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## Cheryl's Birthday

There is a question from the Singaporean math syllabus that has been circulating the Internet, stumping adults and children alike. The question is reproduced below:

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.

 (1) Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know too. (2) Bernard: At first I didn't know when Cheryl's birthday is, but I know now. (3) Albert: Then I also know when Cheryl's birthday is.

So when is Cheryl's birthday?

We can arrive at the answer via the process of elimination. For your convenience, you can click on the dates in the question above to strike them through, thus visually eliminating them. You can try to figure out the answer for yourself, or click below to reveal the full answer and explanation.

Albert is given the month of Cheryl's birthday; i.e. May, June, July, or August. Bernard is given the day of Cheryl's birthday; i.e. 14, 15, 16, 17, 18, or 19.

Looking at Statement 1, if the day given to Bernard is 18 or 19, he would know Cheryl's birthday right away, as there is only one candidate each for those days. Albert's month thus cannot be May or June, as otherwise he cannot confidently state that Bernard doesn't know the date. We can thus eliminate all May and June dates.

Looking at Statement 2, Bernard has figured out the date via Albert's statement, using the same reasoning we did to eliminate May and June as possibilities, and arriving at only one candidate. For that to happen, his day must be unique among the remaining candidates, and therefore must be 15, 16, or 17. We can thus eliminate July 14 and August 14, leaving July 16, August 15, and August 17.

Looking at Statement 3, Albert has now also arrived at one candidate and figured out the date. This means his month is unique among the remaining candidates. Thus his month must be July, and we arrive at the only possibility remaining, July 16.

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