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📌 Section 1 is Logical reasoning specially designed for CAT
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SET 1 — 🔍 CAT Level
The bar graph below provides the number of wars won and lost by four countries – A through D – during the 18th and the 19th centuries. It is known that, during the 18th and 19th centuries, the four countries waged wars only against each other.
Question 1
If D won 16 wars against each of B and C, what is the maximum number of wars that B won against D?
A. 21
B. 22
C. 23
D. 24
Question 2
If B won an equal number of wars against each of A, C and D, what is the maximum number of wars that D could have won against A?
Question 3
Among the four countries, it so happened that a pair of countries (X, Y) were such that all the wars that X waged were only against Y.
How many possibilities exist for (X, Y)?
A. 1
B. 2
C. 3
D. 4
Question 4
Among the four countries, it so happened that a pair of countries (X, Y) were such that all the wars that X waged were only against Y.
What is the maximum number of wars that D won against B?
Answers and Explanation
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Country | Wars Won | Wars Lost |
|---|---|---|
A | 25 | 38 |
B | 36 | 26 |
C | 22 | 28 |
D | 33 | 24 |
Since these four countries fought only among themselves, every win of one country is a loss of another.
Q1. D won 16 wars each against B and C. Maximum number of wars B won against D?
D has 33 wins in total.
Given:
D vs B = 16 wins
D vs C = 16 wins
So D's wins against A:
33 - 16 - 16 = 1
A has 38 losses in total. Since 1 loss was against D, A must have lost:
38 - 1 = 37
wars against B and C together.
C has only 22 wins in total, so C can contribute at most 22 of these 37 losses.
Therefore B must have beaten A at least:
37 - 22 = 15
times.
B has 36 total wins.
To maximise B's wins against D, minimise its wins elsewhere.
So:
Maximum B wins against D = 36 - 15 = 21
✅ Answer: 21
Q2. B won an equal number of wars against A, C and D. Maximum wars D could have won against A?
B won 36 wars in total.
Equal number against three countries means:
36 ÷ 3 = 12
So B won 12 each against A, C and D.
A lost 38 wars in total.
12 of A's losses were against B.
So A lost the remaining:
38 - 12 = 26
wars against C and D.
To maximise D's wins against A, let C's wins against A be as small as possible.
Hence maximum:
D wins against A = 26
✅ Answer: 26
Q3. All wars of X were only against Y. How many possibilities for (X, Y)?
For X to have fought only Y:
Y must have enough losses to account for all of X's wins.
Y must have enough wins to account for all of X's losses.
Check A as X
A has 25 wins and 38 losses.
Y would need at least 25 losses and 38 wins.
No country has 38 wins.
❌ Not possible.
Check B as X
B has 36 wins and 26 losses.
Y must have at least 36 losses and 26 wins.
No country satisfies both.
❌ Not possible.
Check C as X
C has 22 wins and 28 losses.
Y needs at least 22 losses and 28 wins.
B: 26 losses, 36 wins ✓
D: 24 losses, 33 wins ✓
So:
(C, B)
(C, D)
are possible.
Check D as X
D has 33 wins and 24 losses.
Y needs at least 33 losses and 24 wins.
A has:
38 losses and 25 wins ✓
So:
(D, A)
is also possible.
Therefore there are:
3 possibilities
✅ Answer: C. 3
Q4. Given such a pair (X, Y), what is the maximum number of wars D won against B?
The useful cases are:
(C, B) or (C, D).
Case 1: C fought only B
C has 22 wins and 28 losses.
Therefore:
C beat B = 22
B beat C = 28
B has 26 total losses.
Out of these, 22 are against C.
So only:
26 - 22 = 4
losses remain for B against A and D.
Thus:
A wins against B + D wins against B = 4
To satisfy the remaining totals, A must account for at least 1 of these.
Hence maximum:
D wins against B = 4 - 1 = 3
Case 2: C fought only D
C's 22 wins are against D and its 28 losses are against D.
So D already has:
28 wins against C.
D has only 33 wins altogether.
Hence D has just:
33 - 28 = 5
wins left against A and B.
On satisfying the remaining totals, D can beat B at most 3 times.
Therefore:
✅ Answer: 3
Final Answers
Q1 → 21 ✅
Q2 → 26 ✅
Q3 → 3 ✅
Q4 → 3 ✅
⚡ Brain teasers are not just about getting the right answer — they train you to think differently under pressure.
Stay consistent with your daily practice, challenge your assumptions, and keep sharpening your reasoning skills.
See you in tomorrow’s challenge!