The Math Problem That Has Stumped the Entire Internet

[Kilgore Trout]

So, recently there was a math Olympiad in Singapore for high school students and one of the math/logic questions has gone viral over the internet because it's so baffling.

Yet there is a solution.
It's one of those story math questions students are familiar with. This one about a girl who can't give a straight answer to a simple question.

Can you solve it. Or can you understand the explanation of the solution.
Is there even one person on Terb who can solve the problem without looking at the answer first?


Here goes:
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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?



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The New York Times and the Guardian in London explain the solution in different ways below:
My head spins around in circles and I can't understand their reasoning though.

NY Times:
http://www.nytimes.com/2015/04/15/s...gapore-math-problem-cheryl-birthday.html?_r=0

Guardian:
http://www.theguardian.com/science/...rt-bernard-and-cheryls-birthday-maths-problem

 

[Tony2Tap]

Without even checking, I'm going July 16.

When Albert says "I don't know when your birthday is, but I know Bernard doesn't know, either."
- Bernard would know if the Number was 18 or 19
- The month can't be May or June (where 18 and 19 exist)
- The month must be July or August
When Bernard says "I didn't know originally, but now I do."
- He knows it's in July or August based on the above
- If the number were 14 he wouldn't know, so it must be 15, 16 or 17
- That leaves July 16, August 15 and August 17
- At this point Bernard knows (16 = July 16)
When Albert says "Well, now I know, too!"
- If he had August as a month he wouldn't know, so he must have July. He knows it isn't the 14th (or Bernard wouldn't know) so it must be the 16th

 

[oldjones]

As the Times points out, you have to know the unwritten rules that folks in these puzzles live by.:

Prime Directive: They only and always speak in a sequence that brings us and them to the answer, never out of turn or on side issues.

Rule1: As soon as you know, you must announce: It's the only way Albert can 'know' Bernie doesn't know, and thus Bernard's silence eliminates 18, and 19 which each occur only once in the list. If he was told either one, he'd be shouting.

All other numbers and all months still show more than once. Now Bernie thinks about what Month-Man Albert said a little more:

Rule2: No one lies, prevaricates or even exaggerates: So he knows that Cheryl didn't say either May—as in 19—or June—as in 18—when she whispered to Albert, 'cause then he'd have to follow Rule1. Leaving July or August, and 14, 15, 16 and 17 BUT (and it's a big but) Bernie actually knows which number, because Cheryl told him. So he says (Rule1) he now knows the entire birthday, month and day.

Rule3: There'd be no fun if anyone said how: So Albie chortles, "Aha! By now, all BernBrain should know is the number he was told and either of July or August which he then figured out, unless that number only occurs in … JULY! So now I know too!"

Leaving the kids on TERB to Rule 3.

Thanks Kilgore

 

[canada-man]

that's not a math question

 

[oldjones]

that's not a math question

The first guy on the interweebs to say so. Does that mean Boolean Algebra isn't logic and Euclidean Geometry is Art?

 

[Bud Plug]

The first guy on the interweebs to say so. Does that mean Boolean Algebra isn't logic and Euclidean Geometry is Art?

I agree with Canadaman on this one. It is a logic exercise to examine if you can avoid misdirecting your analysis. If you ignore how it is that Bernard is sure that he knows the date, and just accept that he does, the answer is straightforward.

 

[tribunus]

After giving her the correct response, Albert & Bernard ask her if they can double-team her on July 16. She quickly agreed.

 

[Purlieu]

I love Logic Puzzles!

 

[Kilgore Trout]

Okay, thanks for explanation T2T and Oldjones, i get it now.
Finally.

 

[GPIDEAL]

Without even checking, I'm going July 16.

When Albert says "I don't know when your birthday is, but I know Bernard doesn't know, either."
- Bernard would know if the Number was 18 or 19
- The month can't be May or June (where 18 and 19 exist)
- The month must be July or August
When Bernard says "I didn't know originally, but now I do."
- He knows it's in July or August based on the above
- If the number were 14 he wouldn't know, so it must be 15, 16 or 17
- That leaves July 16, August 15 and August 17
- At this point Bernard knows (16 = July 16)
When Albert says "Well, now I know, too!"
- If he had August as a month he wouldn't know, so he must have July. He knows it isn't the 14th (or Bernard wouldn't know) so it must be the 16th

You're right but I think I can explain it simpler.

When Bernard was told the day but still didn't know the month (according to Albert), that eliminated May and June because they contain the only unique days that give away the month.

Bernard realized this, but also narrowed down the day as the 16th when Albert said he still didn't know after implicityly recognizing that it can't be May or June, because July & August contain a common day being the 14th.

After Bernard admits this (implicitly), Albert then knows that this eliminates the 14th (so the 16th is the odd man out).

P.S. I've recanted. See my post #20.

 

[Kilgore Trout]

As GPI says.
Key is once you know you must annouce it and once Bernard wasn't the first one to speak it automatically eliminated everything in May and June.

Also you must always tell the truth and from there everything progresses in a linear fashion as the 2 guys bounce facts back and forth and truth is arrived at.

 

[GPIDEAL]

Isn't there a step missing. How does he eliminate the August dates.

Bernard can eliminate August after learning that it can't be May or June (and he knows it's the 16th).

P.S. I've recanted. See my post #20.

 

[buttercup]

IMO, every step of the thing falls out just fine, logically, without our having to know any (unwritten) rules.

Albert knows the month but not the day

Bernard knows the day but not the month.

If the day is 18, Bernard, who knows the day, can immediately pin down the month as being june. The same if the day is 19 -- Bernard can immediately pin down the month as may.

But Albert, who knows the month, reports that he knows that Bernard does not know the month.

Bernard deduces, from this report, that the month must be july or august, because, if the month were may or june, Albert (who knows the month) could not be certain that Bernard could not pin the month.

So at this point, Albert has deduced that the day is 14, 15, 16, or 17, and Bernard has deduced that the month is july or august.

Next, Bernard reports that he now knows the month.

Now, day 14 is available in both july and august. Therefore, if the day were 14, Bernard would not be able to tell between july and august. But Bernard has reported that he can pin down the month. So Albert knows the day is not 14. That leaves 15, 16, 17 as the possible days.

But Bernard now knows the month. If the day is 15, Bernard cannot pin the month, because 15 appears on Cheryl’s list in both july and august. The same is true of 17, which also appears in both july and august. Only if the day is 16, can Bernard pin the month, i.e july, because 16 appears only in july.

So Albert can deduce, from the fact that Bernard says he knows the month, that the day must be 16.

In other words, Cheryl’s birthday is july 16th.

Every step follows in logical sequence as the new facts are laid out.

 

[mikeeman]

But it does not necessarily have to be as you put it. Both Albert and Bernard could be wrong in their assumptions. Then again, they could just be liars, or liars who are wrong. I still think Cheryl was wearing a gold and white dress. Now, the question is: who is her pimp....Albert or Bernard?

 

[Polaris]

That's why I never liked math.

This is bullshit. Honestly it is.

What are they trying to prove here? What are they teaching kids?

Having said all that, I think I figured it out.

I was not paying attention, but the number 18 is unique.

Therefore, since she did not say 18 was her birth day, then June 18 is struck out.

19 is unique too, so strike that out.

So you left with two less dates.

May 15
May 16

June 17

July 14
July 16

August 14
August 15
August 17

Remember, one guy was told the month, and the other only the day.

Then one declares he still does not know, the guy told the month. Which means since the unique 18 & 19 dates are stuck out, and the way the dialogue goes, they still don't know, means June 17 is out too, along with August 17.

May 15
May 16

July 14
July 16

August 14
August 15

And that is the answer. Cheryl is a girl and she does not want you to know her birthday.

 

[Malika Fantasy]

Simpler answer
Don't be friend with Cheryl. Because Cheryl is an asshole of a friend.

But seriously I always hated those type of problems...

 

[Polaris]

But guys, there is only list, and two guys, each with one piece of information.

I think this test question, is to see if the student taking the test, over does it.

As far as I can tell, you cannot eliminate a month, other than June.

Cheryl is playing hard to get.

:hippie:

 

[Toke]

You're right but I think I can explain it simpler.

When Bernard was told the day but still didn't know the month (according to Albert), that eliminated May and June because they contain the only unique days that give away the month.

Bernard realized this, but also narrowed down the day as the 16th when Albert said he still didn't know after implicityly recognizing that it can't be May or June, because July & August contain a common day being the 14th.

After Bernard admits this (implicitly), Albert then knows that this eliminates the 14th (so the 16th is the odd man out).

Now I understand. Thanks.

 

[GPIDEAL]

Now I understand. Thanks.

Don't thank me yet. I was wrong (no, I didn't look at those solution links, if there is A solution).


Before I left this thread, my own answer was giving me a headache, but I had to log off. Then the problem gnawed at my mind since that time. So I've been thinking about this question again, and it helps to write your thoughts down on paper.

The best answer so far is Mikeeman, and Polaris is right about this question being bullshit and that she's playing hard to get. Malika is also right that Cheryl is an asshole.

The only dates we can eliminate (per Old Jones' Rule #1) are May 19th or June 18th, but we CAN'T eliminate those months because they include other possible days (Bernard could've been told that her birthday was on the 15th, 16th, or 17th).

As for Albert, we don't know what month he was told, so he's right when he says that he doesn't know either due to multiple choices for days.

Lastly, it doesn't matter what these guys THINK the date could be, there are still too many possibilities since there's more than one month with two or more possible days remaining: May 15th or August 15th, May 16th or July 16th, June 17th or August 17th, and July 14th or August 14th.

You can't deduce her birth date from their reactions, which are erroneous if not irrational. They're better off to pay her if they want to get laid.

 

[fuji]

The only dates we can eliminate (per Old Jones' Rule #1) are May 19th or June 18th, but we CAN'T eliminate those months because they include other possible days (Bernard could've been told that her birthday was on the 15th, 16th, or 17th).

Yes you can eliminate those months. Albert KNOWS that Bernard doesn't know the date. How can he know that?

He had to have been told a month where EVERY day was duplicated in another month, and the only months like that are July and August. If Albert had been told May or June then it is possible Bernard heard a unique day and knows her birthday, but Albert says that isn't the case. So, he wasn't told May or June, he was told July or August.

That enabled Bernard to figure out it was July or August which was enough information for him to figure out her birthday. Which proves he wasn't told 14, since in that case he wouldn't be able to figure it out.

After that is where I got stuck. I had to read the answer to understand that it can only be July.