Answer to Puzzle #48: When is Cheryl's Birthday

48. 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 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?

In April 2015 this puzzle began to trend worldwide on Twitter under the hashtag #cherylsbirthday. We don't get much fame in this game so it seems worth our attention. There are 'solutions' all over the internet. A good portion of them are wrong. Let's take a look:

Before reading the answer can I interest you in a clue?

We are going to look at the dates and try to eliminate them until only one is left. This puzzle involves making assumptions based, in a sense, by putting yourself in someone else's position and thinking what conditions must be present for them to say the things they say. We have done this twice before in the somewhat easier man with three daughters and the much easier three men in different colored hats riddles.

It's helpful to lay the dates out in a table so we can identify what is special about different dates.

14th15th16th17th18th19th
MayMay 15May 16May 19
JuneJune 17June 18
JulyJuly 14July 16
AugustAugust 14August 15August 17

Albert who has been told the month but not the day says he does not know the full date. We take that to mean that he is not able to deduce the full date. In fact at this stage the only way he could deduce the date would be if there were a month with only one of Cheryl's possible dates in it and that were the month that he had been told. There is not.

More importantly he says he knows, that Bernard does not know the full date. Bernard could know or rather deduce the full date if, say, the day he had been given were the 18th. The 18th and 19th are the only days that appear in only one month, June and May respectively. Since Albert says definitively that Bernard can not know the full date, or rather there is no possible day that Bernard could have been given that would enable him deduce the full date, we know that the month can not be May or June. And so we can eliminate all the dates in May or June. To clarify, Albert is effectively saying that he knows Bernard can not have been given the days 18 or 19 and the only way he can know this is that the month is not May or June.

14th15th16th17th18th19th
MayMay 15May 16May 19
JuneJune 17June 18
JulyJuly 14July 16
AugustAugust 14August 15August 17

Bernard did not know the full date, now he does. It's important to look at this from Bernard's perspective. Remember unlike us, he does know the day. So Bernard is sat there looking at the same table as us, he knows what possible dates are available and if he had been told, say the the day is the 17th he would immediately see that there is only one possible date. Only if the number he had been told were 14 then would he still not know the full date. Since he does now know the full date we can eliminate July 14 and August 14.

At this stage I think a common mistake is to feel that because Bernard knows the answer we should too. We can not, there is not enough information to solve the puzzle yet. The clue is that Bernard knows. Even though I know this it still catches me, you have to look at the table and think what does Bernard declare and which days, of which he has been told only one, would enable him to do this. Anyway...

14th15th16th17th18th19th
MayMay 15May 16May 19
JuneJune 17June 18
JulyJuly 14July 16
AugustAugust 14August 15August 17

Back to Albert. Albert is keeping up with us and has reduced the table down to the dates we have. He then declares that he now knows the full date. He would not be able to do this if the month he had been told were August as there are two dates in August. We can now eliminate the August dates.

14th15th16th17th18th19th
MayMay 15May 16May 19
JuneJune 17June 18
JulyJuly 14July 16
AugustAugust 14August 15August 17

And so we are left with one. Cheryl's Birthday is July 16

Assumptions

It can not be stated enough that the clue is what the people in the puzzle say. We do not get to solve the puzzle right till the end. They have information we don't, we are not able to eliminate dates like they can, we are able to infer things from the fact that they are able.





© Nigel Coldwell 2004 -  – The questions on this site may be reproduced without further permission, I do not claim copyright over them. The answers are mine and may not be reproduced without my expressed prior consent. Please enquire using the link at the top of the page.
 


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