When is Cheryl’s birthday?

You may have seen the math question that appeared in a high school quiz in Singapore and went viral on the internet. It stumped many because it was more of a logical puzzle than a math problem. The answer to the conundrum lies in the conversation between Albert and Bernard.

Here is my answer and explanation for those who are still wondering about when is Cheryl’s birthday.

The List

Both Albert and Bernard have a list of months and dates that Cheryl gave them:

And then she told Albert and Bernard the month and day of her birth separately. Such that:

  • Albert knows the exact month Cheryl was born.
  • Bernard knows the exact day Cheryl was born.

The Conversation between Albert & Bernard

Albert says: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know too. (In other words I am CERTAIN that Bernard CANNOT know)

It is easy to understand that Albert doesn’t know the exact date of birth since all he knows is the month. But for him to say “Bernard doesn’t know either” means the following must be true:

  1. Cheryl must have told Albert either July or August as her birth month. As neither of these months have unique dates, it gave Albert the confidence that Bernard cannot know the date.
  2. The birth day could not have been 18th or 19th because both of these days only occur once on the list and would make it very easy for Bernard to figure out the month. (hence the statement that Bernard cannot know – would not be possible)
  3. Since the month Cheryl told Albert is not May or June, he knows that 18th or 19th could not be the birthday and this is why he knows for a fact that Bernard does not know either.

Bernard: At first I don’t know when Cheryl’s birthday is, but now I know.

Implying the same reasoning, Bernard figured out that the correct month must be either July or August. In the last part of his sentence he makes the claims that he now knows the date; which means the following must be true:

  1. Bernard knows the month is either July or August
  2. Bernard must have been told 15th, 16th, or 17th.
  3. 14th is eliminated because it appears in both July and August. And if it was the correct date then Bernard would have never figured out the correct month.

Albert: Then I also know when Cheryl’s birthday is.

The only way Albert could make such statement is if he was told the month of July as being the correct one. This is because out of the three remaining dates (15th, 16th and 17th) only the 16th belongs to a single month.

The answer:

The answer is July 16th.

Deciphering the conversation (for fun)

and… here is what would actually be going on in Albert and Bernard’s minds.

Albert: I know it’s July, and Bernard probably knows either 14th or 16th. Both of which appear multiple times on the list. So, while I don’t know whether the date is 14th or 16th, I am 100% sure Bernard wouldn’t know either.

Bernard: hmmm… I do know the day is the 16th which appears in May and July. So the birthday is either in May or July. Furthermore, the only way Albert would know that I, myself, don’t know the full date is if the months were either July or August, because unique dates appear in the months of May and June. And… since I know the day is 16th, which only appears in May and July; and since I already eliminated May from what Albert said, the month must have been July. So, I now know!

Albert: Hmm… the only way Bernard would’ve figured out it’s July, is if the day was 16th,  since 14th appears in both July and August. So, I now know that it is July 16th.

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