Birthday Puzzle

In case you missed it, the following is supposedly a math question posed to 14 year-old students in the Singapore and Asian Schools Math Olympiads.

This test is less about math and more about logic. If you want to take a minute to solve it, go ahead.

It’s initially confusing but turns out to be simpler than it looks. Here is the solution.

Albert says: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.

According to the test, Albert is told the month – May, June, July, or August, and Bernard is told the day – 14, 15, 16, 17, 18, or 19. If Cheryl’s birthday is on the 19th, then Bernard would know her birthday is May 19, as that is the only 19 in the list. Likewise, if her birthday is on the 18th, then Bernard would know her birthday is June 18. For Albert to know that Bernard doesn’t know the date, the day can’t be 18 or 19. For Albert to be certain that the day isn’t 18 or 19, Albert must have been told that the month is either July or August. That leaves 5 possible dates.

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

At first, Bernard doesn’t know the month but when he hears Albert say, “I know that Bernard does not know too” he knows Albert must have been told the month is July or August. For Bernard to know both the month and the day, the day can’t be 14, because there are two 14s remaining – in July and in August – and Bernard wouldn’t know which of the two was Cheryl’s birthday. That leaves three dates: July 16, August 15, and August 17. And because Bernard knows the day, he also knows the month.

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

Albert must have been told July, because if he had been told August he still wouldn’t know the date – there are two dates in August. There is only one date remaining in July, so he knows Cheryl’s birthday is July 16.