CAT 2008 — QA Question 25
Each question is followed by two statements, A and B. Answer each question using the following instructions:
Mark option (1) if the question can be answered by using statement A alone but not by using statement B alone.
Mark option (2) if the question can be answered by using statement B alone but not by using statement A alone.
Mark option (3) if the question can be answered by using either statement alone.
Mark option (4) if the question can be answered by using both the statements together but not by either of the statements alone.
Mark option (5) if the question cannot be answered on the basis of the two statements.
In a single elimination tournament, any player is eliminated with a single loss. The tournament is played in multiple rounds subject to the following rules:
a. If the number of players, say n, in any round is even, then the players are grouped in to n/2 pairs. The players in each pair play a match against each other and the winner moves on to the next round.
b. If the number of players, say n, in any round is odd, then one of them is given a bye, that is, he automatically moves on to the next round. The remaining (n − 1) players are grouped into (n − 1)/2 pairs. The players in each pair play a match against each other and the winner moves on to the next round. No player gets more than one bye in the entire tournament.
Thus, if n is even, then n/2 players move on to the next round while if n is odd, then (n + 1)/2 players move on to the next round. The process is continued till the final round, which obviously is played between two players. The winner in the final round is the champion of the tournament.
If the number of players, say n, in the first round was between 65 and 128, then what is the exact value of n?
A. Exactly one player received a bye in the entire tournament.
B. One player received a bye while moving on to the fourth round from the third round
Answer & solution
- A
1
- B
2
- C
3
4
- E
5
From statement (A) alone:
Exactly 1 player received a bye in the entire tournament. We get many values of n between 65 and 128 that satisfy this condition.
For example, n can have the value 124 in round 1, to follow the pattern, [124-62-31-16-8-4-2-1].
Also, n can have the value 127 in round 1, to follow the pattern, [127-64-32-16-8-4-2-1].
∴ We cannot answer the question on the basis of statement (A) alone.
From statement (B) alone:
One player received a bye while moving on to the fourth round from the third round.
Here also, we get multiple values of n.
For example, n can have the value 124 in round 1, where 1 player received a bye while moving from round 3 to round 4 following the pattern, [124-62-31-16-8-4-2-1].
Also, n can have the value 122 in round 1, where 1 player received a bye while moving from round 3 to round 4 following the pattern, [122-61-31-16-8-4-2-1].
∴ We cannot answer the question on the basis of statement (B) alone.
From statements (A) and (B) together:
n can only have the value 124 in round 1, where exactly 1 player received a bye while moving from round 3 to round 4 following the pattern [124-62-31-16-8-4-2-1].
∴ We can answer the question using both the statements (A) and (B) together.
Hence, option (d).
Note: An analysis of how 124 was arrived at when using both conditions together:
Let the number of players in the first round be n. Since only one player gets a bye, and that too when moving from the third to the fourth round, hence we have the following conditions:
1. There will be n players in the first round, where n is even.
2. There will be n/2 players in the second round, where n/2 is even.
3. There will be n/4 players in the third round, where n/4 is odd.
4. There will be players in the fourth round, where should be even.
5. All numbers of players in the subsequent rounds should also be even.
From condition 3, we can conclude that:
where k is an integer
Hence, n = 16k – 4; so, within the given range, n could be 76 or 92 or 108 or 124.
Writing the pattern for each of the above possible values of n, we have:
76: [76-38-19-10-5-3-2-1]
92: [92-46-23-12-6-3-2-1]
108: [108-54-27-14-7-4-2-1]
124: [124-62-31-16-8-4-2-1]
We see that only 124 satisfies condition 5.
Hence, option (d).