When is Cheryl's birthday?

[sj-roc]

You may have seen the story of this puzzle in social media and on the news:

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?

The answer is July 16. Most of the explanations I've seen are hard to follow; here's my version of it.

Suppose A were told the bday month were May. This would leave the 15, 16 and 19 as possible dates. The 15 and 16 dates appear with other months, but if B were told the bday date was 19, then B would immediately know May 19 as the bday since there are no other 19 dates in the set of ten.

That is to say, if the bday month were May, there is a non-zero chance (1 in 3 to be exact, but all that matters is that it's non-zero) that B would know the bday from the day alone. But A in his first comment says he knows there is NO chance B knows the bday from the date alone. This means the assumption of May as the bday month is FALSE.

One reaches a similar conclusion when assuming June as the bday month (because June 18 is the only one with an 18 in it), so this month is also ruled out in similar fashion.

This leaves July or August as the bday month, and the date possibilities are 14, 15, 16, 17. We are told B knows which of these it is.

Let's suppose B were told the bday date is 14. But B says in his comment that he now knows from A's first comment the full bday date. But the 14 possibility has two different months attached to it: July and August. This means 14 cannot be the date, which leaves 15, 16 and 17 as the bday date candidates, each of which have only one matching month and so all three remain as candidates; the three remaining possibilities are July 16, August 15 and August 17. Since B knows (was told) which of 15, 16, 17 is the bday date, he now knows the bday in full, consistent with his comment.

Now let's suppose the bday month is August and consider A's second comment, in which says he has figured out the bday in full on the basis of B's remark. But there are two possibilities for August so this conclusion would be wrong. Therefore the bday cannot be in August.

This leaves the last remaining possibility of July 16, which is the solution.

I wonder if the problems on the Wonderlic ever get this challenging?

[TheLionKing]

Whaaaaat ?

[sj-roc]

Whaaaaat ?

That puzzle's been all over the news and social media the last 24 hours, all sorts of discussion about it on facebook and such places. It recently appeared as a problem posed in a mathematics competition in Singapore for elite high school students and was never intended to be particularly straightforward but rather a fairly challenging brain teaser. The funny thing is, it isn't really a math problem per se, but actually one of logic. The three comments between A and B have to be carefully considered in turn, in order to successively rule out various candidates from the given set of ten until only one possibility — the correct one — remains.

[sj-roc]

Remember how the Hamilton Tiger-Cats' Pete Dyakowski won CBC’s Canada’s Smartest Person contest in 2012? There's a 4-min video of him solving this problem (correctly) on Maclean's website. It's basically the same solution I gave:

http://www.macleans.ca/multimedia/video ... cation=ufi

Yeah, that problem should go on the Wonderlic.

EDIT: additional comment from Yahoo's CFL blogger Andrew Bucholtz:

https://ca.sports.yahoo.com/blogs/cfl-5 ... 29549.html

Hamilton Tiger-Cats' offensive lineman Peter Dyakowski is known for his brawn and toughness on the field, but also the brains he's shown off it. He won CBC's Canada's Smartest Person show in 2012, appeared on Jeopardy! last year and has even been serving as Maclean's "resident genius," taking their brutally-tough current-events quiz weekly and doing very well. It's an extension of that latter role where he showed off his smarts again this week, using a whiteboard at the Ticats' facility to diagram the solution to a viral logic puzzle about Cheryl's birthday that's been going around the internet. ...

That's not only impressive thinking by Dyakowski, but a very solid and efficient logical explanation of the solution. ... He's a regular Captain Holt, and it's good to see that he's exercising his brain as well as his muscles in the offseason. This is yet more proof that not all football players fit the "dumb jock" stereotype; there are lots of smart people out there who just happen to play football for a living. This is more evidence that Dyakowski's one of those.

[Robbie]

Another good and detailed answer:
http://www.cbc.ca/news/trending/when-is ... -1.3032029

[Toppy Vann]

Quite frankly I found how this was worded complete crap and the online video explanation complete rubbish.

Last night I read dozens of online attempts and very smart folks outlined their reasoning but the assumptions can't be found in the wording of the conversations.

This is an entirely bogus question in my view. Borders on mental *beep*.

Singapore needs to do more to educate their people in critical thinking and reasoning and a more global world view instead of seeing everything in black and white.

[sj-roc]

Quite frankly I found how this was worded complete crap and the online video explanation complete rubbish.

Last night I read dozens of online attempts and very smart folks outlined their reasoning but the assumptions can't be found in the wording of the conversations.

This is an entirely bogus question in my view. Borders on mental *beep*.

Singapore needs to do more to educate their people in critical thinking and reasoning and a more global world view instead of seeing everything in black and white.

I'll respectfully disagree on this being a "bogus" question. The only place I felt the wording was a bit clunky was where Bernard says, "At first I don't know...", which really should have been, "At first I didn't know..."

I suspect the problem was originally devised by an ESL person whose mother tongue lacks any forms of English-like past tense in its verbs, where tense is instead conveyed with use of adverbs (e.g., "I write a letter yesterday", "I write a letter today", "I write a letter tomorrow"). I understand that this is the case for languages like Mandarin; someone like Robbie would likely have a better handle on this matter. The problem was perhaps even first composed in said mother tongue. In some accounts of the problem statement, the phrase "just became" in the first line was rendered as "just become", which I think is how it was initially presented as per this scanned copy:



Every verb here appears only in what we English speakers would call its present tense form, and it's worth noting that this would pose no comprehension issues if the students to whom the problem was assigned shared the same language background. Otherwise the problem as worded presents a well-posed exercise in logic with a precise solution. I would argue in fact that the problem requires critical, logical thinking about what information is conveyed in its statement and care in how to use it to arrive at the solution. Frankly, the fact that Asian students are attacking this type of problem (presumably with some success) aligns well with the commonly encountered narrative that western students chronically lag behind them in academic performance.

I would agree that some explanations of the solution out there aren't as well presented as they could be. I found a video at the Globe & Mail wherein the part about how May & June are eliminated was so weakly explained that I wasn't even convinced the woman truly understood it herself as she was telling it. It's relatively clear why May 19 and June 18 can be eliminated right away and she got that across properly, but her explanation didn't get across well enough to my satisfaction why those months can be entirely ruled out altogether, which is a much more subtle observation.

[Toppy Vann]

SJ, I am totally convinced that people are deluding themselves as the logic in this question just doesn't add up - and I love these sorts of things.

[sj-roc]

I've thought this through again and it still sounds like a perfectly logical puzzle. Is there any one particular step in the solution that you feel is flawed?

It should be understood that this wasn't made up by some no-account kid on Imgur or 4chan who set out to troll the world. This problem was devised by a mathematics competition committee that sets problems for some of the most gifted high school students in the world, and they've been doing it for decades. This was only ONE question in a competition that contained 25 of them.

There's a vaguely Bayesian quality to the problem, in the sense that it involves a state of incomplete information, and then this information state changes (approaches completion) as each statement by Albert or Bernard is given.

It's important to understand that there are three agents involved here: Albert, Bernard, and you (the reader) trying to solve the problem, each of whom starts from a different state of incomplete information. The reader is given ten possibilities. Albert is told only the month. Bernard is told only the date. It's also important to understand that all three agents' states of knowledge evolve differently through the course of the statements by A & B, yet all three reach the same conclusion.

Considering the comments again post hoc (i.e. from the fourth perspective of an all-knowing observer who knew all along the birthday was July 16), Albert's first comment is based on knowing the birthday is in July (even though the reader doesn't yet know this). Since he knows it's July, he knows the day must be either 14 or 16, one of which was told to Bernard. This means Albert knows Bernard has four possibilities for the birthday: July 14, July 16, May 16, August 14. Hence Albert's first comment that Bernard doesn't have it fully worked out either.

Both Bernard and the reader can use this comment to rule out May or June, in the manner already explained (Albert also knows that his comment divulges this — this is important later). Bernard of course only needs to rule out May since he knows (was told) it's the 16th (which means he only has May 16 and July 16 to consider from the outset), but the reader who lacks this information is still also able to rule both May and June. At this point Bernard can say he knows when the birthday is as he is now only left with July 16. The reader meanwhile from Bernard's conclusion would be able to rule out July 14 and August 14, leaving July 16, August 15, August 17.

Albert's reaction can now be considered. He has the two possibilities of July 14, July 16. He knows that his first comment divulged only July or August as month possibilities as already mentioned, so he's aware that this comment reduced Bernard to three possibilities: July 14, July 16, August 14. Upon hearing Bernard assert that he now knows the birthday, Albert can only deduce that is must be on the 16th. Bernard could not have been this certain had he been told the 14th. This means Albert can also dismiss the 14th from his candidates and deduce July 16 as the birthday. Finally, Albert's statement that he knows the birthday allows the reader to eliminate August as the birthday month from his list of three options, which again leaves July 16 as the answer.

If you start from the assumption of any one of the other days as the actual birthday, the statements by A & B will not follow logically. Only July 16 works.

The presence of numbers in the birthday might be distracting from the fact that this is purely a logic puzzle and doesn't require any conventional mathematics, so to speak. The following problem is logically equivalent and can be solved in like manner:

Cheryl just bought her dad some clothes as a gift, but doesn't tell Albert or Bernard exactly what it is. Cheryl gives them a list of 10 possibilities.

Red Tie, Red Socks, Red Shoes

Yellow Shirt, Yellow Jacket

Blue Pants, Blue Socks

Green Pants, Green Tie, Green Shirt

Cheryl then tells Albert only the colour of the clothes and tells Bernard only the type.

Albert to Bernard: "I don’t know exactly what Cheryl bought her dad, but I also know for a fact that you don't know exactly what it is, either."

Bernard to Albert: "Oh, really? At first, I didn’t know what she bought, either, but now I do know, after what you just said."

Albert to Bernard: "Is that so? Well, now I know what she bought, too."

So exactly what did Cheryl buy for her dad?

You can apply the same logic as in the birthday version. Albert's first comment — that he is confident Bernard doesn't know exactly what the gift is — means it can't be Red or Yellow, because otherwise there is a chance Bernard was told it was either Shoes or a Jacket, in which case he would immediately know the item's full description as there is only one colour for each of these items.

So the gift has to be either Blue or Green. Bernard then, having heard this comment, says that now he can figure it out fully. That means the gift can't be Pants, as these have two colour possibilities. It has to be one of the candidates for which there is only one clothing type, which leaves Blue Socks, Green Tie, Green Shirt.

Finally Albert responds to this comment by stating that Bernard's comment allows him (Albert) to also figure it out. But there are two possibilities for the gift had he been told it was Green. So it can't be Green or he couldn't have asserted that was able to figure it out. This leaves Blue Socks as the only possibility consistent with all the clues in the puzzle.

[Robbie]

It looks like others have argued that Cheryl's birthday can be on August 17 if Bernard made a statement before Albert's first official statement that made Albert to make his first statement about how Bernard doesn't know either. Bernard's unofficial statement before the conversation was that he did not know Cheryl's birthday is. If that's the case, May 19 and June 18 are eliminated.

Albert then makes his first statement about how he doesn't know Cheryl's birthday and reinforces Bernard's unofficial statement, so June 17 is eliminated as well as that was the remaining day in June and Albert did not receive June from Cheryl. So that leaves:

May 15, 16
July 14, 16,
August 14, 15, 17

Bernard then makes his first official statement about how now he knows Cheryl's birthday. Looking at the remaining list, 14, 15, and 16 are appears more than once in the months, whereas 17 only appears in August. So for Bernard to say that he knows, then he must have received 17 and knows that it's August 17.

Albert confirms that he knows as well now that Bernard must have received 17 so Cheryl's birthday is August 17.

I suspect the problem was originally devised by an ESL person whose mother tongue lacks any forms of English-like past tense in its verbs, where tense is instead conveyed with use of adverbs (e.g., "I write a letter yesterday", "I write a letter today", "I write a letter tomorrow"). I understand that this is the case for languages like Mandarin; someone like Robbie would likely have a better handle on this matter. The problem was perhaps even first composed in said mother tongue. In some accounts of the problem statement, the phrase "just became" in the first line was rendered as "just become"

I suppose a typical confusing issue for SE Asians and ESL learners is when it comes to tag questions and statements as most Asian languages respond to negation questions in a different way, using two good examples:

FACT: I am not an ethnic Korean.
Question asked to me: Aren't you Korean?
Correct English answer: No (meaning No, I'm not Korean. I'm Chinese.)
Asian language-formatted answer: Yes (meaning Yes, you are right in that I'm not Korean.)

FACT: My friend and I saw a movie together and neither of us liked that movie.
Statement by my friend: That was not a good movie.
Correct English response: No (meaning No, that was not a good movie.)
Asian language-formatted response: Yes (meaning Yes, I agree with you in that it was not a good movie.)

It's understandable how ESL find it confusing that inverse questions like Are you Korean? and Are you not Korean? yields the same answer in English.

------------------------------------

These kinds of word problems and puzzles are typical in technical interviews given to candidates to assess their creative and logical approach. Microsoft has a whole bunch of these puzzles and I wonder why this one stood out as viral more than others. I suppose it's because it sounds original.

That said....how about a more traditional one. See if you can solve this without trying to look it up.

A woman named Jane is walking around a movie theatre when she suddenly points to a lady in a movie poster and excitedly exclaims:

I do not have any siblings, but that woman's mommy is my mommy's daughter.

Who is the woman in the movie poster that Jane pointed to?

[sj-roc]

It looks like others have argued that Cheryl's birthday can be on August 17 if Bernard made a statement before Albert's first official statement that made Albert to make his first statement about how Bernard doesn't know either. Bernard's unofficial statement before the conversation was that he did not know Cheryl's birthday is. If that's the case, May 19 and June 18 are eliminated.

Albert then makes his first statement about how he doesn't know Cheryl's birthday and reinforces Bernard's unofficial statement, so June 17 is eliminated as well as that was the remaining day in June and Albert did not receive June from Cheryl. So that leaves:

May 15, 16
July 14, 16,
August 14, 15, 17

Bernard then makes his first official statement about how now he knows Cheryl's birthday. Looking at the remaining list, 14, 15, and 16 are appears more than once in the months, whereas 17 only appears in August. So for Bernard to say that he knows, then he must have received 17 and knows that it's August 17.

Albert confirms that he knows as well now that Bernard must have received 17 so Cheryl's birthday is August 17.

That's interesting, but it's a different problem what with the extra statement at the start. One I haven't considered, so I won't comment at this point.

I suspect the problem was originally devised by an ESL person whose mother tongue lacks any forms of English-like past tense in its verbs, where tense is instead conveyed with use of adverbs (e.g., "I write a letter yesterday", "I write a letter today", "I write a letter tomorrow"). I understand that this is the case for languages like Mandarin; someone like Robbie would likely have a better handle on this matter. The problem was perhaps even first composed in said mother tongue. In some accounts of the problem statement, the phrase "just became" in the first line was rendered as "just become"

I suppose a typical confusing issue for SE Asians and ESL learners is when it comes to tag questions and statements as most Asian languages respond to negation questions in a different way, using two good examples:

FACT: I am not an ethnic Korean.
Question asked to me: Aren't you Korean?
Correct English answer: No (meaning No, I'm not Korean. I'm Chinese.)
Asian language-formatted answer: Yes (meaning Yes, you are right in that I'm not Korean.)

FACT: My friend and I saw a movie together and neither of us liked that movie.
Statement by my friend: That was not a good movie.
Correct English response: No (meaning No, that was not a good movie.)
Asian language-formatted response: Yes (meaning Yes, I agree with you in that it was not a good movie.)

It's understandable how ESL find it confusing that inverse questions like Are you Korean? and Are you not Korean? yields the same answer in English.

I have indeed noticed this language pattern from some ESL Asians in their English! One could argue pedantically that the Asian rendering is in fact more correct, but the wide consensus on the conventions of the English language dictate otherwise.

These kinds of word problems and puzzles are typical in technical interviews given to candidates to assess their creative and logical approach. Microsoft has a whole bunch of these puzzles and I wonder why this one stood out as viral more than others. I suppose it's because it sounds original.

That said....how about a more traditional one. See if you can solve this without trying to look it up.

A woman named Jane is walking around a movie theatre when she suddenly points to a lady in a movie poster and excitedly exclaims:

I do not have any siblings, but that woman's mommy is my mommy's daughter.

Who is the woman in the movie poster that Jane pointed to?

Hmmm... sounds like the woman on the poster is Jane's daughter. Let's call Jane's daughter Anne and Jane's mother Betty. Then:

(that woman's mommy) is (my mommy's daughter) = (Anne's mother) is (Betty's daughter), where (Anne's mother) is Jane herself. Considering (Betty's daughter) by itself, this can only be Jane since she self-proclaims having no siblings, so everything is consistent.

[Toppy Vann]

What is happening is that people are working this problem backwards from the solutions and the explanations I have read over and over are illogical.

What they say means it's this date is nuts.

This comes from Singapore where their primary language is not English so you have to cut some slack but only to a point.

Others keep coming to an alternate date.

To trick students with poorly worded questions is shameful. This type of wording in business would get you fired.

[sj-roc]

What is happening is that people are working this problem backwards from the solutions and the explanations I have read over and over are illogical.

What they say means it's this date is nuts.

This comes from Singapore where their primary language is not English so you have to cut some slack but only to a point.

Others keep coming to an alternate date.

To trick students with poorly worded questions is shameful. This type of wording in business would get you fired.

Some people might be working it backwards (I'm not even 100% sure what you mean by this) but that doesn't invalidate the correct solution that has widely circulated. As for the alternate date, that's coming from a different formulation of the problem — a different problem, really — from what was given.

Criticising the language is a bit of a red herring IMHO. I would think the students to whom the problem was assigned probably have the same Asian-formatting to their English (which Robbie described) as the person/people who set it. Besides, for all we know, the competition in which this problem appeared may have originally been conducted in their own language anyway, with this English version only circulating in English-language media after the fact, which would make the whole language issue moot. I don't think any participants in the competition were "tricked". And I'd be willing to bet a good handful of them even managed to figure it out, too.

At any rate, there are plenty of highly talented mathematicians in North America and elsewhere who speak English as their mother tongue. If the syntax of the problem wording had created any issues, or if the problem itself, worded even in precise Queen's English, were somehow flawed and ill-posed so as to have no true, valid solution, I think we would have heard about it by now from at least one of them.

There's another aspect of this problem that seems to have gotten lost in the acknowledgement of its difficulty level. And that is, as difficult as solving this problem might be, creating a problem like this is an even more difficult challenge.

To explain what I mean, forget for a moment you ever heard the original problem that started this thread and consider instead this different but related one:

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

Her birthday is actually July 16. She then tells Albert and Bernard separately the month and the day of her birthday respectively.

Is it possible for Albert and Bernard to determine together when Cheryl's birthday is, without disclosing to each other in any way the partial information about it that they have been given?

This IMHO is a MUCH more difficult, open-ended problem that the original one. The answer to it is yes, and it would take the three statements given in the original one for each of them to arrive at the correct answer (as per the solution of the original problem) without either of them directly disclosing what that already know.

The person who CREATED the original problem essentially solved this far more difficult one.

[Toppy Vann]

I have yet to see a decent mathematician / educator give this question any kudos.

And I'm in Asia where the English is just crap that they feed these students or at such a high British level not a lot of Canadians would pass.

[sj-roc]

I have yet to see a decent mathematician / educator give this question any kudos.

That's not something they typically do, though. But if the problem had some fundamental flaw with it, they would have been all over it.

It's like long snappers in football. No one ever lauds them for getting the job done all season long, but then they wear the goat horns if they suddenly sail the ball five feet over the punter's head in the last two minutes of a playoff game.