Five shooters – P, Q, R, S and T – took part in a three-round shooting competition. In every round, each shooter received an integer score, and the five scores in any round were exactly 6, 7, 8, 9 and 10 in some order (so no two shooters received the same score in the same round). A shooter's final total is the sum of the scores in the three rounds, and the final ranking is by total, highest first.
The following facts are known:
1. No two shooters had the same final total. R won the competition and T finished last.
2. S scored 6 in Round 1, and T scored 8 in Round 1.
3. R had the highest score in Round 2.
4. The shooter who scored 10 in Round 1 scored 6 in Round 3.
5. P's score in Round 3 was 4 more than P's score in Round 2.
6. T was the only shooter who received the same score in two different rounds.
The following facts are known:
1. No two shooters had the same final total. R won the competition and T finished last.
2. S scored 6 in Round 1, and T scored 8 in Round 1.
3. R had the highest score in Round 2.
4. The shooter who scored 10 in Round 1 scored 6 in Round 3.
5. P's score in Round 3 was 4 more than P's score in Round 2.
6. T was the only shooter who received the same score in two different rounds.
Question 1
Single correct option
Who finished second in the competition?
- A. Q
- B. P
- C. S
- D. R
Show answer and solution
Answer: (B) P
Deduction. From (5), with scores between 6 and 10, P scored 6 in R2 and 10 in R3. From (4), the R1 topper scored 6 in R3: not P (10 in R3), not S or T (6 and 8 in R1), and not R (R has 10 in R2, so 10 in R1 would be a repeat, violating (6)). So Q scored 10 in R1 and 6 in R3, and P, R share 7 and 9 in R1. In R2 and in R3, the remaining scores 7, 8, 9 go to (Q, S, T) and (R, S, T) respectively.
T must finish below S: 8 + t₂ + t₃ < 6 + s₂ + s₃. S cannot repeat a score, so s₂ + s₃ ≤ 8 + 9 = 17, giving t₂ + t₃ ≤ 14. The only possibility is t₂ = t₃ = 7, so T = 22 (and T's repeat satisfies (6)). S then needs more than 22, so S has 8 and 9 in some order.
- If S = 8 (R2), 9 (R3): Q = 10 + 9 + 6 = 25 and R = r₁ + 18. R must beat 25, so r₁ = 9, making P = 7 + 16 = 23 = S – a tie ✗.
- If S = 9 (R2), 8 (R3): Q = 10 + 8 + 6 = 24 and R = r₁ + 19. r₁ = 9 gives P = 23 = S ✗; so r₁ = 7, R = 26, P = 9 + 16 = 25 ✓.
| Shooter | R1 | R2 | R3 | Total |
|---|---|---|---|---|
| P | 9 | 6 | 10 | 25 |
| Q | 10 | 8 | 6 | 24 |
| R | 7 | 10 | 9 | 26 |
| S | 6 | 9 | 8 | 23 |
| T | 8 | 7 | 7 | 22 |
Ranking: R (26), P (25), Q (24), S (23), T (22).
T must finish below S: 8 + t₂ + t₃ < 6 + s₂ + s₃. S cannot repeat a score, so s₂ + s₃ ≤ 8 + 9 = 17, giving t₂ + t₃ ≤ 14. The only possibility is t₂ = t₃ = 7, so T = 22 (and T's repeat satisfies (6)). S then needs more than 22, so S has 8 and 9 in some order.
- If S = 8 (R2), 9 (R3): Q = 10 + 9 + 6 = 25 and R = r₁ + 18. R must beat 25, so r₁ = 9, making P = 7 + 16 = 23 = S – a tie ✗.
- If S = 9 (R2), 8 (R3): Q = 10 + 8 + 6 = 24 and R = r₁ + 19. r₁ = 9 gives P = 23 = S ✗; so r₁ = 7, R = 26, P = 9 + 16 = 25 ✓.
| Shooter | R1 | R2 | R3 | Total |
|---|---|---|---|---|
| P | 9 | 6 | 10 | 25 |
| Q | 10 | 8 | 6 | 24 |
| R | 7 | 10 | 9 | 26 |
| S | 6 | 9 | 8 | 23 |
| T | 8 | 7 | 7 | 22 |
Ranking: R (26), P (25), Q (24), S (23), T (22).
Question 2
Numerical answer
What was Q's score in Round 2?
Show answer and solution
Answer: 8
Final grid:
| Shooter | R1 | R2 | R3 | Total |
|---|---|---|---|---|
| P | 9 | 6 | 10 | 25 |
| Q | 10 | 8 | 6 | 24 |
| R | 7 | 10 | 9 | 26 |
| S | 6 | 9 | 8 | 23 |
| T | 8 | 7 | 7 | 22 |
In R2, R has 10 (clue 3) and P has 6 (clue 5). T must repeat a score and can only do so with 7 in both R2 and R3. S takes 9 in R2 (S cannot repeat a score, and S's R3 score is 8 or 9), leaving 8 for Q.
| Shooter | R1 | R2 | R3 | Total |
|---|---|---|---|---|
| P | 9 | 6 | 10 | 25 |
| Q | 10 | 8 | 6 | 24 |
| R | 7 | 10 | 9 | 26 |
| S | 6 | 9 | 8 | 23 |
| T | 8 | 7 | 7 | 22 |
In R2, R has 10 (clue 3) and P has 6 (clue 5). T must repeat a score and can only do so with 7 in both R2 and R3. S takes 9 in R2 (S cannot repeat a score, and S's R3 score is 8 or 9), leaving 8 for Q.
Question 3
Single correct option
How many shooters scored more in Round 3 than in Round 1?
- A. 1
- B. 2
- C. 3
- D. 4
Show answer and solution
Answer: (C) 3
Final grid:
| Shooter | R1 | R2 | R3 | Total |
|---|---|---|---|---|
| P | 9 | 6 | 10 | 25 |
| Q | 10 | 8 | 6 | 24 |
| R | 7 | 10 | 9 | 26 |
| S | 6 | 9 | 8 | 23 |
| T | 8 | 7 | 7 | 22 |
Compare R3 with R1: P 10 > 9 ✓, Q 6 < 10 ✗, R 9 > 7 ✓, S 8 > 6 ✓, T 7 < 8 ✗.
So 3 shooters (P, R, S) improved from Round 1 to Round 3.
| Shooter | R1 | R2 | R3 | Total |
|---|---|---|---|---|
| P | 9 | 6 | 10 | 25 |
| Q | 10 | 8 | 6 | 24 |
| R | 7 | 10 | 9 | 26 |
| S | 6 | 9 | 8 | 23 |
| T | 8 | 7 | 7 | 22 |
Compare R3 with R1: P 10 > 9 ✓, Q 6 < 10 ✗, R 9 > 7 ✓, S 8 > 6 ✓, T 7 < 8 ✗.
So 3 shooters (P, R, S) improved from Round 1 to Round 3.
Question 4
Single correct option
Which shooter had the highest cumulative score at the end of Round 2?
- A. Q
- B. R
- C. P
- D. S
Show answer and solution
Answer: (A) Q
Cumulative after two rounds (R1 + R2): P 9 + 6 = 15, Q 10 + 8 = 18, R 7 + 10 = 17, S 6 + 9 = 15, T 8 + 7 = 15.
Q led after two rounds with 18, one point ahead of R. R only overtook Q in Round 3 (R scored 9 against Q's 6).
| Shooter | R1 | R2 | R3 | Total |
|---|---|---|---|---|
| P | 9 | 6 | 10 | 25 |
| Q | 10 | 8 | 6 | 24 |
| R | 7 | 10 | 9 | 26 |
| S | 6 | 9 | 8 | 23 |
| T | 8 | 7 | 7 | 22 |
Q led after two rounds with 18, one point ahead of R. R only overtook Q in Round 3 (R scored 9 against Q's 6).
| Shooter | R1 | R2 | R3 | Total |
|---|---|---|---|---|
| P | 9 | 6 | 10 | 25 |
| Q | 10 | 8 | 6 | 24 |
| R | 7 | 10 | 9 | 26 |
| S | 6 | 9 | 8 | 23 |
| T | 8 | 7 | 7 | 22 |
Question 5
Single correct option
Which of the following statements is true?
- A. P scored 10 in Round 1
- B. S scored 9 in Round 2
- C. T scored 8 in Round 3
- D. R scored 9 in Round 1
Show answer and solution
Answer: (B) S scored 9 in Round 2
Final grid:
| Shooter | R1 | R2 | R3 | Total |
|---|---|---|---|---|
| P | 9 | 6 | 10 | 25 |
| Q | 10 | 8 | 6 | 24 |
| R | 7 | 10 | 9 | 26 |
| S | 6 | 9 | 8 | 23 |
| T | 8 | 7 | 7 | 22 |
- 'P scored 10 in Round 1' – false; P scored 9 (Q scored 10).
- 'S scored 9 in Round 2' – true.
- 'T scored 8 in Round 3' – false; T scored 7.
- 'R scored 9 in Round 1' – false; R scored 7 in Round 1 and 9 in Round 3.
Tip: In 'which is true' questions on a fully solved grid, check each option straight against the table.
| Shooter | R1 | R2 | R3 | Total |
|---|---|---|---|---|
| P | 9 | 6 | 10 | 25 |
| Q | 10 | 8 | 6 | 24 |
| R | 7 | 10 | 9 | 26 |
| S | 6 | 9 | 8 | 23 |
| T | 8 | 7 | 7 | 22 |
- 'P scored 10 in Round 1' – false; P scored 9 (Q scored 10).
- 'S scored 9 in Round 2' – true.
- 'T scored 8 in Round 3' – false; T scored 7.
- 'R scored 9 in Round 1' – false; R scored 7 in Round 1 and 9 in Round 3.
Tip: In 'which is true' questions on a fully solved grid, check each option straight against the table.
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