Question

If the number of players in the first round was between 65 and 128, 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.

• A. If A alone but not B alone is sufficient
• B. If B alone but not A alone is sufficient
• C. If both A and B together are sufficient
• D. If A alone is sufficient and B alone is sufficient
• E. If not even A and B together are sufficient

Hint:

In bye-related tournament problems, the combination of total byes and their specific round can uniquely determine the initial player count.

Answer 1

The Correct Option is C

From A: Knowing only that exactly one bye occurred in the tournament narrows possibilities, but without knowing which round, multiple \(n\) values are possible in the 65–128 range. From B: Knowing only that the bye occurred in round 3 → round 4 also does not fix \(n\) uniquely, because different initial \(n\) can produce a bye in that round. Together: The round of the bye plus its uniqueness allows exact backtracking of eliminations to find initial \(n\). This yields a unique \(n\). Thus, both together are sufficient, neither alone is.