Cheryl's Birthday

Classic Problem
Albert and Bernard have just become friends with Cheryl.

They both want to know when her birthday is.

Cheryl tells both of them that it is one of the following $10$ possible dates:


 * $15$ May, $16$ May, $19$ May


 * $17$ June, $18$ June


 * $14$ July, $16$ July


 * $14$ August, $15$ August, $17$ August.

Cheryl then separately and privately tells Albert which month she was born in, and Bernard which day of the month she was born on.

Albert and Bernard then compare notes, as follows:


 * Albert: I don't know when Cheryl's birthday is, but I do know that you don't know either, Bernard.


 * Bernard: You're right, I didn't know at first, but now you've told me that, actually now I do know when Cheryl's birthday is.


 * Albert: And now you've told me that, now I know when Cheryl's birthday is as well.

When is Cheryl's birthday?

Solution
Cheryl's birthday is July $16$th.

Proof
Albert's first statement is that he knows that Bernard cannot know what Cheryl's birthday is.

That means he knows that Bernard has not been told either $18$ or $19$.

Albert knows this because he has been told that the month is neither May nor June.

(If he had been told either May or June, then Albert cannot be certain that Bernard had not been told $18$ or $19$, and so in that case he cannot know that Bernard did not know the birthday.)

This means that Albert has been told either July or August.

Bernard has just been told that Albert knows he does not know the birthday.

Bernard uses the above reasoning to work out that Albert must have been told July or August.

Bernard then announces to Albert that now he does know when Cheryl's birthday is.

This means that he has been told one of the dates which belongs either to July or August, but not both.

That is, Bernard has not been told $14$.

So he must have been told $15$, $16$ or $17$.

Hence Cheryl's birthday is on the $16$th of July, or the $15$th or $17$th of August.

Albert, having heard Bernard announce when the birthday is, deduces the above.

If he had been told August, then he still would not know whether the date was $15$ or $17$.

But he announces that he does know.

So he must have been told July.

Hence the result.

Wrong answer
There is a popular wrong answer of $17$ August, which is a result of the following reasoning based on an inaccurate interpretation of the premises.

To recap, this is the problem: Albert is given the month. Bernard is given the number. We are then given the following conversation:


 * Albert: I don't know when the birthday is, but I know Bernard doesn't know either.


 * Bernard: At first I didn't know when the birthday is, but now I know.


 * Albert: Then I know the birthday too.

The reasoning is as follows:


 * $(1): \quad$ Albert knows that Bernard doesn't know. (Maybe Cheryl told him this).


 * $(2): \quad$ Albert deduces Bernard can’t have a unique date such as $18$ or $19$.


 * $(3): \quad$ Albert smugly taunts Bernard, announcing Bernard doesn't know. This is the first statement of the problem.


 * $(4): \quad$ Bernard realises what Albert has realised, which is that Bernard does not have $18$ or $19$. Now if Albert was holding June he would know the answer, because there is only one remaining date in June, namely June $17$. So Bernard deduces it is not June.


 * $(5): \quad$ Bernard announces he knows the answer. This is the second statement of the problem.


 * $(6): \quad$ If Bernard is so confident, he must have a unique date. We know it is not $18$ or $19$. What other unique date can it be? There are two $14$s, two $15$s, two $16$s and two $17$s -- but Bernard has eliminated June $17$ - leaving him with August $17$ only. That's how he worked it out.


 * $(7): \quad$ Albert is furious Bernard beat him to the answer. Albert puts himself in Bernard's shoes, running through the $6$ steps above. Finally Albert reaches the same conclusion we have, Bernard must have $17$. Albert announces he knows the answer too.

So August $17$ is a valid answer.

Explanation
Bernard's reasoning that only June can be eliminated is based on the fact that he assumes that Albert has found out independently (for example, by Cheryl telling him) that Bernard does not know the birthday.

Hence he has not ruled out May.

May is ruled out only if Albert has worked out that Bernard cannot know the birthday purely based on the fact of knowing the date.

If he had been told that Bernard did not know the birthday, then he already knows that Bernard has not been told $19$.

Suppose Albert had been told May.

If Albert did not know independently of the given information that Bernard does not know the birthday, then he could not know that Bernard had not been told $19$.

Hence he could not say I know Bernard doesn't know either.

If Albert did not know independently of the given information that Bernard does not know the birthday, then that means it would be possible for Bernard to have been told $15$ or $16$.