1) and (2). What is the probability, for each team, of winning the tour nament? What is the probability, for each team, of being the big loser?

Necessary skills or knowledge. Students should be studying, or have studied, probability and be famil iar with basic concepts of probability, including addition and multiplication rules for probabilities.

Answer

Seed

Probability of Winning

Tournament

Probability of Being the Big Loser

#1 #2 #3 #4

0.512 0.276 0.144 0.068

0.132 0.256 0.324 0.288

Note that the number-three seed is most likely to be the big loser, as expected. Recall that if every team beats the teams that it is expected to beat, the number-three seed will be the big loser. Yet, although dramatic differences in the probabilities of winning occur, because each team is nearly twice as

likely to win the tournament as the team seeded im

mediately below it, the same is not true of the big loser probabilities: the likelihoods of the number-two, -three, or -four seed's being the big loser are nearly

the same, and only the number-one seed stands out as being unlikely to be the big loser. As noted at the end of the article, something even more surprising happens as the number of teams increases.

Method of solution. Students can work in groups or individually on this activity, which can be assigned as homework. No shortcut is available to determine these probabilities. Since three games will be

played in the tournament, 23, or eight, tournament outcomes are possible, all with different probabili ties. Students must list them all. In the chart that follows, the shorthand "1/4, 2/3; 1/2" stands for the outcome, "number-one seed beats number four, number two beats number three, and in the cham

pionship, number one beats number two." In each outcome, the big loser is shown in boldfaced type; the winner is always to the left of the slash.

Outcome Probability 1/4, 2/3; 1/2

1/4,2/3; 2/1

1/4, 3/2; 1/3

1/4, 3/2; 3/1

4/1, 2/3; 2/4

4/1, 2/3; 4/2

4/1,3/2; 3/4

4/2, 3/2; 4/3

(0.8X0.6X0.6) = 0.288 (0.8X0.6X0.4) = 0.192 (0.8X0.4X0.7) = 0.224 (0.8X0.4X0.3) = 0.096 (0.2X0.6X0.7) = 0.084 (0.2X0.6X0.3) = 0.036 (0.2X0.4X0.6) = 0.048 (0.2X0.4X0.4) = 0.032

Answering the question is now a matter of adding probabilities. The number-one seed is the winner in two of the outcomes, for example, whose combined

probability is 0.288 + 0.224 = 0.512; similarly, the number-one seed is the big loser in two of the out comes, with a combined probability of 0.084 + 0.048 = 0.132.

LEVEL 3 TASK: THE GENERAL CASE Question. In the case where the higher-seeded team wins every game, what is the general formula, for any size tournament consisting of 2n teams, that says which team will be the big loser?

Necessary skills or knowledge. Students should be in second-year algebra or beyond and have a

knowledge of functions, in particular, , the ceil

ing function; lx] = the least integer n>x, for exam

ple, 16.71 = 17.

Answer. In a tournament with teams (T = 2n), in which the higher-seeded team wins every game, the big loser will be seed

~2r

3J

The number

THE MATHEMATICS TEACHER



itself is never an integer, since

29 ott+l

3i "3 "

3 *

The numerator therefore has no factors of 3.

Method of solution. Students should work either

individually at home or in groups in class. Students should be told that the ceiling function is involved in the answer. The teacher may suggest that stu dents start by extending the level 1 task one step, determining the big loser in a thirty-two-team tour nament, by using a sheet similar to figure 2.

Round 1 Round 2 Round 3 Round 4 Round 5 Winner

1 1 -16 16

1

1

12 12

.13. 13

14 14

11 11

JUL IO

AS 15

In every round, seed #1 plays the worst team still

remaining, seed #2 plays the second-worst, and so on. Who will seed #1 play in the first round? Who will seed #16 play? Fill in the blanks in the first round. Who will be the big loser?

Fig. 2 A thirty-two-team tournament

Students should fill in the blanks in the chart

according to the seeding rule given previously. After

completing that task, students will discover that the number-twenty-two seed is the big loser in a

thirty-two-team tournament. The requested formu la can be found by observing the following pattern:

Number of Teams Probability

16 32

3 6 11 22

As the number of teams increases, the big loser appears to come closer and closer to (2/3)T, yet the answer obviously must be an integer. The facts that 2/3 of 4 is -2.67, that 2/3 of 8 is -5.33, that 2/3 of 16 is -10.67, that 2/3 of 32 is -21.33?in every case, the big loser is the next integer higher than (2/3)T? suggest the formula. Some students may need hints, and the teacher can encourage students to extend the sequence of big losers to still larger tournaments. Any student who finds a potential solution should be encouraged to check it by testing it on the next-larger-sized tournament, since it is

possible to find false solutions that work for a small tournament but not for a larger one.

LEVEL 4 TASK: PROVING THE FORMULA Problem. In a tournament with = 2n teams, with each game won by the higher seed, prove the fol

lowing formula:

Big loser =

Necessary skills or knowledge. A knowledge of mathematical induction is needed. Because of the

difficulty of the proof, it should be assigned, possi bly for extra credit, only to those few students in an advanced high school mathematics class who dis

play a real talent for mathematical reasoning. It could also be assigned in a college-level course in discrete or finite mathematics.

Solution. Let = 2n, for = 1, 2, 3,... Obviously the formula holds for = 1, as well as for = 2, 3, 4, and 5. Let be any positive integer, and assume that the formula holds for any tournament with 2n teams. It is necessary to show that the formula also holds for a tournament with 2n+l teams.

After the first round has been completed in the 2"+1-team tournament, only seeds 1 through 2n will remain. From that point, the tournament is a 2"-team tournament, so by the induction hypothe sis, seed

2n

will start a string of losses that will continue to the end of the tournament. So the question becomes, Who did seed

play in the first round? That team will be the big loser; call that team seed L.

Note that in each round of a tournament, the teams are matched so that the sum of their seeds is the number of remaining teams plus 1, as shown in

figures 1 and 2. That is,

Vol. 92, No. 3 ? March 1999



L + 2"

so

(1) L = 2"+1 + l

= 2n+1 + 1,

2"

for the ceiling function for any nonintegral , < [Yl < + 1. As mentioned previously,

2"

is indeed not an integer; therefore,

2"< 2" <#-2" + l.

Subtracting from 2 + 1 gives

(2) 2"+1 + 1 -|?

2" > 2"+1 + 1 - 2. 2n

>2n+1 + l 2n + l .

Simplifying the expression on the far right of

inequality (2) yields

2"+1 + 1 - (2. 2" + = 2"+1 + 1-2^-1

2"+1|l

Similarly, the expression on the left of inequality (2) simplifies to

2"+1 + 1.

Therefore,

?? 2B+1 + 1 > 2"+1 + 1 - ?

> =-? 2n+1.

By equation (1), the expression in the middle is ir that is,

|-2"+1 + l>L>

L is an integer, and the foregoing inequalities show that it is the smallest integer larger than

(2/3)-2?+1;thatis,

3 ?

which completes the proof.

ADDITIONAL COMMENTS As previously mentioned, unexpected results occur in the calculations of big-loser probabilities as the number of teams in the tournament increases. Un

fortunately, determining the probabilities of win

ning and losing is beyond the scope of most high school classes. In a sixteen-team tournament, fif

teen games are played, so the tournament has 215, or 32 768, different possible outcomes, all with dif ferent probabilities. When reasonable assumptions are made about the probability of a given team's

beating any other team, the overall probabilities can be calculated by the same method that was used in the level 2 task, except that a computer is required, along with someone to program it to calculate each of the 32 768 outcomes. The author's desktop com

puter handled the task in about seventy minutes, although of course a mainframe computer could do it in seconds. The results are certainly worth not

ing. In determining the probabilities, the assump tion was made that if seed A is higher than seed B, then the probability of A's beating is

0 5 , Q-4(A-B) a5+ 15 *

The formula is, of course, completely arbitrary, but it seems reasonable and does fit the rules given previously for a probability formula: the probability that A will beat is always more than 0.5, and the

probability grows as A - grows. In other words, the farther apart A and are in the seedings, the more likely A is to beat B. The maximum probabili ty occurs when the number-one seed plays the six teenth seed in the first round: the probability that the number-one seed will win is

0.5 + 0.4(16-1) 15

or 0.9. However, when the number-eight seed plays the ninth seed, the probability that the eighth seed will beat the ninth is

0.5 + 0.4(9-8) 15

'

or 0.523. Of course, the probability that the six teenth seed will beat the number-one seed is then

0.1, and the probability that the ninth seed will beat the eighth seed is 0.477. On the basis of this

assumption about the relative strength of any two

teams, table 1 summarizes the probabilities of each team's being the winner or big loser.

No surprises occur in the winners' column: the

higher a team is seeded, the more likely it is to win the tournament. The identity of the big loser, how ever, has done something very odd: it has drifted down the seeding chart. The most likely big loser is neither the eleventh seed nor the sixteenth seed; it is the fourteenth seed.

The results, of course, would be different if the formula

0.5 + 0.4(A-B) 15

were changed. Those who possess the necessary programming knowledge might wish to determine

THE MATHEMATICS TEACHER



#11 #12

' 0.0095

#14 0.0043 #15 0.0026 #16 0.0014

0,0790^ 0.?792 0.0805 0.0837 0.0835 0.0801

whether another assignment of probabilities might make a team other than the fourteenth seed the most likely big loser.

Ironically, as if to verify these calculations in real life, in the NCAA men's basketball competi tion's four sixteen-team regional tournaments, the four big losers included two number-fourteen seeds, one of them Butler; one number-eleven seed; and

unexpectedly, a number-two seed.

CONCLUSION If students are to take one lesson away from this set of tasks, it should be that mathematical questions can have answers that are contrary to expectations. If a tournament has been played and a team has been identified that has lost its first game, with its oppo nent then losing, and its opponent losing, all the way to the end of the tournament, it would almost seem obvious to describe that team as the worst competing in the tournament. Yet when every team in a tour nament plays as expected, the big loser is not the worst team in the tournament. When upsets occur, the most likely big loser is a different team from the one that would have otherwise claimed that title, and that team is not the worst in the field either.

Most students who take mathematics courses in school never once perceive the beauty that mathe

maticians love in the subject. Often that beauty is in the unanticipated. Good teachers often devote a

great deal of effort toward making mathematics seem useful, and nothing is wrong with that

approach?mathematics is useful, and many stu dents never notice its utility. Rarely is a teacher able to communicate the aesthetics of mathematics: that it has the power to please, to humble, and to

surprise. That is part of the subject, too. @

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the big loser

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