# Cute logic problem

I used to quite enjoy doing those logic problems that were popular when I was an adolescent and a couple of days ago I came across one that is similar to those but for some reason has captured the imagination of the inhabitants of the internet and has been discussed widely. Here it is:

Albert and Bernard just met Cheryl. “When’s your birthday?” Albert asked Cheryl.

Cheryl thought a second and said, “I’m not going to tell you, but I’ll give you some clues.” She wrote down a list of 10 dates:

May 15 — May 16 — May 19

June 17 — June 18

July 14 — July 16

August 14 — August 15 — August 17

“My birthday is one of these,” she said.

Then Cheryl whispered in Albert’s ear the month — and only the month — of her birthday. To Bernard, she whispered the day, and only the day.

“Can you figure it out now?” she asked Albert.

Albert: I don’t know when your birthday is, but I know Bernard doesn’t know, either.

Bernard: I didn’t know originally, but now I do.

Albert: Well, now I know, too!

When is Cheryl’s birthday?

With most logic puzzles of this type, once the answer is figured out or revealed, people tend to agree that it is correct. But apparently there has been a controversy over this one which I don’t understand since the solution seems unambiguous to me.

The real issues over which there could be some contention involve the social dynamics on display here. For example, would you like to have Cheryl as a friend? She seems to be one of those annoying people who likes to complicate even the most mundane things and it would not surprise me if Bernard and Albert decided that if this is how she responds to a simple question at a first meeting, then being friends with her would be too much work.

On the other hand, it seems a bit presumptuous on their part to ask Cheryl when her birthday is when they had just met her and she may have responded that way to signal that it was inappropriate. Or maybe I am just out of touch with contemporary conversational starters among the younger generation.

All in all, I would give all three of them a wide berth.

### Comments

1. Jazzpirate says

I see a unique solution. What’s the supposed controversy about? The link provided only says that it “stumped” the internet, which isn’t hard. Ask the internet “what is 1+1+1+1+1+1+1+1+1-1+1*0?” and they’ll be “stumped” as well. Or “does 0.9999…=1?”

2. says

When my students ask when my birthday is, I say, “Somewhere between January and December.”

Does that mean I should stop posting on your blog?

3. Trebuchet says

Seems straight forward enough. The 19th, and only the 19th, appears just once.

4. Jazzpirate says

@Trebuchet both the 18th and the 19th appear just once, and if it were one of those, the guy who only knows the day would know IMMEDIATELY, i.e. it’s the wrong answer.

5. OverlappingMagisteria says

@Trebuchet – nope the 18th appears once as well. An in either case, if the answer was the May 19th Bernard would have said “Yea I knew that as soon as she told me 19. Duh!” But he claims he didn’t know at first.

I’ve got an answer… but I’m not sure if I’m allowed to post it this soon. Don’t want spoilers!

6. OverlappingMagisteria says

I can’t help but post my answer with an explanation. In order to avoid spoiling it for others I’ve encoded it. To decode it, go to http://www.rot13.com and copy and paste the below gibberish into the box:

Gur nafjre V tbg jnf Whyl fvkgrragu. V sbhaq vg urycshy gb betnavmr gur cbffvovyvgvrf vagb n gnoyr:

Znl 15 16 19
Wha 17 18
Why 14 16
Nht 14 15 17

Abgvpr gung gurer vf bayl bar “rvtugrra” naq bar “avargrra”. Guvf zrnaf gung VS ure oveguqnl vf rvgure bs gubfr qngrf, gura Oreaneq jbhyq unir xabja nf fbba nf fur gbyq uvz gur qnl. Ohg ur qvq abg. Jvgu gung va zvaq, urer’f n jnyxguebhtu:

Nyoreg xabjf gur zbagu naq fnlf ur qbrfa’g xabj gur nafjre (abg fhecevfvat fvapr rnpu zbagu unf zhygvcyr cbffvovyvgvrf). Ubjrire, ur fnlf gung Oreaneq PNA’G xabj rvgure. Guvf zrnaf gung gur zbagu gung Nyoreg xabjf vf rvgure Whyl be Nhthfg. Jul? Orpnhfr vs vgf Znl be Whar, gura gurer vf n cbffvovyvgl gung Oreaneq jbhyq vafgnagyl xabj gur nafjre. Vs Purely unq gbyq uvz Znl be Whar, Nyoreg’f erfcbafr jbhyq unir orra “V qba’g xabj, ohg Oreaneq zvtug xabj.”

Fb jr ryvzvangr gur gbc gjb ebjf, Znl naq Whar, sebz bhe Puneg. Oreaneq, jub vf n travhf, sbyybjf gur ybtvp nobir, naq nyfb ryvzvangrf Znl naq Whar sebz uvf cbffvovyvgvrf.

Oreaneq abj nafjref, “V qvqa’g xabj ng svefg” (pbasvezvat gung vg jnfa’g 18 be 19) “ohg abj V qb.” Fb ur jnf noyr gb qrgrezvar gur shyy qngr bayl nsgre ryvzvangvat Znl naq Whar. Guvf zrnaf gung gur ahzore Purely gbyq uvz jnf ABG sbhegrra. Ubj qb jr xabj? Orpnhfr vs Purely unq fnvq sbhegrra, gura Oreaneq jbhyq fgvyy or fghpx, naq ur jbhyq unir fnvq “V qvqa’g xabj, naq V’z fgvyy fghpx orgjrra Whyl sbhegrra naq Nht sbhegrra.” Fb Purely zhfg unir gbyq uvz rvgure 15, 16 be 17.

Fb ng guvf cbvag bhe cbffvovyvgvrf ner Whyl fvkgrra, naq Nhthfg svsgrra be friragrra.

Ng guvf cbvag, Nyoreg, jub jnf rkcregyl sbyybjvat nybat jvgu gur nobir ybtvp, nyfb ryvzvangrf obgu sbhegrraf sebz uvf gnoyr bs cbffvovyvgvrf. Naq abj ur fnlf “V xabj gbb!” Guvf zrnaf gung Purely zhfg unir gbyq uvz gur zbagu bs Whyl. Ubj qb jr xabj? Orpnhfr vs fur unq gbyq uvz Nhthfg, gura Nyoreg’f ercyl jbhyq unir orra “V qba’g xabj…. V’z fgvyy fghpx orgjrra Nhthfg svsgrra naq friragrra.” Fb Purely zhfg unir gbyq uvz Whyl.

Naq gur bayl cbffvovyvgl yrsg va Whyl vf gur fvkgrragu. DRQ!

7. doublereed says

That’s a fun problem 😀

8. doublereed says

I think it just takes people a little bit to parse what “I don’t know when your birthday is, but I know Bernard doesn’t know, either” means.

9. Chiroptera says

But apparently there has been a controversy over this one which I don’t understand since the solution seems unambiguous to me.

The controversy doesn’t seem to be the answer; the controversy is that people had misread the purpose of the problem (a test question for a high level mathematics competition) and thought it was to be included as a “typical” problem for a younger grade level.

Or maybe I just out-of-touch with contemporary conversational starters.

My experience is that you get a lot of weird behaviour in any group whose initials are the first consecutive letters of the alphabet.

10. brucegee1962 says

Well explained, Overlapping Magisteria! I got the same answer, but you explained it better than I probably could have.

11. Rob Grigjanis says

Mano,

The ‘Alternate solution’ looks completely incoherent to me.

12. Who Cares says

friragu bcgvba. gung vf friragu zbagu fvkgrragu qnl
Nyoreg fnlf gung vg vfa’g gur svsgu be fvkgu zbagu be ryfr ur pbhyqa’g fgngr vg qrsvavgryl gung Oreaneq qbrfa’g xabj rvgure frrvat gung bayl gubfr gjb zbaguf unir havdhr qngrf.
Guvf yrnirf sbhe qngrf.
Oreaneq fgngrf ur xabjf nsgre urnevat gung Nyoreg rkpyhqrf gur svsgu naq fvkgu zbagu.
Gung yrnirf gur qnl orvat gur svsgrragu, fvkgrragu be friragrragu.
Fvapr Nyoreg fgngrf ur xabjf nsgre urnevat Oreaneq fnl ur abj xabjf gung zrnaf vg vf gur friragu zbagu fvkgrragu qnl fvapr ur pnaabg xabj vs gur zbagu jbhyq or gur rvtug zbagu bs gur lrne.

13. Who Cares says

That alternate solution is bunk.
Albert must know that Bernard is told the day or he can’t state that authoritatively that Bernard doesn’t know. The only way he can do that is by knowing that the information Bernard has is not enough to uniquely identify the birthday of Cheryl. And that requires knowing at least the type of information given to Bernard. The same goes for Bernard. Bernard must know that Albert has been given the month or he can’t state like that that he knows after the first statement from Albert.

14. OverlappingMagisteria says

Yea.. I’m having a real hard time figuring out what the Alternate explanation on wikipedia is trying to say. It seems to be saying that we made the assumption that Albert knows what type of information Bernard received and vice-versa. (Ie. Albert knows that Bernard got one date, and Bernie knows that Al got one month.) I admit, this is an assumption I made and it was not stated in the problem.

But I think removing that assumption would just make the problem unsolvable. Albert would not be able to say “I know that Bernard doesn’t know.” He would say “I have no idea what she told Bernard. For all I know she told him the same exact thing she told me, or maybe she told him a choice of 2 dates…or maybe she even told him both the month and day and to keep quiet! Maybe Bernard knows, or maybe not.”

15. Rob Grigjanis says

OverlappingMagisteria:

It seems to be saying that we made the assumption that Albert knows what type of information Bernard received and vice-versa.

Yes, but then it goes ahead and makes that assumption anyway later on, followed by gibberish, followed by August 17! Must’ve been written by underwear gnomes.

16. Who Cares says

@OverlappingMagisteria(#15):
It isn’t an assumption we need to make. The first statement by Albert can only be made if he knows the type of information given to Bernard. The reply by Bernard can only be made if Bernard knows the type of information given to Albert.

17. OverlappingMagisteria says

Here’s a better explanation of the Alternate solution: http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/apr/15/why-the-cheryl-birthday-problem-turned-into-the-maths-version-of-thatdress

The alternate solution makes the assumption that Albert, perhaps through divine revelation, just knew that Bernard did not know the answer in step 1. Not that he used logic in order to deduce that Bernard did not know. I have no idea why someone would make that assumption, but if you do, I think it works out. But that is such a bizarre assumption to make, IMO.

They apparently do know what type of info the other has. My attempt at understanding the mess on wikipedia was mistaken.

18. Who Cares says

It breaks on point 4.
Seeing that Albert knows the month Bernard can remove 2 of them after the first statement of Albert not one as that alternate assumption does.

19. OverlappingMagisteria says

Seeing that Albert knows the month Bernard can remove 2 of them after the first statement of Albert not one as that alternate assumption does.

Not if Bernard knows that Albert received divine revelation saying “Bernie does not know!” If that is the case, Bernard might think that Albert was given May and thus can’t eliminate May.

Normally, Bernard would think “Al can’t have May because then he wouldn’t have had any way to deduce that I don’t know.” But in the bizzare alternate solution, Bernard would think “Al, might have May, and only knows that I don’t know because God told him, not because he was able to deduce it.”

Again.. I have no idea why anyone would assume that Albert magically knows Bernies state of mind without deduction. the Alternate solution is pretty poor.

20. DonDueed says

Well, I got the official correct answer pretty quickly. I agree that it seems unambiguous since the alternate answer requires knowledge not explicitly stated in the problem.

As to Mano’s question about whether I’d like to know any of these folks, I’d say the answer has to be “mu”. These are not real people, nor even fictional people. They are chess pieces, only here to present the problem. Mano’s question is like asking what happened to the engineers of those trains that left Chicago and Phoenix at different times, only to collide somewhere in Kansas.

21. Rob Grigjanis says

OverlappingMagisteria @18: Ahah! A much clearer explanation, which does work. It’s a very different problem, but as you say, a bizarre assumption.

Who Cares @19: Sorry if I’m misunderstanding, but if you’re saying that point 4 in the Guardian link is in error, I think you’re wrong. There’s a huge difference between A deducing that B can’t know (that means it can’t be May or June), and A being told that B can’t know (that leads to the deduction that the date can’t be 18 or 19).

22. Who Cares says

@Rob Grigjanis(#22):
Ah, it seems I misread what they meant.

23. fentex says

The original formulation of the puzzle – which was posed to Singaporean students in an exam – is a bit controversial because the language used to set it was ambiguous.

It was not only a logic but a grammar puzzle as well in practice and the grammar used was incorrect.

The way in which it was explained who knew what was slightly (but not impossibly) unclear but worse they got the tense in one of the three statements of fact wrong.

So some of the controversy is not about the answer but the quality of the question and it’s place in a math exam.

24. says

I also came up with the “alternative” answer.

The two obvious dates are eliminated – May 19, June 18, which also eliminates June 17, leaving only seven. Six of them are in ambiguous pairs. It could be either, hence why the boys knew it wasn’t them.

Only the “alternative” solution is unambiguous.

25. Rob Grigjanis says

left0ver1under @25:

Only the “alternative” solution is unambiguous.

I disagree. The first solution only assumes that we have been provided with all necessary information to solve the problem. That seems eminently reasonable to me. In that case, the only way that A can know that B doesn’t know is if the month is not May or June, and the rest follows unambiguously.

If, on the other hand, A only knows that B can’t know because A has been told, you have to assume that we haven’t been given everything to start with; namely, that someone told A!

26. Henry Gale says

It’s interesting to note that in the problem that Mano posted Cheryl tells Albert the month of her birth.

In the version I saw (original?) it did not specify what Albert was told.

http://www.cnn.com/2015/04/15/living/feat-cheryl-birthday-math-problem-goes-viral/

I noticed this when I read OverlappingMagisteria solution at #6. The decoded version states:

“Albert knows the month”

We don’t know that Albert knows the month.

27. Henry Gale says

Nevermind. Respectively got me.

28. lanir says

I’m usually pretty smart about using logic. When I get presented something like this however, I sometimes forget I’m not limited to a one move solution (to use a chess analogy). Then I get frustrated for a bit and go in circles until it occurs to me to think a couple moves in either direction. Once I start thinking that way I can usually come up with a solution. In some respects realizing how I think about problems like this is as interesting to me as solving them. Thanks for the puzzle. 🙂