THE CORRECT ANSWER OF THIS PUZZLE IS:

JULY 16

Explanation:

Let’s go through it line by line.

Rahul: I don’t know when Mehak’s birthday is, but I know that Sachin doesn’t know too.

All Rahul knows is the month, and every month has more than one possible date, so of course he doesn’t know when her birthday is. The first part of the sentence is redundant.

The only way that Sachin could know the date with a single number, however, would be if Mehak had told him 18 or 19, since of the ten date options only these numbers appear once, as May 19 and June 18.

For Rahul to know that Sachin does not know, Rahul must therefore have been told July or August, since this rules out Sachin being told 18 or 19.

Line 2) Sachin: At first I did not know when Mehak’s birthday is, but now I know.

Sachin has deduced that Rahul has either August or July. If he knows the full date, he must have been told 15, 16 or 17, since if he had been told 14 he would be none the wiser about whether the month was August or July. Each of 15, 16 and 17 only refers to one specific month, but 14 could be either month.

Line 3) Rahul: Then I also know when Mehak’s birthday is.

Rahul has therefore deduced that the possible dates are July 16, Aug 15 and Aug 17. For him to now know, he must have been told July. Since if he had been told August, he would not know which date for certain is the birthday.

The answer, therefore is July 16.