Karan remembers that his sister's birthday is not after 18th August, while his mother remembers that it is before 20th August but after 17th August. On which date in August must Karan's sister's birthday fall?

Introduction / Context:
This is a logical calendar puzzle that focuses on combining two pieces of information about a date range. Instead of computing weekdays, you must intersect two sets of possible dates given by different people's memories. Such questions test your ability to reason logically about inequalities and constraints on dates.

Given Data / Assumptions:

• Karan remembers that his sister's birthday is not after 18th August (so it is on or before 18th August).
• Karan's mother remembers that the birthday is before 20th August but after 17th August (so it is on 18th or 19th August).
• We must determine the exact date in August that satisfies both Karan's and his mother's statements.
• We assume all statements are correct and refer to the same year.

Concept / Approach:
This problem can be solved by interpreting each statement as a range of possible dates:

• "Not after 18th August" means 1st to 18th August (≤ 18).
• "Before 20th August but after 17th August" means only 18th or 19th August (17 < date < 20).

The actual birthday must satisfy both conditions simultaneously, so we find the intersection of these two sets of dates.

Step-by-Step Solution:
Step 1: Translate Karan's information: "not after 18th August" means the birthday could be on any date from 1st to 18th August, inclusive, but not on 19th, 20th, or later. Step 2: Translate the mother's information: "before 20th August but after 17th August" restricts the birthday to 18th or 19th August only. Step 3: Now find the intersection of these two sets of possible dates. Step 4: Karan's range (1st–18th) and the mother's range (18th–19th) overlap only at 18th August. Step 5: Therefore, the only date that satisfies both pieces of information is 18th August.

Verification / Alternative check:
To verify, test 18th August directly:

• Is 18th August "not after 18th August"? Yes, it is exactly 18th.
• Is 18th August "before 20th August but after 17th August"? Yes, 18 is greater than 17 and less than 20.

Now test 19th August:

• 19 is not "on or before 18", so it fails Karan's condition.

Any date before 18th (such as 17th) fails the mother's condition. Hence, 18th August is uniquely correct.

Why Other Options Are Wrong:
19: Satisfies the mother's range but does not satisfy Karan's "not after 18th" condition.
17: Satisfies Karan's condition, but it is not after 17th as required by the mother.
16: Also satisfies Karan's condition but is not after 17th August, so it contradicts the mother's statement.
20: Fails both conditions directly, as it is after 18th and not before 20th.

Common Pitfalls:
A common mistake is to misinterpret phrases like "not after", "before", and "after", confusing inclusive and exclusive boundaries. Another pitfall is failing to convert the verbal statements into clear inequality ranges and then forgetting to take the intersection of the sets. Carefully writing down the ranges for each statement and marking them on a number line can help avoid such errors.

Final Answer:
Karan's sister's birthday must fall on 18th August.