Sixteen players, seeded 1 (strongest) to 16, took part in a knockout badminton tournament. There were four rounds: Round 1, the quarterfinals (QF), the semifinals (SF) and the final. Every match produced a winner, who advanced, and a loser, who was eliminated. The draw was fixed as follows.
| Round 1 match | Players |
|---|---|
| M1 | 1 vs 16 |
| M2 | 8 vs 9 |
| M3 | 5 vs 12 |
| M4 | 4 vs 13 |
| M5 | 3 vs 14 |
| M6 | 6 vs 11 |
| M7 | 7 vs 10 |
| M8 | 2 vs 15 |
In the quarterfinals, the winner of M1 played the winner of M2, M3's winner played M4's winner, M5's winner played M6's winner and M7's winner played M8's winner. In the semifinals, the winners of the first two quarterfinals played each other, as did the winners of the last two. A match is called an upset if the player with the higher seed number (i.e. the weaker seed) wins it.
The following is known:
1. Exactly three Round 1 matches were upsets, and there were exactly six upsets in the whole tournament.
2. No player seeded 13 to 16 won any match.
3. Seeds 3 and 4 both lost in the quarterfinals, each to a player whose seed number is greater than 10.
4. Seed 7 won exactly one match.
5. The final was played between a player seeded in the top 4 (seed number 1 to 4) and a player whose seed number is greater than 10.
6. Seed 1 was knocked out by the player who finished as runner-up.
| Round 1 match | Players |
|---|---|
| M1 | 1 vs 16 |
| M2 | 8 vs 9 |
| M3 | 5 vs 12 |
| M4 | 4 vs 13 |
| M5 | 3 vs 14 |
| M6 | 6 vs 11 |
| M7 | 7 vs 10 |
| M8 | 2 vs 15 |
In the quarterfinals, the winner of M1 played the winner of M2, M3's winner played M4's winner, M5's winner played M6's winner and M7's winner played M8's winner. In the semifinals, the winners of the first two quarterfinals played each other, as did the winners of the last two. A match is called an upset if the player with the higher seed number (i.e. the weaker seed) wins it.
The following is known:
1. Exactly three Round 1 matches were upsets, and there were exactly six upsets in the whole tournament.
2. No player seeded 13 to 16 won any match.
3. Seeds 3 and 4 both lost in the quarterfinals, each to a player whose seed number is greater than 10.
4. Seed 7 won exactly one match.
5. The final was played between a player seeded in the top 4 (seed number 1 to 4) and a player whose seed number is greater than 10.
6. Seed 1 was knocked out by the player who finished as runner-up.
Question 1
Single correct option
Which seed won the tournament?
- A. Seed 1
- B. Seed 2
- C. Seed 11
- D. Seed 12
Show answer and solution
Answer: (B) Seed 2
Deduction: Clue 2 means seeds 1, 4, 3 and 2 win their Round 1 matches. Seed 7 wins M7 (clue 4). So the three Round 1 upsets must be M2, M3 and M6: 9, 12 and 11 win. Clue 3: seed 4 loses in the QF to 12 (the only player >10 in that quarter), and seed 3 loses to 11. Seed 7 won only one match, so seed 2 beats 7 in the QF. In the first QF, 1 must beat 9 (otherwise seed 1 could not be knocked out by the runner-up, since 9 is not >10 and could not reach the final under clue 5). The final needs one top-4 seed and one seed >10; the SFs are 1 vs 12 and 11 vs 2. Seed 1 is knocked out by the runner-up, so 12 beats 1 and then loses the final. Hence 12 is runner-up and the other finalist, a top-4 seed, is 2 (beating 11). Upset count: 3 (R1) + 2 (QF) + 1 (12 over 1) = 6 ✓, so the final (2 beats 12) is not an upset ✓.
| Round | Results (winner first) |
|---|---|
| Round 1 | 1–16, 9–8, 12–5, 4–13, 3–14, 11–6, 7–10, 2–15 |
| QF | 1–9, 12–4, 11–3, 2–7 |
| SF | 12–1, 2–11 |
| Final | 2–12 |
The champion is seed 2.
| Round | Results (winner first) |
|---|---|
| Round 1 | 1–16, 9–8, 12–5, 4–13, 3–14, 11–6, 7–10, 2–15 |
| QF | 1–9, 12–4, 11–3, 2–7 |
| SF | 12–1, 2–11 |
| Final | 2–12 |
The champion is seed 2.
Question 2
Numerical answer
How many matches did seed 12 win in the tournament?
Show answer and solution
Answer: 3
Deduction: Clue 2 means seeds 1, 4, 3 and 2 win their Round 1 matches. Seed 7 wins M7 (clue 4). So the three Round 1 upsets must be M2, M3 and M6: 9, 12 and 11 win. Clue 3: seed 4 loses in the QF to 12 (the only player >10 in that quarter), and seed 3 loses to 11. Seed 7 won only one match, so seed 2 beats 7 in the QF. In the first QF, 1 must beat 9 (otherwise seed 1 could not be knocked out by the runner-up, since 9 is not >10 and could not reach the final under clue 5). The final needs one top-4 seed and one seed >10; the SFs are 1 vs 12 and 11 vs 2. Seed 1 is knocked out by the runner-up, so 12 beats 1 and then loses the final. Hence 12 is runner-up and the other finalist, a top-4 seed, is 2 (beating 11). Upset count: 3 (R1) + 2 (QF) + 1 (12 over 1) = 6 ✓, so the final (2 beats 12) is not an upset ✓.
| Round | Results (winner first) |
|---|---|
| Round 1 | 1–16, 9–8, 12–5, 4–13, 3–14, 11–6, 7–10, 2–15 |
| QF | 1–9, 12–4, 11–3, 2–7 |
| SF | 12–1, 2–11 |
| Final | 2–12 |
Seed 12 beat 5 (Round 1), 4 (QF) and 1 (SF) and lost the final: 3 wins.
| Round | Results (winner first) |
|---|---|
| Round 1 | 1–16, 9–8, 12–5, 4–13, 3–14, 11–6, 7–10, 2–15 |
| QF | 1–9, 12–4, 11–3, 2–7 |
| SF | 12–1, 2–11 |
| Final | 2–12 |
Seed 12 beat 5 (Round 1), 4 (QF) and 1 (SF) and lost the final: 3 wins.
Question 3
Single correct option
Who knocked out seed 9?
- A. Seed 1
- B. Seed 8
- C. Seed 12
- D. Seed 2
Show answer and solution
Answer: (A) Seed 1
From the bracket: 9 beat 8 in Round 1 (an upset) and then met the M1 winner, seed 1, in the quarterfinal.
If 9 had won that QF, 9 would have played 12 in the SF, and seed 1 would have been knocked out by 9, who cannot be the runner-up (a finalist must be top-4 or >10 by clue 5, and 9 is neither). So seed 1 beat 9: seed 1 knocked out seed 9.
If 9 had won that QF, 9 would have played 12 in the SF, and seed 1 would have been knocked out by 9, who cannot be the runner-up (a finalist must be top-4 or >10 by clue 5, and 9 is neither). So seed 1 beat 9: seed 1 knocked out seed 9.
Question 4
Numerical answer
What is the sum of the seed numbers of the four semifinalists?
Show answer and solution
Answer: 26
Deduction: Clue 2 means seeds 1, 4, 3 and 2 win their Round 1 matches. Seed 7 wins M7 (clue 4). So the three Round 1 upsets must be M2, M3 and M6: 9, 12 and 11 win. Clue 3: seed 4 loses in the QF to 12 (the only player >10 in that quarter), and seed 3 loses to 11. Seed 7 won only one match, so seed 2 beats 7 in the QF. In the first QF, 1 must beat 9 (otherwise seed 1 could not be knocked out by the runner-up, since 9 is not >10 and could not reach the final under clue 5). The final needs one top-4 seed and one seed >10; the SFs are 1 vs 12 and 11 vs 2. Seed 1 is knocked out by the runner-up, so 12 beats 1 and then loses the final. Hence 12 is runner-up and the other finalist, a top-4 seed, is 2 (beating 11). Upset count: 3 (R1) + 2 (QF) + 1 (12 over 1) = 6 ✓, so the final (2 beats 12) is not an upset ✓.
| Round | Results (winner first) |
|---|---|
| Round 1 | 1–16, 9–8, 12–5, 4–13, 3–14, 11–6, 7–10, 2–15 |
| QF | 1–9, 12–4, 11–3, 2–7 |
| SF | 12–1, 2–11 |
| Final | 2–12 |
Semifinalists: 1, 12, 11 and 2. Sum = 1 + 12 + 11 + 2 = 26.
| Round | Results (winner first) |
|---|---|
| Round 1 | 1–16, 9–8, 12–5, 4–13, 3–14, 11–6, 7–10, 2–15 |
| QF | 1–9, 12–4, 11–3, 2–7 |
| SF | 12–1, 2–11 |
| Final | 2–12 |
Semifinalists: 1, 12, 11 and 2. Sum = 1 + 12 + 11 + 2 = 26.
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