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from the it-seemed-like-the-logical-thing-to-do-at-the-time dept.
A couple of months ago, it was a color-changing dress that blew out the neural circuits of the Internet. Now Kenneth Chang reports in the NYT that a problem from a math olympiad test for math-savvy high school-age students in Singapore is making the rounds on the internet that has perplexed puzzle problem solvers as they grapple with the simple question: "So when is Cheryl's birthday?"
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?
Logical puzzles like this are common in Singapore. The Singapore math curriculum, which has a strong focus on logic-based problem solving, has been so successful that it's been adopted around the world. According to Terrance F. Ross, US students have made strides in math proficiency in recent years, but they still lag behind many of their peers internationally, falling at the middle of the pack in global rankings. In the same PISA report the U.S. placed 35th out of 64 countries in math. "And even though the "Cheryl's Birthday" question may be atypical of the average Singaporean classroom, perhaps it's still worth asking: Are you smarter than a (Singaporean) 10th-grader?"
(Score: 5, Informative) by TheB on Thursday April 16 2015, @02:16AM
Correct.
*** SPOILERS ***
"Albert: I don’t know when your birthday is, but I know Bernard doesn’t know, either."
Since there is only one 18th and 19th.
This eliminates May and June from the possible months.
This leaves Bernard with
July 14 — July 16
August 14 — August 15 — August 17
"Bernard: I didn’t know originally, but now I do"
If Bernard was told 14th he wouldn't know which month.
so Albert knows that it is either
July 16, August 15, or August 17
"Albert: Well, now I know, too!"
If Albert was told August he wouldn't know which day.
This leaves July 16 as the only possible day.
(Score: 5, Funny) by b on Thursday April 16 2015, @04:06AM
Nice answer! The last step foiled me. The riddle also reminds me of this "joke".
Three logicians walk into a bar. The bartender asks "does everyone want beer?" The first says "I don't know", the second says "I don't know" and the third answers "Yes".
(Score: 1, Informative) by Anonymous Coward on Thursday April 16 2015, @07:09AM
Because "everyone" is a logical AND.
If the first one didn't want a beer, he would know the answer. No, not everyone wants a beer. Even if two of them would want a beer, that's not everyone.
So, when he answers "I don't know", the other two can deduce that he wants a beer.
The same thing goes for the second person.
As both said "I don't know", the third person only needs to answer himself.
(Score: 5, Insightful) by KilroySmith on Thursday April 16 2015, @04:32AM
Your logic is lacking.
Albert doesn't know anything other than what Cheryl told him ("To Bernard, she whispered the day, and only the day"). As far as Albert is concerned, Cheryl could have told Bernard the exact month, day, hour, and second. When he says "I don’t know when your birthday is, but I know Bernard doesn’t know, either", he's making a statement that has no basis in logic, and as a result gives no information to Bernard.
Albert doesn't know that Cheryl told Bernard the day. Any conclusions you draw can't assume that knowledge.
(Score: 1, Insightful) by Anonymous Coward on Thursday April 16 2015, @05:53AM
Yes, yes, I thought of that objection too, but it's also easy figure out what the people who designed the question wanted you to answer.
(Score: 3, Interesting) by TheB on Thursday April 16 2015, @07:11AM
You are right that without an assumption this puzzle is unsolvable.
It is a common error in puzzles and tests. I've seen similar "must read between the lines" questions on college entrance exams. According to one instructor, it was an intentional omission to test ability to make reasonable conclusions of intent.
(Score: 2, Funny) by Anonymous Coward on Thursday April 16 2015, @08:02AM
Well, here's what actually happened:
(Score: 0) by Anonymous Coward on Thursday April 16 2015, @08:04AM
Err ... s/Alice/Cheryl/ of course …
(Score: 1, Funny) by Anonymous Coward on Thursday April 16 2015, @08:25AM
More like they looked it up on facebook etc even if cheryl doesn't list it on facebook, sometimes you can tell from the birthday greetings in her timeline ;).
(Score: 2) by wonkey_monkey on Thursday April 16 2015, @09:49AM
Any or all them could also have been lying at any point, since that's not explicitly stated.
Cheryl's birthday could be August 15, May 16, or January 4.
She might even be a fictional character, and this entire thing is just a tissue of lies, in which case she doesn't even have a birthday.
systemd is Roko's Basilisk
(Score: 0) by Anonymous Coward on Thursday April 16 2015, @04:35PM
Technically you are correct. Also note https://xkcd.com/1475/ [xkcd.com]
(Score: 0) by Anonymous Coward on Thursday April 16 2015, @08:40PM
Incorrect.
Albert knows the month.
Albert knows that Bernard knows the date.
Albert makes the statement "I don’t know when your birthday is, but I know Bernard doesn’t know, either" and passes critical information to Bernard.
By saying that "Bernard doesn't know, either", Albert affirms that for the month that was spoken by Cheryl, all the days in the month are replicated in other months.
Let's take the answer as an example. July has two dates: 14th and 16th. The 14th is replicated in August, and the 16th is replicated in May. Therefore Albert knows that Bernard has no idea what Month the birthday is just by Bernard using his private information (16th).
This is also true of August as all the days (14th, 15th, 17th) are repeated in other months.
Albert could not have made the absolute statement "Bernard doesn't know, either" if the month was either May or June. Both of these months contain a unique day, thus Bernard could have known the birthday using just his private information if he was given 18th or 19th as the day.
If May or June was the correct month, Albert would have said "Bernard could know the birthday" and a different logic path would follow.
(Score: 2) by EQ on Thursday April 16 2015, @09:46PM
What most folks miss is this : By saying that "Bernard doesn't know, either", Albert affirms that for the month that was spoken by Cheryl, all the days in the month are replicated in other months.
It doesn't necessary follow - its a matter of semantics to get that much inference (affirms) out of such a simple statement. The broadest meaning does NOT include that information, and requires a contextual jump that may not be justified in normal conversation. "doesn't know" could be stating the simple fact that he does not know the birthday meaning the month AND day -- which is an allowable and perfectly lgical semantic interpretation of the statement. In that case you cannot draw the inference which the problem assumes that you do. Once you get past this, and make the non-colloquial semantic change in the processing of the statements, the problem is easy. Its not making the logic that's tough, its the contextual jump. For many, this isnt a logic problem, its a trick of semantics problem.
(Score: 4, Interesting) by bzipitidoo on Thursday April 16 2015, @04:35AM
I don't follow it. I guess we have to assume several things. First, Albert and Bernard communicate with each other in some fashion. They don't pool their knowledge, because if they did they'd know immediately. Maybe Albert only sees that Bernard is still stumped? Or knows that Cheryl would not give Bernard a clue that gives away the information? How does Albert know Bernard was told the day? Cheryl could have told them both the same information, the month. Or she could have given Bernard a formula, something like "day = month + 9" or "day = 23 - month".
But, if everyone knows that Cheryl told Bernard which day, we can eliminate May 19 and June 18, otherwise Bernard would know. June 17 can also be eliminated, because if Albert knew the month was June, and knew that Bernard did not know Cheryl's birthday, then Albert would know that it has to be June 17. But I don't yet see why May 15 and 16 are out.
(Score: 3, Insightful) by Anonymous Coward on Thursday April 16 2015, @04:51AM
The first assumption is that Cheryl was even born. How does Albert know that Cheryl was not from her mother's womb untimely ripped?
(Score: 0) by Anonymous Coward on Thursday April 16 2015, @07:42AM
From the puzzle:
“My birthday is one of these,” she said.
This already implies that Cheryl has a birthday, because if she didn't have a birthday, none of the given dates would be her birthday.
(Score: 0) by Anonymous Coward on Thursday April 16 2015, @08:18AM
Albert assumes Cheryl has a birthday, before she confirms that she has a birthday.
(Score: 2) by Anal Pumpernickel on Thursday April 16 2015, @02:23PM
What if she was lying?
(Score: 3, Funny) by fritsd on Thursday April 16 2015, @08:30AM
Singapore probably doesn't have room for the Forest of Dunsinane.
(Score: 2) by TheB on Thursday April 16 2015, @07:42AM
Without assuming that both Albert and Bernard are aware of Cheryl whispering the month to one and the day to the other this puzzle is unsolvable.
Given that the reader is given this information it is reasonable to assume that both Albert and Bernard also know this.
May 15th and 16th are out for the same reasoning you used to eliminate June 17th.
If Cheryl's birthday was May 19th then Albert's statement "I know Bernard doesn’t know, either." would be invalid.
Since there is only one 19th Bernard could deduce it must be May 19th. Albert must know the month is not May for his statement to be correct.
(Score: 4, Insightful) by bzipitidoo on Thursday April 16 2015, @02:16PM
Oh, I see it now. Albert knows that Bernard does not know, not because Bernard or Cheryl told Albert that, but because Albert was told a month for which Cheryl did not give unique days. I was thinking that Albert saw only that Bernard did not know, which eliminates only June. So all of May and June are out. After Albert announces that he knows Bernard does not know, Bernard can also make that connection.
Then, the 14th is out, because Bernard would still not know which month Albert was told if the day was the 14th, and he says he does know after hearing Albert. We're down to July 16, Aug 15, and Aug 17.
Finally, August is eliminated because Albert says that because Bernard now knows, he knows too. If Albert knew the month was August, he still could not tell if it was the 15th or 17th. So Albert must have been told that the month was July, for him to be able to announce (truthfully) that now he knows too.
(Score: 0) by Anonymous Coward on Thursday April 16 2015, @07:48AM
Yes, you have to assume they both know that Bernard was told the day and Albert was told the month. The only other communication between them is in what they explicitly said in this problem.
Now if Albert can say he knows Bernard doesn't know when Cheryl was born, then as Both May and June have unique days in it, Albert knows it is neither May nor June, that lets Bernard know it isn't May or June, now Bernard says he does know with this new information, which leaves only one option.
(Score: 2, Insightful) by hoeferbe on Thursday April 16 2015, @04:50AM
OK, I understand how this eliminates June as a possible month from Albert's point of view, but I don't understand how it removes May.
Taking away 18 as a candidate would leave June 17th as the only June choice. If Albert had been told "June" by Cheryl, then Albert would now know her birthday is June 17th. But since Albert doesn't know that, it removes June as a candidate.
In Albert's mind, these should still be the possibilities:
May 15 16
Jul 14 16
Aug 14 15 17
What is the reasoning that Albert can eliminate the entire month of May before Bernard says "I didn’t know originally, but now I do"?
(Score: 4, Informative) by KilroySmith on Thursday April 16 2015, @05:48AM
Well, if you ignore my comment above and assume that Bernard knows that Albert knows the month, and Albert knows that Bernard knows the day, then...
Following TheB's analysis,
"Albert: I don’t know when your birthday is, but I know Bernard doesn’t know, either"
If Cheryl had told Bernard a unique date (i.e. the 18th or 19th which only occur in one month), then Bernard would immediately know both the month and date. With this statement, Albert is telling Bernard that the Month that Albert was told doesn't have a unique possible birthday date - that for the month he was told, each of the possible dates also occur in a second month. That tells Bernard that Albert must have been told either July or August, because both May and June have unique dates (the 19th of May or the 18th of June).
I think you're getting a bit confused at this step. Albert couldn't make his statement "I know Bernard doesn't know either" if it's possible that Bernard was given either the 18th or 19th. Albert knows, based on the month he was given, that Bernard couldn't possibly have been given a unique date, so Albert must have been given a month that doesn't have a unique date.
Bernard now knows two pieces of information - a date that occurs in at least two different months, and the fact that May and June have been eliminated. The 14th occurs in both July and August, but can't be the right date - if it was, Bernard wouldn't be able to make the statement that "I didn’t know originally, but now I do". If the date was the 14th, the information that Albert has given him wouldn't be sufficient to choose one or the other. So, we can eliminate the 14th.
July 16 is a possibility. The 16th occurs in both May and July, so Bernard wouldn't be able to tell the difference originally, but would be able to by using Albert's revelation to eliminate May 16.
August 15 is a possibility. The 15th occurs in both August and May, and Bernard should be able to eliminate May using Albert's revelation.
August 17 is also a possibility. The 17 occurs in both August and June, and Bernard should be able to eliminate June using Albert's revelation.
So how do we choose between these three possibilities?
Bernard reveals the next clue - "Well, now I know too". So, of the three possibilities, only one can be possible
If Albert had been told August, then he couldn't determine whether Aug 15 or Aug 17 was the correct one, so it can't be either of those or he couldn't make the statement.
If Albert had been told July, then he could make the statement. This is the only possibility left, so it must be the correct date.
(Score: 4, Informative) by Ryuugami on Thursday April 16 2015, @05:50AM
If B was told "18th", he would know it was May 18th.
If A was told "May", there would still be a possibility of the day being May 18th, so he wouldn't know B has insufficient information.
In other words, as the first step you can strike all months that have any unique days.
If a shit storm's on the horizon, it's good to know far enough ahead you can at least bring along an umbrella. - D.Weber
(Score: 2) by fritsd on Thursday April 16 2015, @08:32AM
Cheryl also says aloud to both Albert and Bernard: "I have told Albert the month and I have told Bernard the day"
(Score: 0) by Anonymous Coward on Thursday April 16 2015, @08:48AM
"Albert: I don’t know when your birthday is, but I know Bernard doesn’t know, either."
How does A know B doesn't know? C hasn't even asked B anything and thus B hasn't even said anything yet!
(Score: 0) by Anonymous Coward on Thursday April 16 2015, @01:11PM