Question: 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
Explanation:
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 n 4 + 1 2 = n + 4 8 players in the fourth round, where n + 4 8 should be even.
5. All numbers of players in the subsequent rounds should also be even.
From condition 3, we can conclude that:
n + 4 8 = 2 k , 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).