# Cheryl’s Birthday

Here is a more clearly worded version of a logic puzzle that went viral in April 2015. The puzzle was given to high school math students in Singapore.

Albert and Bernard have just met Cheryl, and they ask her when her birthday is. Cheryl does not reveal the date of her birthday. Instead, she gives the two a list of ten different dates, saying that one of these dates is her birthday.

• May 15, May 16, May 19
• June 17, June 18
• July 14, July 16
• August 14, August 15, August 17

Then Cheryl whispers to Albert the month, and only the month, of her birthday. Separately, she whispers to Bernard the day, and only the day, of her birthday.

• Albert says, “I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know either.”
• Bernard says, “I didn’t know at first, but now I do.”
• Albert says, “Then I know now too.”

When is Cheryl’s birthday?

See Answer
Albert heard only the month of Cheryl’s birthday, but from this he knows that Bernard doesn’t know. Bernard, who was only told the day, could only know Cheryl’s birthday if he was told the 18th or the 19th, since those are the days that only appear once. So Albert must know that Bernard was not told these days, which he would only know if Albert was told a month other than May or June. So Albert must have been told either July or August, and Bernard can deduce this. After Bernard hears Albert’s first statement, he says he knows Cheryl’s birthday. Therefore he must have been told a day that is on the list for either July or August, but not the other. That eliminates July 14 and August 14, leaving only July 16, August 15, and August 17. If Albert were told the month of August, he would not now be able to say that he knows Cheryl’s birthday. So he must have been told July, making Cheryl’s birthday July 16.

## 2 thoughts on “Cheryl’s Birthday”

1. “which he would only know if Albert was told a month other than May or June. ”
I admit I can’t see why this is so. Can you expand on th elogic?

1. Sure! If Albert was told May or June, then he would not be able to make the statement, “I know that Bernard doesn’t know either,” because those months each include possibilities that allow for Bernard to know Cheryl’s birthday. For instance, if Albert was told May, he would look at the May options and see May 19. He would think, “If Cheryl whispered ’19’ to Bernard, then he would know her birthday is May 19, because there are no other 19s.” June also has a unique date as a possibility: June 18. But all the dates in July and August are not unique, so only with those months could Albert confidently state, while knowing only the month, “I know that Bernard does not know either.”