Four batsmen – Arjun, Bala, Chirag and Dev – played in all five matches (M1 to M5) of a one-day series for their team. The table lists, for each match, the four scores made by these batsmen, in descending order; it does not say who made which score.
| Match | Scores (runs) |
|---|---|
| M1 | 70, 60, 45, 23 |
| M2 | 88, 50, 12, 5 |
| M3 | 78, 40, 34, 20 |
| M4 | 92, 41, 30, 26 |
| M5 | 101, 55, 15, 9 |
Their series totals were: Arjun 220, Bala 218, Chirag 214 and Dev 242.
The following is also known:
1. Dev scored the only century of the series.
2. In M1, Dev was the top scorer among the four and Chirag scored the fewest.
3. In M2, Bala was the top scorer among the four and Dev scored the fewest.
4. Arjun was the top scorer among the four in exactly one match, M3.
5. Chirag was the top scorer in M4, and that was his highest score of the series.
| Match | Scores (runs) |
|---|---|
| M1 | 70, 60, 45, 23 |
| M2 | 88, 50, 12, 5 |
| M3 | 78, 40, 34, 20 |
| M4 | 92, 41, 30, 26 |
| M5 | 101, 55, 15, 9 |
Their series totals were: Arjun 220, Bala 218, Chirag 214 and Dev 242.
The following is also known:
1. Dev scored the only century of the series.
2. In M1, Dev was the top scorer among the four and Chirag scored the fewest.
3. In M2, Bala was the top scorer among the four and Dev scored the fewest.
4. Arjun was the top scorer among the four in exactly one match, M3.
5. Chirag was the top scorer in M4, and that was his highest score of the series.
Question 1
Numerical answer
How many runs did Arjun score in M4?
Show answer and solution
Answer: 30
Deduction: The clues fix M1: Dev 70, Chirag 23 (Arjun/Bala share 60, 45); M2: Bala 88, Dev 5 (Arjun/Chirag share 50, 12); M3: Arjun 78; M4: Chirag 92; M5: Dev 101.
Dev: 70 + 5 + 101 = 176, so M3 + M4 = 66. From {40, 34, 20} + {41, 30, 26}, only 40 + 26 works.
Chirag: 23 + 92 = 115, so M2 + M3 + M5 = 99 with M2 ∈ {50, 12}, M3 ∈ {34, 20}, M5 ∈ {55, 15, 9}: only 50 + 34 + 15 works. So Arjun has 12 in M2.
Arjun: 12 + 78 = 90, so M1 + M4 + M5 = 130 with M1 ∈ {60, 45}, M4 ∈ {41, 30}, M5 ∈ {55, 9}: only 45 + 30 + 55 works. Bala gets the rest (total 218 ✓).
| Batsman | M1 | M2 | M3 | M4 | M5 | Total |
|---|---|---|---|---|---|---|
| Arjun | 45 | 12 | 78 | 30 | 55 | 220 |
| Bala | 60 | 88 | 20 | 41 | 9 | 218 |
| Chirag | 23 | 50 | 34 | 92 | 15 | 214 |
| Dev | 70 | 5 | 40 | 26 | 101 | 242 |
Arjun scored 30 in M4.
Dev: 70 + 5 + 101 = 176, so M3 + M4 = 66. From {40, 34, 20} + {41, 30, 26}, only 40 + 26 works.
Chirag: 23 + 92 = 115, so M2 + M3 + M5 = 99 with M2 ∈ {50, 12}, M3 ∈ {34, 20}, M5 ∈ {55, 15, 9}: only 50 + 34 + 15 works. So Arjun has 12 in M2.
Arjun: 12 + 78 = 90, so M1 + M4 + M5 = 130 with M1 ∈ {60, 45}, M4 ∈ {41, 30}, M5 ∈ {55, 9}: only 45 + 30 + 55 works. Bala gets the rest (total 218 ✓).
| Batsman | M1 | M2 | M3 | M4 | M5 | Total |
|---|---|---|---|---|---|---|
| Arjun | 45 | 12 | 78 | 30 | 55 | 220 |
| Bala | 60 | 88 | 20 | 41 | 9 | 218 |
| Chirag | 23 | 50 | 34 | 92 | 15 | 214 |
| Dev | 70 | 5 | 40 | 26 | 101 | 242 |
Arjun scored 30 in M4.
Question 2
Single correct option
In how many matches did Bala score more runs than Chirag?
- A. 1
- B. 2
- C. 3
- D. 4
Show answer and solution
Answer: (B) 2
Deduction: The clues fix M1: Dev 70, Chirag 23 (Arjun/Bala share 60, 45); M2: Bala 88, Dev 5 (Arjun/Chirag share 50, 12); M3: Arjun 78; M4: Chirag 92; M5: Dev 101.
Dev: 70 + 5 + 101 = 176, so M3 + M4 = 66. From {40, 34, 20} + {41, 30, 26}, only 40 + 26 works.
Chirag: 23 + 92 = 115, so M2 + M3 + M5 = 99 with M2 ∈ {50, 12}, M3 ∈ {34, 20}, M5 ∈ {55, 15, 9}: only 50 + 34 + 15 works. So Arjun has 12 in M2.
Arjun: 12 + 78 = 90, so M1 + M4 + M5 = 130 with M1 ∈ {60, 45}, M4 ∈ {41, 30}, M5 ∈ {55, 9}: only 45 + 30 + 55 works. Bala gets the rest (total 218 ✓).
| Batsman | M1 | M2 | M3 | M4 | M5 | Total |
|---|---|---|---|---|---|---|
| Arjun | 45 | 12 | 78 | 30 | 55 | 220 |
| Bala | 60 | 88 | 20 | 41 | 9 | 218 |
| Chirag | 23 | 50 | 34 | 92 | 15 | 214 |
| Dev | 70 | 5 | 40 | 26 | 101 | 242 |
Bala vs Chirag: M1 60 > 23 ✓, M2 88 > 50 ✓, M3 20 < 34, M4 41 < 92, M5 9 < 15. So 2 matches.
Dev: 70 + 5 + 101 = 176, so M3 + M4 = 66. From {40, 34, 20} + {41, 30, 26}, only 40 + 26 works.
Chirag: 23 + 92 = 115, so M2 + M3 + M5 = 99 with M2 ∈ {50, 12}, M3 ∈ {34, 20}, M5 ∈ {55, 15, 9}: only 50 + 34 + 15 works. So Arjun has 12 in M2.
Arjun: 12 + 78 = 90, so M1 + M4 + M5 = 130 with M1 ∈ {60, 45}, M4 ∈ {41, 30}, M5 ∈ {55, 9}: only 45 + 30 + 55 works. Bala gets the rest (total 218 ✓).
| Batsman | M1 | M2 | M3 | M4 | M5 | Total |
|---|---|---|---|---|---|---|
| Arjun | 45 | 12 | 78 | 30 | 55 | 220 |
| Bala | 60 | 88 | 20 | 41 | 9 | 218 |
| Chirag | 23 | 50 | 34 | 92 | 15 | 214 |
| Dev | 70 | 5 | 40 | 26 | 101 | 242 |
Bala vs Chirag: M1 60 > 23 ✓, M2 88 > 50 ✓, M3 20 < 34, M4 41 < 92, M5 9 < 15. So 2 matches.
Question 3
Single correct option
Who was the second-highest scorer among the four in M5?
- A. Arjun
- B. Bala
- C. Chirag
- D. Cannot be determined
Show answer and solution
Answer: (A) Arjun
Deduction: The clues fix M1: Dev 70, Chirag 23 (Arjun/Bala share 60, 45); M2: Bala 88, Dev 5 (Arjun/Chirag share 50, 12); M3: Arjun 78; M4: Chirag 92; M5: Dev 101.
Dev: 70 + 5 + 101 = 176, so M3 + M4 = 66. From {40, 34, 20} + {41, 30, 26}, only 40 + 26 works.
Chirag: 23 + 92 = 115, so M2 + M3 + M5 = 99 with M2 ∈ {50, 12}, M3 ∈ {34, 20}, M5 ∈ {55, 15, 9}: only 50 + 34 + 15 works. So Arjun has 12 in M2.
Arjun: 12 + 78 = 90, so M1 + M4 + M5 = 130 with M1 ∈ {60, 45}, M4 ∈ {41, 30}, M5 ∈ {55, 9}: only 45 + 30 + 55 works. Bala gets the rest (total 218 ✓).
| Batsman | M1 | M2 | M3 | M4 | M5 | Total |
|---|---|---|---|---|---|---|
| Arjun | 45 | 12 | 78 | 30 | 55 | 220 |
| Bala | 60 | 88 | 20 | 41 | 9 | 218 |
| Chirag | 23 | 50 | 34 | 92 | 15 | 214 |
| Dev | 70 | 5 | 40 | 26 | 101 | 242 |
In M5: Dev 101, Arjun 55, Chirag 15, Bala 9. The second-highest is Arjun. Every score is uniquely fixed, so 'cannot be determined' is wrong.
Dev: 70 + 5 + 101 = 176, so M3 + M4 = 66. From {40, 34, 20} + {41, 30, 26}, only 40 + 26 works.
Chirag: 23 + 92 = 115, so M2 + M3 + M5 = 99 with M2 ∈ {50, 12}, M3 ∈ {34, 20}, M5 ∈ {55, 15, 9}: only 50 + 34 + 15 works. So Arjun has 12 in M2.
Arjun: 12 + 78 = 90, so M1 + M4 + M5 = 130 with M1 ∈ {60, 45}, M4 ∈ {41, 30}, M5 ∈ {55, 9}: only 45 + 30 + 55 works. Bala gets the rest (total 218 ✓).
| Batsman | M1 | M2 | M3 | M4 | M5 | Total |
|---|---|---|---|---|---|---|
| Arjun | 45 | 12 | 78 | 30 | 55 | 220 |
| Bala | 60 | 88 | 20 | 41 | 9 | 218 |
| Chirag | 23 | 50 | 34 | 92 | 15 | 214 |
| Dev | 70 | 5 | 40 | 26 | 101 | 242 |
In M5: Dev 101, Arjun 55, Chirag 15, Bala 9. The second-highest is Arjun. Every score is uniquely fixed, so 'cannot be determined' is wrong.
Question 4
Numerical answer
In how many of the five matches did Arjun score more runs than Dev?
Show answer and solution
Answer: 3
Deduction: The clues fix M1: Dev 70, Chirag 23 (Arjun/Bala share 60, 45); M2: Bala 88, Dev 5 (Arjun/Chirag share 50, 12); M3: Arjun 78; M4: Chirag 92; M5: Dev 101.
Dev: 70 + 5 + 101 = 176, so M3 + M4 = 66. From {40, 34, 20} + {41, 30, 26}, only 40 + 26 works.
Chirag: 23 + 92 = 115, so M2 + M3 + M5 = 99 with M2 ∈ {50, 12}, M3 ∈ {34, 20}, M5 ∈ {55, 15, 9}: only 50 + 34 + 15 works. So Arjun has 12 in M2.
Arjun: 12 + 78 = 90, so M1 + M4 + M5 = 130 with M1 ∈ {60, 45}, M4 ∈ {41, 30}, M5 ∈ {55, 9}: only 45 + 30 + 55 works. Bala gets the rest (total 218 ✓).
| Batsman | M1 | M2 | M3 | M4 | M5 | Total |
|---|---|---|---|---|---|---|
| Arjun | 45 | 12 | 78 | 30 | 55 | 220 |
| Bala | 60 | 88 | 20 | 41 | 9 | 218 |
| Chirag | 23 | 50 | 34 | 92 | 15 | 214 |
| Dev | 70 | 5 | 40 | 26 | 101 | 242 |
Arjun vs Dev: M1 45 < 70, M2 12 > 5 ✓, M3 78 > 40 ✓, M4 30 > 26 ✓, M5 55 < 101. Arjun outscored Dev in 3 matches, even though Dev's total is 22 runs higher.
Dev: 70 + 5 + 101 = 176, so M3 + M4 = 66. From {40, 34, 20} + {41, 30, 26}, only 40 + 26 works.
Chirag: 23 + 92 = 115, so M2 + M3 + M5 = 99 with M2 ∈ {50, 12}, M3 ∈ {34, 20}, M5 ∈ {55, 15, 9}: only 50 + 34 + 15 works. So Arjun has 12 in M2.
Arjun: 12 + 78 = 90, so M1 + M4 + M5 = 130 with M1 ∈ {60, 45}, M4 ∈ {41, 30}, M5 ∈ {55, 9}: only 45 + 30 + 55 works. Bala gets the rest (total 218 ✓).
| Batsman | M1 | M2 | M3 | M4 | M5 | Total |
|---|---|---|---|---|---|---|
| Arjun | 45 | 12 | 78 | 30 | 55 | 220 |
| Bala | 60 | 88 | 20 | 41 | 9 | 218 |
| Chirag | 23 | 50 | 34 | 92 | 15 | 214 |
| Dev | 70 | 5 | 40 | 26 | 101 | 242 |
Arjun vs Dev: M1 45 < 70, M2 12 > 5 ✓, M3 78 > 40 ✓, M4 30 > 26 ✓, M5 55 < 101. Arjun outscored Dev in 3 matches, even though Dev's total is 22 runs higher.
Question 5
Single correct option
For which batsman was the difference between his highest and lowest scores in the series the smallest?
- A. Arjun
- B. Bala
- C. Chirag
- D. Dev
Show answer and solution
Answer: (A) Arjun
Deduction: The clues fix M1: Dev 70, Chirag 23 (Arjun/Bala share 60, 45); M2: Bala 88, Dev 5 (Arjun/Chirag share 50, 12); M3: Arjun 78; M4: Chirag 92; M5: Dev 101.
Dev: 70 + 5 + 101 = 176, so M3 + M4 = 66. From {40, 34, 20} + {41, 30, 26}, only 40 + 26 works.
Chirag: 23 + 92 = 115, so M2 + M3 + M5 = 99 with M2 ∈ {50, 12}, M3 ∈ {34, 20}, M5 ∈ {55, 15, 9}: only 50 + 34 + 15 works. So Arjun has 12 in M2.
Arjun: 12 + 78 = 90, so M1 + M4 + M5 = 130 with M1 ∈ {60, 45}, M4 ∈ {41, 30}, M5 ∈ {55, 9}: only 45 + 30 + 55 works. Bala gets the rest (total 218 ✓).
| Batsman | M1 | M2 | M3 | M4 | M5 | Total |
|---|---|---|---|---|---|---|
| Arjun | 45 | 12 | 78 | 30 | 55 | 220 |
| Bala | 60 | 88 | 20 | 41 | 9 | 218 |
| Chirag | 23 | 50 | 34 | 92 | 15 | 214 |
| Dev | 70 | 5 | 40 | 26 | 101 | 242 |
Range (highest − lowest): Arjun 78 − 12 = 66; Bala 88 − 9 = 79; Chirag 92 − 15 = 77; Dev 101 − 5 = 96. Smallest: Arjun.
Dev: 70 + 5 + 101 = 176, so M3 + M4 = 66. From {40, 34, 20} + {41, 30, 26}, only 40 + 26 works.
Chirag: 23 + 92 = 115, so M2 + M3 + M5 = 99 with M2 ∈ {50, 12}, M3 ∈ {34, 20}, M5 ∈ {55, 15, 9}: only 50 + 34 + 15 works. So Arjun has 12 in M2.
Arjun: 12 + 78 = 90, so M1 + M4 + M5 = 130 with M1 ∈ {60, 45}, M4 ∈ {41, 30}, M5 ∈ {55, 9}: only 45 + 30 + 55 works. Bala gets the rest (total 218 ✓).
| Batsman | M1 | M2 | M3 | M4 | M5 | Total |
|---|---|---|---|---|---|---|
| Arjun | 45 | 12 | 78 | 30 | 55 | 220 |
| Bala | 60 | 88 | 20 | 41 | 9 | 218 |
| Chirag | 23 | 50 | 34 | 92 | 15 | 214 |
| Dev | 70 | 5 | 40 | 26 | 101 | 242 |
Range (highest − lowest): Arjun 78 − 12 = 66; Bala 88 − 9 = 79; Chirag 92 − 15 = 77; Dev 101 − 5 = 96. Smallest: Arjun.
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