Cheryl's Birthday Problem - It depends on your point of view

Cheryl's Birthday problem was a question in a recent Singapore Maths Olympiad. Here is the problem:

You have Albert, Bernard and Cheryl. Cheryl says "my birthday is one of these ten dates"

May 15 May 16 May 19
June 17 June 18
July 14 July 16
August 14 August 15 August 17

She gives Albert the month of her birthday. And Bernard the number.

Then the following conversation occurs:

1) Albert: I don’t know when the birthday is, but I know Bernard doesn’t know too.
2) Bernard: At first I don’t know when the birthday is, but now I know.
3) Albert: Then I know the birthday too.

From that information, work out Cheryl's birthday.

[The English is the original, slightly dodgy, English of the original question].

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Something from the article I wrote, which I forgot to say in the video, the alternative solution completely changes the nature of the problem, because the alternative answer can be worked out from the first two statements only. The expected answer does need all three statements.

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A bit more about that third answer. I think the reader is treating it as just a mystery date (made up of a month and number) which they have to work out. This fails to take into account that Albert has the month only, and Bernard the number only.

Oh, another thing I forgot to say: the third answer is also what you get if you mix up the roles of Albert and Bernard - in which case the first statement is information about the date.

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More about the idea of "knowledge":
Let p be a statement, it's either true or false. Logic is the maths of these statements. We can modify them, for example we can take not(p) which has the opposite truth value. We can add and multiply statements as well using "or" and "and". Another action is knowledge(p) which might be true or false, but is not the same as p, and the following deductions are different.

So if we remove the story it becomes the difference between reading the first statement as "Bernard doesn't know" and reading it as "Albert knows Bernard doesn't know". And what you can then deduce from that.

For definitions and properties of knowledge in logic, here is the wikipedia page

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The other point I forgot to make, if this was a serious question in mathematical logic it would be written symbolically in formal logic - and there would be no ambiguity. However, this is not meant to be high level maths, it's a puzzle. So by making it accessible, and dressing it up as a puzzle, ambiguity was accidentally introduced. I sympathise with that! Writing puzzles is hard.

Daniel Gjörwell in the comment made a very good point: "The intention of the author was KNOWS as REALIZE. If the first statement would have read: "I don't know but I realize that Bernard doesn't know too"; then I would have got [the correct answer]"

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If you are reading this you are in a special group of highly intelligent people who read the description in the video. Well done you. Your mission, if you choose to accept it, is to work a fish related pun into a comment without other people noticing. Good luck!

[WilliamDP]

Thank you for this explanation. I kept coming up with the 'August 17' answer, and I found it reassuring that you felt that was a valid answer. But after reading the Wikipedia article on this problem, I no longer agree with you that August 17 is a valid answer. The only way August 17 is a valid answer is if Bernard said at the beginning, "I do not know when Cheryl's birthday is", or if Cheryl said "Bernard does not know when my birthday is". I now see that when Albert said, "I don't know when Cheryl's birthday is, but I know Bernard doesn't know too", that meant that Albert knew it was impossible for Bernard to know Cheryl's birthday. Albert did not know what date Bernard was told, and he was not told by anyone that Bernard did not know. Albert knew that the only way Bernard could know the date was that if he were told the date was the 18th or 19th. The only way Albert could know that Bernard was not told the date was the 18th or 19th was that Albert was either told the month was July or the month was August.

[Sdawkminn]

I got June 17th when I did it. My argument for that and my argument against the question not being for the reader is that Albert and Bernard don't seem to be talking to each other. They sound like they are talking to the reader. Albert says, "I know Bernard doesn't know." Who is he talking to if not the reader?

[Dombowerphoto]

Blackboard? That looks distinctively green!

[datenegassie]

Why didn't Albert and Bernard just tell each other the month and number? It would be way easier than each thinking individually.

[pegy]

Maybe Cheryl is just fishing for gifts--if she makes them work out her birthday, they're not likely to forget it.

[mismag]

The more I watch this the more sense you make. I'm a little slow. Great explanation!

[Bismuth]

If the solution was August 17, the conversation could have gone a very interesting way:

B: I don't know.
A: I don't know either.
B: Really? Well now I know!
A: Oh, then I know too!

[Paul_Kielty]

James, I really enjoy these videos, but I was just wondering if you still plan on uploading to this channel. I understand you may be busy, or numberphile videos are priority or something like that, but I hope this channel stays alive! Even as more of a puzzle channel or whatever you like, I do really enjoy both!
Sincerely,
A 16 year old fan from Ireland 😊

[mjoto]

Dr. Grimes. Do you have an apprentice? Please get one. The world needs more people like you. Strong math. Likability. affable. colloquial. overall you get the point across minus negligible-important details, added humor and intrigue. Do not leave this world without making sure there is someone to take your place. Thank you, sir.

[TristanBomber]

Coddamn, I feel like the last answer is a bit fishy.

[Tempo_Topos]

Great explanation of a great puzzle! I just discovered your channel, it looks like it's packed to the gills with cool math vids.

[KiddlesP]

It just seems like everyone is fishing for a way to make them right, or make it so that the problem seems too ambiguous.

[Pferdekopfnebel]

I couldn't wrap my mind around how to interpret the first statement in order to get the
alternative solution, August 17. I finally did, so I thought I'd share my explanation of it
(despite James having explained it - perhaps a different explanation helps someone
else as well).

1. The intended "logical path".
We assume that Albert looks on his imaginary card that tells him what month the
birthday is in. He then concludes from this knowledge that Bernard cannot know the
date. He then goes on to tell us/Bernard about his knowledge.
From this we/Bernard can conclude that it cannot be May or June. On Albert's card 
must be July or August.

2. The alternative "logical path".
Albert has his imaginary card with the month on it but got an extra card from Cheryl[1]
that says: "Bernard doesn't know my birthday." From this Albert/we can conclude 
that Cheryl's birthday cannot be on May 19 or June 18 (since 19 and 18 are unique 
numbers in the date list). However Albert could still have May or June (or July or
August) on his card (as of yet). As he goes on to tell us/Bernard about the additional
card and that he still doesn't know the birthday,  we can exclude June 17 as well.
[1] Alternatively: Bernard stays silent and so Albert concludes that Bernard doesn't 
know the answer.

[Epicaq]

I'm too dumb for this.

I'll go back to looking at the wall or something.

[TheHarryBlock]

The reading of Albert's statement "I don't know, but I know Bernhard doesn't know, too" as "I know without reasoning Bernhard doesn't know. But even given that fact, I can't deduce it", leading to Bernhard's: "Aha, then it can't be in June. Now I know", giving Albert the clue "Aha, than it's the only remaining unique day" leads correctly to Aug 17, but is a far stretch from the implied reading in the original wording. I would never have considered that 'creative' reading when I solved the problem with July 16. but it's interesting and the logic following from it flawlessly yielding Aug 17. Thank you, great video.

[dom_h]

This is very similar to the sum product riddle, where one person is given the sum, the other is given the number, and you need to calculate the two numbers based on the conversation.

[andriyfadyeyenko]

I got August 17 when I solved it so I'm glad that you let minnow that my reasoning wasn't faulty.

[papa]

As always, brilliant, and an excellent discussion of the alternate interpretations of the statements.

[EPICPACKOPENINGSXD]

So the different interpretations here are:
1. Albert was alerted that Bernard doesn't know Cheryl's birthday. This means the dates June 18 and August 19 are eliminated because 18 and 19 are unique numbers in the dates given.
2. Albert knows Bernard can't possibly know the birthday. This means the month can't be May or June, because if it was there would be a possibility that Bernard was given the number 18 or 19, which means Bernard could possibly know Cheryl's birthday and then Albert wouldn't be able to know for sure that Bernard doesn't know Cheryl's birthday.

[sereda]

I got August 17 too, together with a lecturer at my university! We assumed Albert was told that Bernard doesn't know. Not that he deduced it. "But I deduced that Bernard doesn't know" would have been a much better choice if phrasing! :P