Games and Tournaments - Ravi Handa

• In Logical Reasoning very often we encounter problems based on games of tournaments. The first thing that as a CAT taker you need to realize is that such tournament based format offers the examiner a multitude of options. So, there cannot be a set formula for solving such kind of questions. However, if you look at the CAT papers of past few years – a pattern seems to emerge. Let us discuss couple of them.

Type 1:

The questions are typically in a set where the data will be either in the standard tabular format or a format which you would never find on Cricinfo or for that matter any other ESPN website. The ‘different for the sake of being different’ format essentially tests a CAT taker’s ability to infer data in newer formats.

An example of this would be: (From CAT 2004)

Coach John sat with the score cards of Indian players from the 3 games in a one-day cricket tournament where the same set of players played for India and all the major batsmen got out. John summarized the batting performance through three diagrams, one for each game. In each diagram, the three outer triangles communicate the number of runs scored by the three top scorers from India, where K, R, S, V, and Y represent Kaif, Rahul, Saurav, Virender, and Yuvraj respectively. The middle triangle in each diagram denotes the percentage of total score that was scored by the top three Indian scorers in that game. No two players score the same number of runs in the same game. John also calculated two batting indices for each player based on his scores in the tournament; the R-index of a batsman is the difference between his highest and lowest scores in the 3 games while the M-index is the middle number, if his scores are arranged in a non-increasing order.

Q1) How many players among those listed definitely scored less than Yuvraj in the tournament?
a) 0
b) 1
c) 2
d) More than 2

Q2) Which of the players had the best M-index from the tournament?
a) Rahul
b) Saurav
c) Virender
d) Yuvraj

Q3) For how many Indian players is it possible to calculate the exact M-index?
a) 0
b) 1
c) 2
d) More than 2

Q4) Among the players mentioned, who can have the lowest R-index from the tournament?
a) Only Kaif, Rahul or Yuvraj
b) Only Kaif or Rahul
c) Only Kaif or Yuvraj
d) Only Kaif

I will not get into the detail of solving this particular set. Once you interpret the information, the questions are really simple. The catch in this question (this type of questions) is to interpret the given data.
Let us look at this information in a different way:

Let us look at this information in a different way:

As you can see, the triangular format is no different from providing the same information to the student in a table. It is just a little more intimidating in a pressure situation – and that intimidation is exactly what you should avoid. With some very simple addition and calculation you will be solve this problem set.

Bottom line: Even if it takes a couple of minutes, it is best to represent information in a format that you are comfortable with.

Type 2:

For some reason, Tennis appears to be a favorite among exam setters. Actually Tennis does offer some very interesting possibilities – such as seeds, an unconventional scoring and the knockout feature. Knockouts are inherent in the sport of Tennis and hence used frequently by exam setters.

Note: In a knockout tournament, No. of matches = No. of players – 1

Let us look at few ideas related to questions on seeded players. Let’s say in a tournament there are ‘n’ players and they are seeded (ranked) from 1 to n. Typically this ‘n’ is a power of 2 like 32 or 64 or 128.
In the first round the highest seeded player plays the lowest seeded player, the second highest seeded player plays the second lowest seeded player and so on. To put it into perspective:
Round 1 – Match 1 – Seed 1 Vs Seed n
Round 1 – Match 2 – Seed 2 Vs Seed (n-1)
Round 1 – Match 3 – Seed 3 Vs Seed (n-2)
.
.
Round 1 – Match n/2 – Seed n/2 Vs Seed n/2 + 1
In the second round, winner of Match 1 plays winner of Match n/2; winner of Match 2 plays winner of Match n/2 – 1 and so on.
In this kind of questions, an ‘upset’ comes into the picture which essentially means that a lower seeded plays beat a higher seeded player.
The questions are typically of the format:

Question: Who will play match 36 in Round 1?
Ans: It will be played between the 36th highest seed and the 36th lowest seed.
The 36th lowest seed can be sometimes difficult to figure out but you can figure it out easily by calculating (n+1) – 36.

Note: The rth match in Round 1 will be played between Seed ‘r’ and Seed ‘n+1-r’

Question: If there are no upsets, then in Round 2 who will play the 5th match?
Ans: One way of solving this question would be figuring out the winners of Round 1 and then figuring out the 5th from the top and the bottom. If there is no upset, then seed 5 will be there. The other player would be (n/2+1 – 5)

Question: Who will meet Seed 37 in the Quarterfinals of a tournament in which 64 players are taking part? Other than Seed 37’s matches, there were no other upsets.
We first need to analyse which round would be the quarterfinal:
Round 1 (32 matches), Round 2 (16 matches), Round 3 (8 matches – pre-quarter), Round 4 (4 matches – quarterfinals).
In Round 1, Seed 37 must have defeated 64 + 1 – 37 = 28
In Round 2, Seed 37 played the match that Seed 28 would have played. Seed 28 would have played against Seed 32 + 1 – 28 = Seed 5
In Round 3(pre-quarters), Seed 37 played the match that Seed 5 would have played. Seed 5 would have played against Seed 16 + 1 – 5 = Seed 12 and won it.
In Round 4 (quarterfinals), Seed 37 would meet the player that Seed 5 would have met. Seed 5 would have met 8+1-5 = 4.
Hence, Seed 37 will meet Seed 4 in the quarterfinals.
As a matter of fact, even the above solution is not the most optimal one. Because once you realize that Seed 37 defeated Seed 5, he would keep meeting the opponents that Seed 5 would have met.

Type 3

There is another popular type of questions with respect to Games & Tournaments and that is – Football / Hockey tournament questions in which we have to find out Goals scores, winners, ties, etc. In such tournaments, all competitors play a fixed number of matches. Points are awarded for wins / draws / losses. Then an overall ranking is decided by total points or average points per match. Sometimes other factors such as goals scored / goals faced also come into the picture to resolves ties in ranking.

Let us look at a question from CAT 2000.

Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated; the team that remains undefeated in the second stage is declared the winner and claims the Gold Cup. The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage a team earns one point for each win and no points for a loss. At the end of the first stage teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.

Q1) What is the total number of matches played in the tournament?
a. 28
b. 55
c. 63
d. 35

Q2) The minimum number of wins needed for a team in the first stage to guarantee its advancement to the next stage is:
a. 5
b. 6
c. 7
d. 4

Q3) What is the highest number of wins for a team in the first stage in spite of which it would be eliminated at the end of first stage?
a. 1
b. 2
c. 3
d. 4

Q4) What is the number of rounds in the second stage of the tournament?
a. 1
b. 2
c. 3
d. 4

Q5) Which of the following statements is true?
a. The winner will have more wins than any other team in the tournament.
b. At the end of the first stage, no team eliminated from the tournament will have more wins than any of the teams qualifying for the second stage.
c. It is possible that the winner will have the same number of wins in the entire tournament as a team eliminated at the end of the first stage.
d. The number of teams with exactly one win in the 2nd stage of the tournament is 4.

Now questions were asked on – Total number of matches, minimum number of wins required for a team to guarantee advance (or possible advance) to next stage, maximum number of matches that a team can win in the first stage without advancing, etc.

In first stage, teams are divided into two groups of 8 teams each. There they play a match against everyone exactly one ie 8C2 matches in every group. So 2 * 8C2 = 56 matches for the first stage.
In second stage, there are 8 teams in a knockout stage. There will be one winner, so 8 – 1 = 7
So, total number of matches is 56 + 7 = 63

For a team to advance to the second stage, it should be among the top 4 in its group. Total points on stake in a group is the same as the total number of matches which is 8C2 = 28. To guarantee advance, it can have 3 teams with the same or more points. There can be 5 teams with 5 wins or 5 points. So, 5 wins is not good enough to ensure a birth in round 2. However, 6 wins will guarantee its advance. This also tells us that a team might have 5 wins but still not advance.

To figure out the minimum wins required to possibly advance, let us look at the method for ‘n’ teams.
n/2 – 1 teams should win maximum no. of matches.
n/2 + 1 teams should have exactly the same number of wins.
So in this question, the top 3 teams can have a maximum of 7 + 6 + 5 = 18 points.
All other teams (5) have a combined score of 28 – 18 = 10 points. Their individual score is 2 points each and one of these five teams will advance to second stage.
So, minimum wins required to advance is 2.

Let us look at another type of question in which we are given a table and we have to fill it. Given below is a random table at the end of hockey tournament. For each win two points were awarded and for a draw one point was given. We also know that the South Africa – Spain match was a draw. No two teams have the exact same count for Win/Draw/Loss and Australia has won more matches than Spain. Figure out the result of every match from the table given below

Each team has played 4 matches.
A team can get a score of 6 in two ways:
3 Wins and 1 loss or 2 wins and 2 draws

India did not lose, so it will have 2 wins and 2 draws whereas on the other hand Pakistan will have 3 wins
No. of matches played will be 5C2 = 10
The total no. of points at stake is 20. South Africa has the left-over points which is 2.
We also know that the South Africa – Spain match was a draw.
So, now our table looks like this:

2 Points can be achieved by 1 Win, 0 Draw, 3 Loss OR 2 Draw, 2 Loss.
We know that both Spain & South Africa have at least 1 Draw. This means that South Africa’s 2 points are by 2 Draw, 2 Loss

3 Points can be achieved by 1 Win, 1 Draw, 2 Loss OR 3 Draw, 1 Loss.
As no two teams have the same Win/Draw/Loss count, one of the above applies to Australia whereas the other one applies to Spain. As Australia has won more matches, it will get the 1 Win, 1 Draw, 2 Loss.
Our final table will look like this:

Now, let us try and analyze the match results for the 10 matches (in no special order):

Pakistan has won 3 and lost 1. Pakistan cannot win against India as India did not lose a match. So,

Match 1 – India beat Pakistan
Match 2 – Pakistan beat Australia
Match 3 – Pakistan beat Spain
Match 4 – Pakistan beat South Africa
We also know the result of South Africa Vs Spain
Match 5 – South Africa & Spain drew the match.
Spain has lost against Pakistan and it needs to draw all other matches.
Match 6 – India & Spain drew the match
Match 7 – Australia & Spain drew the match
Australia cannot draw another match as it has only 1 draw. It cannot win against India as India has no losses. So, it must have lost against India and the win must have come against the remaining team ie South Africa.
Match 8 – India beat Australia
Match 9 – Australia beat South Africa
The only match remaining between India & South Africa must have been a draw as India scored wins against Pakistan and Australia.
Match 10 – India & South Africa drew the match

Phew!! I hope that you lasted this long without actually playing the game of “Banging head against the wall”.

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