## Games and Tournamnents

Theory

This is one of the most important topics in the CAT LRDI section. Since it is tough to predict the topics of the sets that will come in the exam, this topic becomes all the more critical, as CAT has been asking for at least one set from this topic for the last couple of years. Some of the types of games and tournaments questions that can be asked in the exam are:

ROUND ROBIN:

In these types of tournaments, each contestant(or team) meets every other participant, usually in turn. A round-robin contrasts with an elimination tournament, in which participants/teams are eliminated after a certain number of losses. The group stage of the Indian Premier League(IPL) follows this pattern, where each team plays the other teams twice, and the top teams advance to the next rounds.

For n teams, the total number of matches played in that stage is $$^nC_2$$ when each team plays against the other only once. Each team is awarded points based on the results(win, loss or tie). The top teams with the maximum points qualify for the next rounds.

KNOCKOUT TOURNAMENTS:

In these types of tournaments, once defeated, a contestant(or team) gets automatically eliminated from the tournament, and the winning team advances to the next rounds. A Tennis Grand Slam is a typical example of a knockout tournament. It is to be noted that in a knockout tournament, any two given players can play against each other only once.

Knockout tournaments generally have seeded(ranked) players. A seed is a competitor or a team in sports or another tournament that is given a preliminary ranking for the purposes of the draw. The matches are then organised on the basis of the seeds of the players.

Solved Example

Five teams, A, B, C, D, and E, participated in Indian Cricket League, and each team played against every other team exactly once. In a case of a win, three points were awarded to the winning team, and no points were awarded to the losing team. In case of a draw, both teams were awarded one point each. The points scored by the teams are recorded in the following table.

Q1: Against which team/teams did D win?

i) E

ii) A

iii) B

Iv) More than one of the above.

Sol: Based on the information given,
A has one point, so it should have drawn one match and lost other matches
B has 12 points, so it should have won all the matches
C has 3 points, so it could have won 1 match or drawn 3 matches
D has 7 points, so it should have won 2 matches and drawn one match
E has 6 points, so it could have won 2 matches or won one match and drawn 3 matches

Since B has won all matches, E should have lost at least one match. So the case where E wins one match and draws 3 matches is not possible. Hence, E won 2 matches and lost others.
C could not have drawn 3 matches as it would require 3 other teams with a draw but only two teams (A and D) have a drawn match. Hence, C won one match.
Since D drew with A and lost against B, it should have won the other two matches, against C and E.
E lost to D and B, so it should have won against A and C.
Filling the following table to find the results of every match, we get:

D won against E and C. Hence, option B

Q2: How many teams lost at least two matches?

i) 2

ii) 3

iii) 4

iv) 1

Sol:

Based on the information given,
A has one point, so it should have drawn one match and lost other matches
B has 12 points, so it should have won all the matches
C has 3 points, so it could have won 1 match or drawn 3 matches
D has 7 points, so it should have won 2 matches and drawn one match
E has 6 points, so it could have won 2 matches or won one match and drawn 3 matches

Since B has won all matches, E should have lost at least one match. So the case where E wins one match and draws 3 matches is not possible. Hence, E won 2 matches and lost others.
C could not have drawn 3 matches as it would require 3 other teams with a draw but only two teams (A and D) have a drawn match. Hence, C won one match.
Since D drew with A and lost against B, it should have won the other two matches, against C and E.
E lost to D and B, so it should have won against A and C.
Filling the following table to find the results of every match, we get:

A, C and E have at least two loses. Hence, option B.

Q3: Which of the following is not true?

i) D won against C.

ii) C lost against E.

iii) A drew with C.

iv) E lost against D.

Sol:

Based on the information given,
A has one point, so it should have drawn one match and lost other matches
B has 12 points, so it should have won all the matches
C has 3 points, so it could have won 1 match or drawn 3 matches
D has 7 points, so it should have won 2 matches and drawn one match
E has 6 points, so it could have won 2 matches or won one match and drawn 3 matches

Since B has won all matches, E should have lost at least one match. So the case where E wins one match and draws 3 matches is not possible. Hence, E won 2 matches and lost others.
C could not have drawn 3 matches as it would require 3 other teams with a draw but only two teams (A and D) have a drawn match. Hence, C won one match.
Since D drew with A and lost against B, it should have won the other two matches, against C and E.
E lost to D and B, so it should have won against A and C.
Filling the following table to find the results of every match, we get:

A drew with D, hence option C