How did Albert and Bernard solve Cheryl's puzzle?

In the recent viral mathematics problem, readers were invited to solve the mystery of Cheryl’s birthday based on the ten possible dates and the responses of Albert and Bernard. The solution to that puzzle is readily available on the Internet.

This post addresses the other question: How did Albert and Bernard solve the mystery?

Here’s the setup. The ten possible dates are May 15, 16, 19; June 17, 18; July 14, 16; August 14, 15, 17. Cheryl tells Albert the month: July. Then she tells Bernard the date: 16.

Albert thinks: It is July, so Bernard has either 14 or 16. If Bernard has 14, Bernard will think Jul or Aug. If Bernard has 16, Bernard will think May or Jul. Hence: Bernard does not know the month.

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

Bernard thinks: It is 16, so May or Jul. Albert can say I do not know birthday only if all dates in the birth month (which Albert knows) have duplicates in other months. So Jul or Aug, since dates in only these months are all duplicated in other months. Only Jul has 16. So Jul 16.

Bernard says: At first I don’t know (see "Albert thinks" above) when Cheryl’s birthday is, but I know now.

Albert thinks: From my remark ("Albert says" above), Bernard can shortlist Jul or Aug. It is Jul, so Bernard has either 14 or 16. If Bernard has 14, Bernard will still be unsure (both Jul and Aug have 14). So Bernard has 16. Only Jul has 16. Hence, Jul 16.

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

That is how Albert and Bernard solve the puzzle of Cheryl’s birthday.

By the way, this is not a mathematics problem; it is a logic problem. And I daresay a careful Primary Five student could have a decent crack at solving it.

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