In this article, I will solve a set that appeared in CAT 2008 using the technique of solving tournament-based DI sets. You can try it out on your own and then figure out the areas of improvement, if any by referring back to the article:
In a sports event, six teams (A, B, C, D, E and F) are competing against each other. Matches are scheduled in two stages. Each team plays three matches in stage-1 and two matches in stage-2. No team plays against the same team more than once in the event. No ties are permitted in any of the matches. The observations after the completion of stage-1 and stage-2 are as given below.
Stage – 1
- One team won all the three matches
- Two teams lost all the matches
- D lost to A but won against C and F
- E lost to B but won against C and F
- B lost at least one match
- F did not play against the top team of stage-1
Stage – 2
- The leader of Stage-1 lost the next two matches
- Of the two teams at the bottom after stage-1, one team won both matches, while the other lost both matches
- One more team lost both matches in stage-2
1. The only teams that won both the matches in stage-2 is (are):
(a) B
(b) E and F
(c) A, E and F
(d) B, E and F
(e) B and F
2. The teams that won exactly two matches in the event are:
(a) A, D and F
(b) D and E
(c) E and F
(d) D, E and F
(e) D and F
3. The team(s) with the most wins in the event is (are):
(a) A
(b) A and C
(c) F
(d) E
(e) B and E
4. The two teams that defeated the leader of stage-1 are:
(a) F and D
(b) E and F
(c) B and D
(d) E and D
(e) F and D
Solution:
Each team plays 5 matches in total out of which three are in stage-1 and two in stage-2. So, there would be a total of 15 matches across the two stages. The table has to be formulated only in terms of wins and losses. The first stage had 9 matches in total and the second stage had 6 matches.
From the stage-1 data, we get the following matches:
D vs. A (A)
D vs. C (D)
D vs. F (D)
E vs. B (B)
E vs. C (E)
E vs. F (E)
From the stage-2 data:
Nothing specific is given and we have to figure out the stage-1 matches first.
Now, going to the ‘solving’ part:
We know that each team played 3 matches in stage-1. We know all the matches that D and E have participated in. We also know two matches each of C and F and one match each for A and B. So, we know that in the remaining 3 matches, A and B would each be involved in 2 matches and C and F each in 1 match.
The first bullet says that one team won all the matches. We know that either of D, E, C and F cannot be this team. Also, from bullet number 5, we understand that B lost at least one match. So, it is obvious that A won all the three matches it played.
From bullet 2, it can be understood that two teams lost all their matches. As, from the given data, we know that A, B, E and D have won at least one match each, C and F have to be the teams that have lost all their matches and so, C vs. F is not possible.
Also, from bullet 6, it can be deduced that A vs. F was not one of the matches.
So, the 3 missing matches would be: F vs. B won by B, A vs. B won by A, A vs. C won by A.
So, we have got the entire first stage scenario table here:
| Match | Winner |
| D vs. A | A |
| D vs. C | D |
| D vs. F | D |
| E vs. B | B |
| E vs. C | E |
| E vs. F | E |
| F vs. B | B |
| A vs. B | A |
| A vs. C | A |
Going to the second stage, we understand that:
A lost the next two matches that were against E and F
From the second bullet, we understand that either C or F won both of its matches and the other team lost both of its matches. From the first point, it is obvious that F would have been the winner and C would have been the loser. So, C lost against B and F and F won against A and C.
From the third bullet, it can be seen that there was another team in addition to A and C that had lost both its matches in the second stage. Now, we can see that B, E, F have won at least one match and so, it has to be D who lost both the matches.
So, we get the stage two table as follows:
| Match | Winner |
| A vs. E | E |
| A vs. F | F |
| B vs. C | B |
| F vs. C | F |
| B vs. D | B |
| E vs. D | E |
Now, solving the questions is extremely easy and can be simply plugged in from the tables.
You can read the theory regarding tournament-based DI sets in this article.
