Tournament based questions for CAT 2016– Part 2 (Knockouts)

I had covered a bit about tournaments in the first part of the post here. Now, I would be covering the technical aspects of knockout tournaments based questions for CAT 2016 LRDI. In the coming article (in the coming few days), I will be covering the maxima and minima question types for the round robin tournaments. Do drop in a comment if you want us to cover some other topic (and when we cover it, it essentially means in excruciating detail, as always).

Knockout tournaments

The definition of a knockout tournament, according to Wikipedia,

A single-elimination tournament—also called an Olympic system tournament, a knockout (or, knock-out), single penetration, or sudden death tournament—is a type of elimination tournament where the loser of each bracket is immediately eliminated from winning the championship or first prize in the event.

So, the one who loses is out of the tournament immediately and does not get a chance to come back (unlike a round robin tournament wherein, each player will play each other player at least once and won’t be eliminated at the end of the first round in most cases).

Number of matches

In a set on knockout tournaments based questions for CAT 2016, probably the easiest question will be the one that asks you how many matches would be played in the tournament. The simple answer to the question is (n-1). So, if there are say 8 teams in a knockout tournament, it will require 7 matches for us to determine the winner. This will hold true in case of matches that have a ‘bye’ system as well.

In knock-out (single-elimination) tournaments, if the number of participants is not a power of two (e.g. 16 or 32), one of the methods used to make a working bracket are byes which automatically move certain participants into a later round without requiring them to compete in an earlier one.

So, if A, B, C, D and E are participating, the lot could be drawn as:

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In the above image, the curved brackets indicate a match and the straight arrows indicate a ‘bye’ awarded to the corresponding participant.

Concept of seeds and upsets

The one thing that might makes a knockout tournament based question difficult is the presence of seeds and upsets.

A seed are defined as:

A seed is a competitor or team in a sports or other tournament who is given a preliminary ranking for the purposes of the draw. Players/teams are “planted” into the bracket in a manner that is typically intended so that the best do not meet until later in the competition.

In other words, the ranking of the player, the best being 1 and the last being n is the seed of that particular player.

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So, if we look at a real life example in the rankings above, we know that Novak Djokovic is the top ranked player, Andy Murray is ranked 2 and so on till Roger Federer who is ranked 7. Now, if in the Australian open, because of some reason, Rafael Nadal withdraws, the seed that Roger Federer will be conferred upon will be 6. So, in a tournament, the top ranked participants are sorted according to their ranks and the first one is paired with the last one, the second one with the second last one and so on, till we reach the middle two participants.

The second round will again pair the top most and the bottom most winners and this pattern will continue till the very end where the top 2 participants can face off. This is primarily to ensure that the top two seeds do not meet earlier in the competition and the final becomes as closely fought as is possible.

Upsets

Technically an upset is defined as:

An upset occurs in a competition, frequently in electoral politics or sports, when the party popularly expected to win (the favorite), is defeated by an underdog whom the majority expects to lose, defying the conventional wisdom. The underdog then becomes a giant-killer.

Essentially, if a lower ranked seed defeats a higher seed, it is deemed to be an upset.

So, essentially, in say a 16 member tournament, the draw would look like this (without any upsets)

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The first thing to observe here is that in a round, the sum of the seeds will be equal for all pairs. So, in the first round (or round of 16), the sum is always equal to 17. In the second round (or round of 8), the sum is always equal to 9 and so on till the final (round of 2) where the sum is equal to 3. So, in a round of n teams with consecutive seeds from 1 till n, the sum of seeds will always be equal to (n+1).

Now let’s say that 11 reaches the final instead of 2. So, there obviously has to be a few upsets along the way. In this case, 11 beats 6 in the first round, 3 in the second round and 2 in the third round.

In this case, we cannot say that the sum of the seeds of the players in a particular match will be equal to (n+1). But there is a way around this. We can confer the seed of the higher ranked player to the lower ranked one and move ahead with the fixtures. So, we can write the path as 11 beat 6 in the round of 16, 3 (11) beat 6 in the round of 8, and 3 (11) beat 2 in the round of 4. So, we are tracing the path of 3 and not that of 11 although we have kept in mind that 11 will proceed. Having a notation like the one above will help you do it in a quicker and more mechanical manner.

The question types

There could broadly be two question types:

1) A routine tournament in which no upsets happen

2) A tournament with one or more upsets

Let’s look at a question now and see how to solve it.

There are 64 players in a knock out tournament and every player is ranked (seeded) from 1 – 64. The matches are played in such a manner that in round one the 1st seeded player plays with the 64th, 2nd with the 63rd and so on. The players who win move on to the next round whereas others are out of the competition. In second round, the winner of match 1 will play winner of the last match (which was between seed 32 and seed 33), and winner of match 2 will meet the winner of second last match in round 1 and so forth. Thus after N number of rounds winner is declared.

1. Which seeds will play Match no 4 and Match no 9 in Round 1 of the tournament?
2. In a tournament of 128 players who will play 36 in round II if there are no upsets?
3. Who will meet Seed 68 in the Quarterfinal’s of a 128 format tournament, if seed 5 lost in the pre-quarters and there was no other upset?

Answers:

1. First round is essentially the round of 64 and so, the sum of seeds should be 65. So, 4 will play 61 and 9 will play 55 in the first round.
2. In the first round (round of 128), 36 will play against (129 – 36) i.e. 93. In the second round (round of 64), 36 will play against (65 – 36) i.e. 29. We need not find the opponent in the first round as it is given that there are no upsets.
3. A quarterfinal will be played when there are 2 more rounds to go before the final or when there are 8 teams. So, pre quarters will be the round of 16. In the round of 16, 5 would have played 12 and would have lost to 12. Also, 68 reaches the quarterfinals and so, 68 would have caused some upsets by itself. So, we can track the progress of 68 in this case:

68 v. 61 -> 61 (68) v. 4 -> 4 (68) v. 29 -> 4 (68) v 13 -> 4 (68) v. 5

So, 68 should have played 5 in the quarter finals but as 12 has reached in place of 5, 68 will play 5.

This might look difficult and lengthy but with a bit of practice, you can crack it with minimal solving.

Read the other parts of the tournaments series here:

1. Tournaments based DI sets
2. Cracking round-robin tournaments

Try another set from CAT 2005 and post your answers in the comments section below.

In the table below 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 No. 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.

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Q.1 If there are no upsets (a lower seeded player beating a higher seeded player) in the first round, and only match Nos. 6, 7, and 8 of the second round result in upsets, then who would meet Lindsay Davenport in quarter finals, in case Davenport reaches quarter finals?

(1) Justine Henin

(2) Nadia Petrova

(3) Patty Schnyder

(4) Venus Williams

Q.2 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?

(1) Dinara Safina

(2) Justine Henin

(3) Nadia Petrova

(4) Patty Schnyder

Q.3 If, in the first round, all even numbered matches (and none of the odd numbered ones) result in upsets, and there are no upsets in the second round, then who could be the lowest seeded player facing Maria Sharapova in semi-finals?

(1) Anastasia Myskina

(2) Flavia Pennetta

(3) Nadia Petrova

(4) Svetlana Kuznetsova

Q.4 If the top eight seeds make it to the quarterfinals, then who, amongst the players listed below, would definitely not play against Maria Sharapova in the final, in case Sharapova reaches the final?

(1) Amelie Mauresmo

(2) Elena Dementieva

(3) Kim Clijsters

(4) Lindsay Davenport

Do let us know in case you want us to cover some concepts for CAT 2016. All the best!

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