# Cheryl's birthday

Albert and Bernard have become friends with Cheryl and want to know what their birthday is to give her a gift.

Cheryl gives them a list of 10 possible dates:

On May 15, 16 or 19,
On June 17 or 18,
On July 14 or 16,
Or on August 14, 15 or 17.

In addition, Cheryl tells Albert the month and Bernard the day.

Subsequently, the following conversation occurs:

• Albert: I don't know when Cheryl's birthday is, but I know Bernard doesn't know either.
• Bernard: At first I didn't know when Cheryl's birthday was, but now I know.
• Albert: Well then I also know when Cheryl's birthday is.

When is Cheryl's birthday?

#### Solution

If we follow the thread of the conversation we have the following:

-Albert: I don't know when Cheryl's birthday is, but I know Bernard doesn't know either.

Albert knows the month of birth, and since every month Cheryl proposes has more than one date, he cannot know the specific day. Instead, he knows that Bernard would know the specific date in case Cheryl had told him on the 18th or 19th, since the ten numbers on the list are the only ones that only appear once, so we can discard these two dates since the only thing they both know for sure is that neither can know Cheryl's birthday without the other.

-Bernard: At first I didn't know when Cheryl's birthday was, but now I know.

After hearing this, Albert deduces that Bernard knows the month of the birth of the girl that must be August or July. If he knows the specific date, it is because Cheryl has told him 15, 16 or 17, since if he had told her on the 14th he could not know if it is August or July. Days 15, 16 or 17 refer to a single month, but the 14 is in both. Therefore, Albert has also deduced that the possible dates are July 16, August 15 and August 17.

-Albert: Well then I also know when Cheryl's birthday is

Albert now knows that Bernard must have been told on the 16th since if Cheryl had been born any of the other two days, Bernard could not have said that he knew the exact date of his birthday.

Therefore, Cheryl was born on July 16.