Games and Tournaments: Solved Examples - Logical Reasoning (LR) and Data Interpretation (DI) - CAT

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.

Seeding and Logic Around Seeding:

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:

• Every match has a higher seed playing a lower seed. The higher seed is expected to win the match and proceed to the next round. The pairings of who plays whom in the next rounds is based on the assumption that the higher seed wins. Thus, match 1 of round 2, is expected to be between Seed 1 (expected winner of match 1 of round 1) and Seed 16 (expected winner of match 16 of round 1). However, this match would be disturbed in terms of the players playing this match in the event of an upset in round 1.
• (Note: Whenever a lower seed beats the higher seed, it is called an upset – as the result is against the expected result.) You should be able to visualize that match 1 of round 2 can have four possible match ups. 1 vs 16 is the expected match, but it could be 1 vs 17 also. Likewise, the match could also be 32 vs 16 or 32 vs 17 (depending on who wins the first round’s match 1 and 16 respectively).
• Similarly, Match 1 of round 3 could have multiple match ups possible. These can be visualized as follows: Match 1 of round 3 is decided by the winners of matches 1 and 8 of round 2. Match 1 of round 2, as we have already seen, could be won by players 1,16,17 or 32. Similarly, Match 8 of round 1 could be between: 8 vs 9 (expected match); 8 vs 24; 25 vs 9 or 25 vs 24. Thus, for match 1 of round 3, there could be 16 different match ups possible: viz: Anyone of 1, 16, 17 or 32 vs anyone of 8, 9, 24 or 25. 

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 semi-finals. 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 semi-finals, 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.

2. Analysis tools for a Hockey/Football tournament (round robin):

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 5-0; A vs E 6-2; B vs C 2-1;B vs D 1-1; C vs D 4-1; E vs F 0-0, 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:

• Total matches played: Total Number of matches played would be equal to half the total number of matches in the played column. This occurs since every time a match is played between two teams, the played column is incremented by +1 at two places - +1 for each of the two teams that played the match. Thus, in this table: since the sum of all the matches played is 12, it means that there have been 6 matches played in this tournament till date. 
• Number of wins: In the wins column = Number of losses in the loss column. This has to hold since every time a match is decisive, there would be an increment of one match in the wins column and one match in the losses column. 
• Number of draws: Also, the number of decisive matches in such a case would be equal to the sum in either the wins or the losses column. Thus, in this case, since the total number of wins = 4, it means that 4 matches were decisive out of the 6 matches played. 
• Number of draws: The sum of the number of matches in the drawn column, needs to be divided by 2 in order to find out the number of matches that were drawn. Thus, in the table above, the number of matches drawn would be equal to sum of the numbers in the drawn column divided by 2 = 4/2 = 2. 
• Goals for - Goals Against = Goal difference: The ‘goals for’ column represents the number of goals scored by the team, the ‘goals against’ column represents the number of goals scored against the team (by the opponents of the team).
• Sum of the goal difference column would always be zero.
• The sum of goals for column = Sum of goals against column. This is so because whenever a team scores a goal in its match against another team, there is an increment of +1 in the ‘goals for’ of the team scoring the goal and at the same time there is an increment of +1 in the ‘goals against’ for the opponent team. Thus, every time there is a goal scored, both the goals for and goals against columns’ totals would increase by 1.

Besides these certain score specific deductions are also possible. Some of these (indicative list) are as follows:

• If a team has played 2 matches and won two matches, and it has a goals for 2 and goals against 0, it would mean that it has won both its’ matches by 1-0; 
• If a team has played 2 matches and won two matches, and it has a goals for 3 and goals against 1, it would mean that it has won one match by 1-0 and another by 2-1; 
• If a team has played 2 matches and won one match, and drawn another, and it has a goals for 2 and goals against 0, it would mean that it has won its match 2-0 and drawn 0-0; 
• If a team has played 2 matches and won one match, and drawn' another, and it has a goals for 3 and goals against 1, it could mean two possible cases: (i) It has won 2-0 and drawn 1-1 OR won 3-1 and drawn 0-0. 
• If a team has played 2 matches and won one match, and lost another, and it has a goals for 3 and goals against 1, it would mean it has won 3-0 and lost 0-1.

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.

3. Analysis tools for a cricket score card

A typical cricket batting score card consists of the names of the players, with their runs scored given individually and the overall team score.

• At any point of time during the innings, two batsmen would be batting at the crease, each such pair is referred to as a partnership. The runs scored by a pair of batsman while they are batting together, is seen as the runs for that partnership. The first batsman to take strike during an innings is called batsman 1, the batsman who accompanies him is batsman 2, the batsman who comes in at the fall of the first wicket is batsman 3 and so on till batsman 11 who is the last batsman to walk in to bat during an inning.
• The first wicket partnership is the one that is between the two opening batsmen. Once one of the opening batsmen get out, the number 3 batsman of the team comes into bat, with the not out batsman. This partnership is referred to as the second wicket partnership. Once one of them gets out, the number 4 batsman comes out to bat with the batsman who is not out. This partnership is referred to as the third wicket partnership and so on. Note that, the third wicket partnership has 3 possibilities with respect to the batsmen number batting during the time – Possibility 1: Batsman 1 and Batsman 4; Possibility 2: Batsman 2 and Batsman 4 and Possibility 3: Batsman 3 and Batsman 4. The trend continues in the same way till the 10th wicket partnership, which is the last partnership of the innings.
• Notice that batsman 4 would be a necessary constant for the 3rd wicket partnership. Similarly, for the 2nd wicket partnership, batsman 3 is necessary and can pair with either batsman 1 or 2 depending on who got out and who remains not out. Thus, we can visualize the following table of possible partnerships:
  

The inning is terminated at the fall of the 10th wicket.

• It is sometimes possible to trace which particular batsmen would be batting at the fall of a particular wicket. For instance, if the batsman number 1 has scored 100 runs and the fall of the first 4 wickets were at 22,38,47 and 73 runs, then batsman 1 must be a part of the fifth wicket partnership (apart from of course being a part of the first, second, third and fourth wicket partnerships too).
• Strike rate for a batsman: A batsman’s strike rate is considered to be the number of runs he has scored per 100 balls faced. Thus, for a batsman who had a strike rate of 50, and faced 40 balls before getting out, we can deduce that he must have scored 20 runs.
• Run rate is defined as the number of runs scored per over. Thus, a team scoring at 5 runs per over in a 50 over match, would have scored a total of 250 runs – if it played all the 50 overs.
• The bowling score card shows the performance of the bowlers for the bowling side. Typical information you would be able to derive from the bowling scorecard would be: Overs bowled (the total number of overs bowled by a particular bowler) ; Maidens (the number of overs in which there was no run scored); Runs (The runs conceded by the particular bowler); Wickets (the wickets taken by the particular bowler).
• Extras: During a batting inning some runs are given to the batting team, that were not scored by the batsman off his own bat. These runs are called as extras – and are of 4 kinds: Wides, No balls, byes and leg byes. These runs are added to the team score. Thus, the total score of an inning for a batting team = runs scored by all the batsman of the team + the runs in extras awarded to the team.

4. General Analysis tools for a round robin tournament or for a round robin cum knock out tournament.

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: ^NC₂. Thus, for a 6 team round robin tournament with each team playing the other once, there would be a total of ⁶C₂ = 15 matches.

Note: Here we are using the formula for combinations where NCR =

Sometimes tournaments are held on a round-robin 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.