When is Cheryl's Birthday?

(Reblogged from Mind Your Decisions).

This problem went viral after it appeared on a Math Olympiad test for talented students in Singapore. After it was posted to Facebook, the problem went viral generating debate about the correct answer.

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 the month and Bernard 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? Think carefully about it and if you cannot work out what the solution is, click on the "Read More »" link below to watch a video and read about the logic behind the answer.

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.

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