Cheryl's Birthday Problem (from SASMO 2015) and Solution



Hello! Nat here, and in this blog, I am going to discuss and give the solution to the famous Cheryl birthday problem. This was popular because it is a difficult logic puzzle and involving the ability of carefully tracking down hints in the conversation and working with only a small amount of data. Here’s the question:


Albert and Bernard just became 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 then tells Albert and Bernard separately the month and the day of her birthday respectively.
Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know too.
Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.
Albert: Then I also know when Cheryl’s birthday is.
So when is Cheryl’s birthday?

As I just said, this is a logic question, and as such, we can’t look at it in a very mathematical perspective; we have to use our detective skills.

The first thing to do would be to see why Albert would immediately claim that Bernard also doesn’t know when is Cheryl’s birthday. The only way that Albert would be able to claim as such is if the month he was told a month with all possible dates being non-unique dates (dates that have another date with the same number such as May 15 and August 15). From the data above, we can tell that there are 2 months that fit this description: July and August.

Since we know that the month is either July or August, the next thing to do would to narrow it down to only one of the months or to find the day. To do either of them, we have to analyze the next piece of data. When Bernard says that he knows Cheryl’s birthday, we can eliminate July 14 and August 14, as Bernard wouldn’t be able to find out Cheryl’s birthday if the day he was told was 14.

Now, we are left with only 3 possible dates. July 16, August 15, and August 17. The third and last thing to do is analyze the last piece of data, which is Albert saying that he now also knows Cheryl’s birthday. Since Albert knows the month, we can eliminate August, because if he were told that the month was August, he won’t be able to know the precise date: August 15 or August 17. 

Therefore, we are left with one possible date, the answer: July 16. 

This logic puzzle has a background as a Math Olympiad question. It was from a Math Competition called SASMO (Singaporean and Asian Schools Math Olympiad). It went viral several years ago due to its difficulty as a logic puzzle. I have joined this Olympiad myself, and it was a great experience for me. I think anyone who has interest in Math and Logic must try to compete in such a competition!

Thanks for reading! ~Nat

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