Question

A, B, C, D, E and F are cousins. No two cousins are of the same age, but all have birthdays on the same date. The youngest is 17 years old and the oldest, E, is 22 years old. F is somewhere between B and D in age. A is older than B; C is older than D. Which of the following is not possible?

• A. D is 20 years old
• B. F is 18 years old
• C. F is 19 years old
• D. F is 20 years old

Hint:

For such logic puzzles, create an age table or number line and test each possibility while checking all conditions. Eliminate duplicates and contradictions step by step.

Answer 1

The Correct Option is C

Given: - Ages are all distinct, from 17 (youngest) to 22 (oldest).
- E is 22 years old (oldest), so age set = \{17, 18, 19, 20, 21, 22\} 
- Youngest is 17 years old. - A $>$ B - C $>$ D 
- F is between B and D in age (i.e., either: B<F<D or D<F<B) 
Let’s test all options one by one: 
Option (1): D $=$ 20 Let D $=$ 20. Then C $>$ D ⇒ C $>$ 20. So C ∈ \{21, 22\} (Possible). B $<$ A ⇒ Assign B $<$ A (Possible) F between B and D (Possible for some value) → This option is valid. 
Option (2): F = 18 Now try to place B and D such that F is between them: If F = 18, then possible pairs (B, D): (17, 20), (17, 19), etc. A $>$ B, C $>$ D — assign accordingly. → This is possible. 
Option (3): F = 19 Check if F = 19 can be between B and D: F must be strictly between B and D. So B and D must be 17 and 21 or 18 and 20, etc. Let’s suppose B = 18, D = 20 ⇒ F = 19 works Now check all: - A $>$ B $=$ 18 ⇒ A $>$ 18 - C $>$ D $=$ 20 ⇒ C $>$ 20 → C $=$ 21 or 22 - E $=$ 22 ⇒ C $=$ 21 → All values: A = 19 or 20 or 21, C = 21, D = 20, B = 18, F = 19, E = 22 But now F and A both become 19 ⇒ violates “all ages are different.” Try A $=$ 20, C $=$ 21, B $=$ 18, D $=$ 17, F $=$ 19 → F still shares age with A or someone else → Every configuration gives duplicate ages ⇒ Not possible. 
Option (4): F = 20 Then B and D could be (18, 22), (19, 21), etc. Try B = 18, D $=$ 22 ⇒ F $=$ 20 is valid A $>$ B $=$ 18 ⇒ A $=$ 19 C $>$ D $=$ 22 ⇒ Not possible, because E $=$ 22 is the oldest So try D $=$ 19, B $=$ 18 ⇒ F $=$ 20 is between A $>$ B $=$ 18 ⇒ A $=$ 21 C $>$ D $=$ 19 ⇒ C $=$ 22 Values: A $=$ 21, B $=$ 18, C $=$ 22, D $=$ 19, E $=$ 22, F $=$ 20 → But E 4$=$ 22 and C $=$ 22 — same age ⇒ Not allowed Try C $=$ 21, E $=$ 22 ⇒ Possible Thus, Option (4) is valid. 
Therefore, Option (3) is not possible.