In the previous couple of parts we tackled the questions involving general tournaments and knockouts. In this article, I will be focusing on round-robin tournaments based questions for CAT 2016 and the concepts of maxima and minima.

The basic question types that you can anticipate from these tournament questions are:

1) What is the maximum number of wins in spite of which a team could get eliminated after the first round?

2) What is the maximum number of wins in spite of which a team will definitely not enter the next round?

3) What is the minimum number of wins required to ensure progress to the next round?

4) What is the minimum number of wins required to stand a chance to go to the next round?

Although the questions sound similar (the last two), they mean different things. To put it in a precise manner, it is a matter of choosing between **possibility** and **certainty**. Let’s see this with the help of an example.

Say someone tells you that you need to take 75 mocks in a season to guarantee a 99 %ile score in CAT 2016.

a) What is the minimum number of mocks that you need to take to ensure a 99 %ile in CAT 2016?

b) What is the minimum number of mocks that you need to take to get a 99 %ile in CAT 2016?

The first answer is 75, the second is 0. See the difference? The first one hints at a certainty, the second one is full of hope. So, you need a few things to go in your favour (and outside your locus of control, probably) to get that 99 %ile score if you are going for scenario number 2.

Now, let’s get to the theory of round-robin tournaments based questions for CAT 2016

### Number of matches

This is pretty easy to calculate. If there are say 4 teams, A, B, C and D participating in a tournament, we know that the matches would be as follows:

A vs. B

A vs. C

A vs. D

B vs. C

B vs. D

C vs. D

In other words, we are trying to find out all possible combinations of 2 teams among A, B, C and D. So, ^{4}C_{2} = 6 ways in total.

So, for a round-robin tournament with n teams participating and each team playing each other team exactly once, we will have a total of ^{n}C_{2} matches. Similarly if the format is that of a double round-robin where each team plays each other team exactly twice (a home and an away match, a double header in other words). The common examples of this are the English Premier League, the IPL and so on.

The questions generally have a point to be distributed per match. So, if there is a match between team A and team B, and team A wins, then it will get 1 point. In case it ends in a draw both teams will get half a point each. The number of points is not critical in these questions (unless it is an unbalanced winner gets 3 and draw gets 1 in which case, it becomes more of an LR arrangements and scenarios set than a tournaments based one) and we would simply need to understand the logic. I have assumed one point per match in the remainder of the article.

### Maximum number of matches in spite of which a team crashes out

This covers the first two question types discussed earlier:

1) What is the maximum number of wins in spite of which a team could get eliminated after the first round?

Now, to find the required number, we have to even out the scores for the top teams as much as possible. To do that, we need to have as many points as possible to distribute among the bottom few teams. Let’s take an old CAT question:

There are 16 teams and they are divided into 2 pools of 8 each. Each team in a group plays against one another on a round-robin basis. Draws in the competition are not allowed. The top four teams from each group will qualify for the next round i.e. round 2. In case of teams having the same number of wins, the team with better run-rate would be ranked ahead.

Now, the first thing to note is that 4 teams are progressing to the next round from each group. Also, as there are 8 teams in a group, there would be a total of 8c2 = 28 matches in total and so, 28 points at stake.

So, to start with, we know that we want to be the fifth team and have as many points as are possible. Subsequently, we need to be left with as many points as are possible to distribute among the top 5 teams. So, we need to give the bottom 3 teams as few points as are possible.

The last time can be defeated by all other teams and so, can end up at 0 points.

The second last team (7^{th} place) can be defeated by all other teams except the last placed team and so, can end up at 1 point.

The 6^{th} place team can again be defeated by all other teams except the teams ranked 7 and 8 and so, can end up at 2 points.

Easy till now? Next step is to understand that we have already distributed 0 + 1 + 2 = 3 points and have 25 points left to distribute it among the top 5 teams while making sure that the fifth team gets as many points as are possible. The best way to do it is to ensure that each of the top 5 teams ends up with 5 points. So, even if you have earned 5 points, you ** could** still end up getting yourself eliminated from the competition. The important word here is

**which indicates a possibility.**

__could,__2) What is the maximum number of wins in spite of which a team will definitely not enter the next round?

This indicates a certainty. So, we have to find the minimum number of matches won to stand a chance to go to the next round and go one step down. As you would have noticed, we started filling the points table from the bottom and moved to the top in the first case (in case of maxima). Now, we will start from the top and will move to the bottom as we have to find the minima.

The top most team can win a maximum of 7 matches, the second ranked team can win 6 matches and the third team can win 5 matches. Now, if you have to be the 4^{th} ranked team and get through with the minimum number of points, the distribution of points from rank 4 to rank 8 will have to be equal. As we are done distributing 7 + 6 + 5 = 18 points, we would have to distribute 10 points among 5 teams. So, the 4^{th} placed team could get 2 points and still sneak through (depending on the performance of other teams, of course). So, 2 points indicates a possibility of getting through. Consequently, as postulated, if the team would have got just 1 point, **there was no chance** that it would make it to the next round.

### Minimum number of matches in spite of which a team crashes out

This covers the last two question types discussed earlier.

Again, let’s take the same example:

There are 16 teams and they are divided into 2 pools of 8 each. Each team in a group plays against one another on a round-robin basis. Draws in the competition are not allowed. The top four teams from each group will qualify for the next round i.e. round 2. In case of teams having the same number of wins, the team with better run-rate would be ranked ahead.

3) What is the minimum number of wins required to ensure progress to the next round?

For this question, we have to find the maximum number of wins in spite of which a team would be eliminated at the end of the first round and then go one better. As we have already discussed in the first question, a team could get eliminated in spite of winning a maximum of 5 matches. So, a team has to win 6 matches to ** ensure qualification** to the next round.

4) What is the minimum number of wins required to stand a chance to go to the next round?

For this question, we proceed in the same manner as we did in the second question. Just that we stop at the penultimate step and say that even if a team wins 2 matches, it ** could stand a chance** of going to the next round.

Bottom line is, in cases involving maxima you have to go from bottom to top and in cases involving minima, you have to go from top to bottom. Also, it is always better to think of possibility first for maxima and minima and then going for one more win or one less win to find the certainty scenarios like we have done above. Also, remember that unless you have made a cardinal error (something like giving 0 points to the last 2 teams which is impossible), you need not worry about how exactly the result would pan out. If there is a case that the tournament ends in 4-4-4-4-4-4-4-0, there could be multiple possibilities one of which I have shown below.

So, if you feel that a particular distribution is possible, back yourself and do not fret over the exact result table. You would be right most of the time.

I hope this post has managed to clear a few doubts about round-robin tournaments based questions. Do let us know in the comments section about your views on the same. Also, we would love to hear from you in case you want us to discuss some other topic in depth.

Read the first two parts of the series here:

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