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Reasoning situations based on games and tournaments have become one of the common occurrences in Aptitude exams, and are a special favorite of management entrance exams like the CAT, XAT and other top exams. While, the solving of such questions is based on complete common sense based understanding of the situations involved, it might be a good idea to work on extracting certain standard analysis that goes into solving the various kinds of questions that can be classified under the games and tournaments chapter.
1. Analysis tools for a tennis/badminton or a knockout tournament:
A tennis/badminton tournament is played under the knockout format. In such cases matches between players have the format that the loser gets eliminated and the winner moves into the next round. The following structures apply in such cases:
The number of rounds in the tournament is decided on the basis of the number of players entered. Typically, in a Grand Slam tournament in tennis there are 128 players and the tournament consists of 7 rounds. This can be visualized as:
Notice the following:
(i) The tournament as shown in the table, consists of a total of 127 matches (Sum of 64 + 32 + 16 + 8 + 4 + 2 + 1).
(ii) In each round half the remaining players are eliminated, moving the other half (the winners) into the next round.
(iii) The winner of the final is determined at the end of the 127th match in the tournament.
As you can well imagine for a 64 player tournament, there would be 6 rounds and 63 matches and for a 32 player tournament there would be 5 rounds and 31 matches required to determine the winner.
An interesting thing to note is the system of byes in such tournaments and how it affects the numbers. Byes are necessitated when the number of players is not even in any round. Let us consider a tournament of 121 players. In such a case the following structure would apply:
Round 1: 60 matches and 1 bye – so a total of 61 players progress to the second round;
Round 2: 30 matches and 1 bye – so a total of 31 players progress to the third round;
Round 3: 15 matches and 1 bye – so a total of 16 players progress to the fourth round.
Beyond this the tournament would play out normally without any more byes.
What would be the total number of matches in such a tournament?
Lets’ count: 60 + 30 + 15 + 8 + 4 + 2 + 1 = 120. It is still one less than the number of players.
In general, you can make out that: Number of matches in a knockout tournament of n players would always be (n–1). This can also be thought of as: To determine the winner in a 123 players tournament, we need to identify 122 losers. Since every match determines one loser, to determine 122 losers for a 123 players tournament, we need to play 122 matches.
So for a knockout tournament consisting of 13 players, there would be a total of 12 matches played. I would encourage you to create the tournament structures for various number of players and try to work out the rounds for the tournament.
You would notice another interesting thing when you do so. That is with respect to the number of rounds. We have already seen that the number of rounds for a 128 player tournament is 7, for a 64 player tournament is 6, for a 32 player tournament is 5 and so on. If we investigate how many rounds would be required in order to determine a 65 player tournament, we will notice the following:
Round 1: 32 matches and 1 bye – 33 players progress to round 2;
Round 2: 16 matches and 1 bye – 17 players progress to round 3;
Round 3: 8 matches and 1 bye – 9 players progress to round 4;
Round 4: 4 matches and 1 bye – 5 players progress to round 5;
Round 5: 2 matches and 1 bye – 3 players progress to round 6;
Round 6: 1 match and 1 bye – 2 players progress to round 7;
Round 7: 1 match – decides the winner of the tournament.
Thus, you can notice an important logic about the number of rounds. For a tournament with 65 to 128 players, the number of rounds would always be 7. For 33 players till 64 players, 6 rounds would be required to determine the winner and so on. In terms of powers of the number 2, you can think as follows: If the number of players crosses 26, the number of rounds is 7. If it crosses 25, there would be 6 rounds in the tournament and so forth.
Often, in knockout tournaments like tennis and badminton, seedings are given to the top players – based on who is favored to win the tournament. In such cases, Seed Number 1 is often referred to as the top seed. Such tournaments, have their draws based on these seedings only. For instance, suppose there is tournament having 32 players and a Seeding is given for players from 1 to 32. In such a tournament, the first round consists of 16 matches and who plays who in the matches are defined by the seed numbers. The following table would make this clear to you:
The following points need to be understood with respect to the above table:
Consider the following question that appeared in CAT 2005 on this structure:
Directions: In the following table is the listing of players, seeded from highest (#1) to lowest (#32), who are due to play in an Association of Tennis Players (ATP) tournament for women. This tournament has four knockout rounds before the final, i.e., first round, second round, quarterfinals, and semifinals. In the first round, the highest seeded player plays the lowest seeded player (seed #32) which is designated match No. 1 of first round; the 2nd seeded player plays the 31st seeded player which is designated match mo. 2 of the first round and so on. Thus, for instance, match no. 16 of first round is to be played between 16th seeded player and the 17th seeded player. In the second round, the winner of match no. 1 of first round plays the winner of match no. 16 of first round and is designated match no. 1 of second round. Similarly, the winner of match no. 2 of first round plays the winner of match no. 15 of first round, and is designated match No. 2 of second round. Thus, for instance, match no. 8 of the second round is to be played between the winner of match no. 8 of first round and the winner of match no. 9 of first round. The same pattern is followed for later rounds as well.
Q. If Elena Dementieva and Serena Williams lose in the second round, while Justine Henin and Nadia Petrova make it to the semifinals, then who would play Maria Sharapova in the quarterfinals, in the event Sharapova reaches quarterfinals?
(a) Dinara Safina
(b) Justine Henin
(c) Nadia Petrova
(d) Patty Schnyder
Visualise the rounds as:
According to this question seeds 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 and 31 will reach the second round. In such a case, 4th seed is replaced by 29th seed. Hence, the correct answer is Flavia Pennetta. Option (b) is the correct answer.
Hockey and Football tournaments that are played in a round robin format are also interesting cases for building reasoning questions. A typical table of standings for a hockey/football tournament after two round robin rounds with the following results in the Matches Played: A vs F 50; A vs E 62; B vs C 21;B vs D 11; C vs D 41; E vs F 00, would look as follows:
Often in such tables, the individual match results are removed and you are asked to deduce the match results based on the information in the table.
In such cases, the following observations and overall rules would always hold:
Besides these certain score specific deductions are also possible. Some of these (indicative list) are as follows:
These are just an indicative list and I would expect you to make sense of other such cases that you would see regularly in such tables.
A typical cricket batting score card consists of the names of the players, with their runs scored given individually and the overall team score.
The inning is terminated at the fall of the 10th wicket.
For a round robin tournament of n teams, where each team plays the other once, the total number of matches in the tournament would be: ^{N}C_{2}. Thus, for a 6 team round robin tournament with each team playing the other once, there would be a total of ^{6}C_{2 }= 15 matches.
Note: Here we are using the formula for combinations where NCR =
Sometimes tournaments are held on a roundrobin cum knockout basis. In such tournaments, the initial rounds are played in multiple groups on a round robin format – where teams within a group play each other once, and the top teams advance to the next round which is held on a knock out basis, starting with either the semi final or quarter final or pre quarter final. The world cup football is a classic example of this. 8 groups of 4 team each participate in the group stages, followed by the top two teams of each group qualifying to play the pre quarterfinals, which is played in a knock out format.
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