Single Elimination Tournament

CAT 2008 Slot 1 · QA · Hard · Data Sufficiency

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Passage / data set

Directions for Questions 19 and 20: Mark (1) if Q can be answered from A alone but not from B alone. Mark (2) if Q can be answered from B alone but not from A alone. Mark (3) if Q can be answered from A alone as well as from B alone. Mark (4) if Q can be answered from A and B together but not from any of them alone. Mark (5) if Q cannot be answered even from A and B together.

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 nn, in any round is even, then the players are grouped into n/2n/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 nn, 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)(n-1) players are grouped into (n−1)/2(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.

Question 1 of 2

What is the number of Matches played by the champion? A. The entry list for the tournament consists of 83 players. B. The champion received one bye.

  1. A.

    (1) Q can be answered from A alone but not from B alone

  2. B.

    (2) Q can be answered from B alone but not from A alone

  3. C.

    (3) Q can be answered from A alone as well as from B alone

  4. D.

    (4) Q can be answered from A and B together but not from any of them alone

  5. E.

    (5) Q cannot be answered even from A and B together

Answer

D

Explanation

Statement A: With 83 players, rounds go 83→42→21→11→6→3→2→183 \to 42 \to 21 \to 11 \to 6 \to 3 \to 2 \to 1 (7 rounds). But without knowing whether champion received a bye, matches played could be 6 or 7. Statement B: Champion received 1 bye, but nn is unknown. Combining A and B: 7 rounds played and champion got 1 bye, so champion played 7−1=67 - 1 = 6 matches. Both statements together are required. Hence, (4).

Question 2 of 2

If the number of players, say nn, in the first round was between 65 and 128, then what is the exact value of nn? 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.

  1. A.

    (1) Q can be answered from A alone but not from B alone

  2. B.

    (2) Q can be answered from B alone but not from A alone

  3. C.

    (3) Q can be answered from A alone as well as from B alone

  4. D.

    (4) Q can be answered from A and B together but not from any of them alone

  5. E.

    (5) Q cannot be answered even from A and B together

Answer

D

Explanation

Statement A alone: Multiple values of nn give exactly 1 bye (e.g., n=127n = 127 gives bye in round 1, n=96n = 96 gives bye in round 6). Statement B alone: Not sufficient alone. Combining A and B: n=124n = 124 is the unique integer between 65 and 128 that generates exactly 1 bye in total and that bye occurs in round 3. Both statements together are sufficient. Hence, (4).

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