Wednesday, 15 April 2015

Cheryl must be a politician as she can't answer a straight question

There's a little logic puzzle that's been doing the rounds on the internet for the past few days, that probably everyone has already seen by now.  Seemingly it's come from some Singaporean schoolchildren's homework.  It is a rather fiendish puzzle and seems to having been given a lot of adults difficulty.

Here is the question:

"Albert and Bernard 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 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 does not 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?"
If you want to have a go at figuring this out for yourself and don't want to know the solution stop reading now.

There are loads of solutions to this problem posted all over the internet and in various news outlets now, but having seen a few of them and worked through the problem myself, none of the explanations I've seen seem particularly clear or well explained - the best one I've seen so far has been on the NY Times website.

Here is my attempt to explain the solution:

After giving her list of ten possible dates, Cheryl tells Albert the month and Bernard the day of her birthday.

Albert and Bernard both have more information than we have.  The key to figuring this out is to keep this in mind and to pay very close attention to the statements made by Albert and Bernard and the information that they (inadvertently) reveal.

Albert states that he doesn't know the exact date of Cheryl's birthday, but he knows that Bernard doesn't know it either.  However, Bernard could have known the exact date if it had been May 19 or June 18, as in our range of dates these are the sole occurrences of these specific days (18 & 19), therefore if it had been either of these dates Bernard would have known the exact date right away.  The fact that Albert is so confident that Bernard doesn't know the exact date of Cheryl’s birthday reveals (to us and to Bernard) that Albert knows the month isn't May or June (and therefore must be either July or August).

This leaves us with the following possibilities:
July 14, July 16
August 14, August 15, August 17

Bernard then declares that at first he didn't know, but now (following Albert’s first statement) he’s figured it out.  So, Cheryl’s birthday can’t be on the 14th of the month, otherwise Bernard wouldn't have been able to narrow down whether it was in July or August.

This leaves 3 possibilities:
July 16
August 15, August 17

Albert then claims to have figured it out as well, which means that it can’t be either of the dates in August, otherwise he wouldn’t have been able to narrow down the day.  This leaves July 16 as the only possible date of Cheryl’s birthday.

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