This problem appeared on a Math Olympiad question for high school students in Singapore. After it was posted to Facebook, the problem went viral generating debate about the correct answer.
Video Version
I’ve posted the problem and explained the answer on my YouTube channel.
When is Cheryl’s Birthday? Answer To Viral Math Puzzle
Alternately, you can read the problem and its solution below.
Text Version
Here’s the problem, which I have very slightly re-worded for clarity.
Albert and Bernard have just become friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.
May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17
Cheryl tells Albert only the month and Bernard only the day.
Albert says, “I don’t know when Cheryl’s birthday is, but I know that Bernard cannot know either.”
Bernard then says, “At first I didn’t know when Cheryl’s birthday is, but now I do know.”
Albert concludes, “Now I know when Cheryl’s birthday is.”
So when is Cheryl’s birthday?
It seems impossible to know, but you can figure it out from the information given! Read on below for the answer. "All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.
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Answer To Cheryl’s Birthday
Take a look at the dates again.
May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17
Albert is told one of the months May, June, July, or August. And Bernard is told one of the days 14, 15, 16, 17, 18, or 19.
Albert first says, “I don’t know when Cheryl’s birthday is, but I know that Bernard cannot know either.”
The first part of the sentence is meaningless–of course Albert doesn’t know the birthday. But the second part is a clue.
Albert is sure that Bernard cannot know the date. If Bernard was told 19, he would know the birthday was May 19. If Bernard was told 18, he would know the birthday was June 18. How is Albert sure that Bernard didn’t hear 18 or 19? It must be because Albert knows the birthday is not in May or June! In other words, Albert was told either July or August.
So the first sentence reduces our list to 5 dates.
July 14, July 16
August 14, August 15, August 17
Bernard then says, “At first I didn’t know when Cheryl’s birthday is, but now I do know.”
Upon hearing Albert’s statement, Bernard figured out the month must be July or August. He then also said now he could figure out the birthday. If he had been told 14, the month would still be ambiguous. So he must have been told 15, 16, or 17.
So Albert is left with 3 possible dates.
July 16
August 15, August 17
Albert finally explains, “Now I know when Cheryl’s birthday is.”
If Albert was told August, he still would not be sure. So he must have been told July.
Therefore, Cheryl said July to Albert and 16 to Bernard. Cheryl’s birthday is July 16.
This reasoning is valid and it’s the only date that makes sense. This has been confirmed in the official answer key.
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