Cheryl's Birthday

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Cheryl's Birthday

Postby Sizik » Tue Apr 14, 2015 8:05 pm UTC

This has apparently been going viral or something after being featured in a Singaporean math competition, so I thought I'd repost it here:

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?
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Re: Cheryl's Birthday

Postby HonoreDB » Tue Apr 14, 2015 10:12 pm UTC

Easier than the versions of this with knowing the product and the sum of tuples of numbers.

Spoiler:
Albert knows that Bernard was told a day that exists in multiple months. So he must have been given a month that contains only duplicated days. May is out because 19 is unique, and June is out because 18 is unique, so it must be July or August.

Narrowing it down to July or August tells Bernard when it is. So it must be a day that exists only in July, or only in August. 14 exists in both, so it must be either July 16, August 15, or August 17.

Now Albert, who knows the month, knows the date. Since August has two candidates, it must be July, since it has only one remaining possibility.

So the birthday is July 16.

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Re: Cheryl's Birthday

Postby BedderDanu » Tue Apr 14, 2015 10:21 pm UTC

I think I just got it

Spoiler:
Albert: Month
Bernard: Day

Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.
There are two days that are unique, 18 and 19. Those are key to solving the problem.
This means that albert has a month that does not contain unique days, This means that the birthday is in July or August

Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.
There are three days that have had their double deleted: 15, 16, 17. Both 14s are still in play
Bernard is stating that he does not have a 14, as having a 15, 16, or 17 tells him the actual date.

Albert: Then I also know when Cheryl’s birthday is.
Albert, knowing that Bernard found the answer, is stating that he knows the answer now as well.
There is one date in July remaining (16) and 2 in august (15, 17).
Since he knows the month, the only way he could have an answer is if his month only had one day remaining.

Answer: July 16


I saw this before, but couldn't answer it then. It just clicked seeing it here.

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Re: Cheryl's Birthday

Postby ahammel » Tue Apr 14, 2015 10:25 pm UTC

Ha, my wife told me this one just last night!
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Re: Cheryl's Birthday

Postby jac50 » Wed Apr 15, 2015 8:46 am UTC

A slight tangent to the original question, but I know that the problem is asking what Cheryl's birthday is based on you being an outside observer to the problem. I was wondering if there's a way to work out how Albert was able to know the date based on him knowing that Bernard now knows the date.

Any thoughts?

Spoiler:
Once you get down to the three possible dates (Aug 15/17, and July 16), I can't see how Albert can determine the date based on him knowing that Bernard now knows the date.

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Re: Cheryl's Birthday

Postby jaap » Wed Apr 15, 2015 9:48 am UTC

jac50 wrote:A slight tangent to the original question, but I know that the problem is asking what Cheryl's birthday is based on you being an outside observer to the problem. I was wondering if there's a way to work out how Albert was able to know the date based on him knowing that Bernard now knows the date.

Any thoughts?

Spoiler:
Once you get down to the three possible dates (Aug 15/17, and July 16), I can't see how Albert can determine the date based on him knowing that Bernard now knows the date.

Now that we know the answer, we can go through their reasoning.
Spoiler:
The answer is July 16th, so

Albert knows: July, hence 14th or 16th
Bernard knows: 16th, hence May, or July.

Albert reasons that 14 and 16 are both not unique in the list and deduces that Bernard cannot know the full date yet.
"I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too."

Bernard reasons that if it were May, then Albert would not have made that statement. In that hypothetical case, Albert would have had to choose between May, 15th, 16th, and 19th. Albert would not have been able to exclude the possibility of 19th of May, a case where Bernard could have deduced the month from the unique day. Therefore Albert couldn't know for sure that Bernard didn't know the full date.
Therefore, Bernard knows it is July (16th).
"At first I don’t know when Cheryl’s birthday is, but I know now."

Albert reasons that if it were 14th, then Bernard would not have made that statement. In that hypothetical case, Bernard would have had to choose between 14th of July or August but Bernard would not have been able to exclude August with the above reasoning, because all of August's dates, 14, 15, and 17, also occur in other months.
Therefore Albert knows that it is the 16th (of July).
"Then I also know when Cheryl’s birthday is."

Edit: Essentially they are going through the exact same reasoning as we do as outside observers, except that they each start with a smaller initial list of possible dates due to the information they have been given.

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Re: Cheryl's Birthday

Postby douglasm » Wed Apr 15, 2015 8:08 pm UTC

jac50 wrote:A slight tangent to the original question, but I know that the problem is asking what Cheryl's birthday is based on you being an outside observer to the problem. I was wondering if there's a way to work out how Albert was able to know the date based on him knowing that Bernard now knows the date.

Any thoughts?

Spoiler:
Once you get down to the three possible dates (Aug 15/17, and July 16), I can't see how Albert can determine the date based on him knowing that Bernard now knows the date.

Spoiler:
Albert is able to follow the same logic we do to get it down to three possible dates. So, Albert has that list of three possible dates to pick from. But, going back to the original setup, Albert also knows the month of the correct date because Cheryl told it to him. Albert knew from the beginning that the correct date was in July, so given a set of dates that only has one entry in July he can obviously pick out that single July entry as the correct one.

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Re: Cheryl's Birthday

Postby Sizik » Wed Apr 15, 2015 9:25 pm UTC

So it seems that the question went viral because you can come up with different answers depending on how you interpret Albert's first statement, and a lot of people (presumably not accustomed to working with formal logic) came up with
Spoiler:
August 17
as the answer.

Here's a video about it
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Re: Cheryl's Birthday

Postby SPACKlick » Wed Apr 15, 2015 9:41 pm UTC

I missed the word respectively in the question so I had a wider analysis for the first 4 steps which I found quite interesting.

Spoiler:
1) Albert doesn't know the birthday
Albert cannot have 18 or 19 as they are unique.

2) Albert knows Bernard doesn't know
Albert cannot have May or June

3) Bernard didn't know
Bernard cannot have 18 or 19

4)After hearing the above Bernard Does Know
Bernard either had a date unique to July or August (15,16,17) or a Month unique to 14,15,16,17 (none)
Bernard had 15,16 or 17

5) Hearing the above Albert now knows
ALbert cannot have august (it would leave two possibilities). Albert has July

6) The date is July 16th


I think an it could be either way round type puzzle could get quite interesting, this one didn't much. The bad writing did bother me throughout.
Spoiler:
Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know either.
Bernard: At first I didn’t know when Cheryl’s birthday is, but I know now.
Albert: Then I also know when Cheryl’s birthday is.


Edit: I've no idea how the logic for the wrong answer in the linked video works. Can someone explain how they're interpreting the relevant line to make it valid?

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Re: Cheryl's Birthday

Postby Xias » Thu Apr 16, 2015 1:43 am UTC

SPACKlick wrote:Edit: I've no idea how the logic for the wrong answer in the linked video works. Can someone explain how they're interpreting the relevant line to make it valid?


The second part of Alfred's statement can be interpreted as "I am aware of the fact that Bernard doe4sn't know," rather than "It is logically impossible for Bernard to know, based on the month I was given." Since the statement is based on his observation, rather than on the month he was given, it changes the date range available to Bernard when he speaks (and solves it) and therefore changes the solution.

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Re: Cheryl's Birthday

Postby Reydien » Thu Apr 16, 2015 2:39 pm UTC

SPACKlick wrote:Edit: I've no idea how the logic for the wrong answer in the linked video works. Can someone explain how they're interpreting the relevant line to make it valid?


Here's how that thought process works, at least as I understand it:

Spoiler:
  • Alfred and Bernard are given their respective information (Alfred has August, Bernard has 17), and then they're standing in the room together watching each other.
    Alfred notices that Bernard has not immediately proclaimed that he knows the answer, and basically says, "I noticed you haven't already answered it. fascinating..."
  • At this point Alfred knows the answer is in August, and based on his observation knows the answer isn't 18 or 19. He still doesn't know, so he stays quiet after this.
  • Bernard hears Alfred's comment, then notices that Alfred did not immediately proclaim that he knew the answer. He makes the following deduction: I have 17, so it is either June 17 or August 17. After Alfred made his comment, if he was holding June he would have realized that there was only one Date in June left and proclaimed that he knew the answer. he didn't, though, which means he can't be holding June. Therefore he must be holding August.
  • Bernard basically says, "it is fascinating. What's also fascinating is that you didn't then know the answer. That means I know what the answer is now."
  • After Bernard makes his statement, Alfred basically walks through the same logic Bernard must have stepped through. Alfred knows it is in august, so either 14, 15, or 17. If it was 14 or 15 Bernard still couldn't have figured it out, so he must have the 17.

The difference is in the amount of sureness, so to speak, in Alfred's first statement. In the logic of the official answer, Alfred's first statement is basically saying, "based on the month I am holding, I know that Bernard cannot be holding a unique number." In the logic of the alternative answer, Alfred's first statement is just saying, "oh hey, Bernard didn't blurt out that he already knows the answer."

That second statement is independent of what Alfred already knows, and could even have been made by an outside observer. In fact, that might be a way of understanding the logic of the alternative solution: assume that the only thing Alfred and Bernard can say is that they've figured out the answer, and the other statements are made by a third individual, let's say Daniel. it would go as follows:

  • Alfred and Bernard are given their respective information. Neither one speaks up.
  • Daniel says, "I've noticed that Bernard hasn't stated that he knows the answer yet."
  • Both Alfred and Bernard process Daniel's Statement, but neither one speaks up.
  • Daniel says, "I've now also noticed that after my first statement Alfred didn't state that he knows the answer."
  • Both Alfred and Bernard process Daniel's second statement, and Bernard speaks up, "I know the answer now."
  • Alfred hears Bernard's statement, then also speaks up, "I know the answer now too."

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Re: Cheryl's Birthday

Postby TheGrammarBolshevik » Sun Apr 19, 2015 5:36 pm UTC

There's a much harder version here. Be aware that there are answers in the comments.
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Re: Cheryl's Birthday

Postby slinches » Wed Apr 22, 2015 7:33 am UTC

There's a fundamental flaw in the problem, though. How does Albert know what information Cheryl told Bernard about her birth date? There's an implicit assumption you have to make that Cheryl explained what piece of info was given to the other. Without that, Albert couldn't correctly claim anything about Bernard's knowledge of the answer.

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Re: Cheryl's Birthday

Postby TheGrammarBolshevik » Wed Apr 22, 2015 3:33 pm UTC

Why does "implicit assumption" = "fundamental flaw"? Everyone was able to figure out that Albert and Bernard knew the problem setup, so I don't see what the problem is.
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Re: Cheryl's Birthday

Postby slinches » Wed Apr 22, 2015 4:35 pm UTC

Fundamental flaw was a bit of an overstatement, but it's still an ordinary flaw. And it was significantly worse in the version I read first:

the internet wrote: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.


This version has two conflicting implications. One that A and B know nothing about what the other may have been told and the opposite when A states that he knows something about B's knowledge of the answer.

If Cheryl had said something to the effect of "I'll give one of you the date and the other the month and then you can try to figure it out without telling anyone what I told you.", that conflict would be completely resolved.

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Re: Cheryl's Birthday

Postby TheGrammarBolshevik » Wed Apr 22, 2015 4:49 pm UTC

OK, but were you able to figure out pretty quickly what the intended setup was?
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Re: Cheryl's Birthday

Postby slinches » Wed Apr 22, 2015 6:34 pm UTC

Yes I was able to deduce the intent, but then I'm familiar with the format for these types of logic problems. For those who don't do these sorts of things often, the unwritten assumption of "take all these statements as true without regard for real human behavior/fallibility" may not be clear.

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Re: Cheryl's Birthday

Postby Wildcard » Thu Apr 23, 2015 8:23 am UTC

What I find interesting is the selection of the dates to present in the puzzle—the method of choosing them such that a solution is possible and such that only the intended solution works. (I think you could probably model this as a graph of some type....)

But what about extending the number of variables? What if you had a month, day AND a year, and three people who each had only one piece of information?

Or as another variation, what about the number of statements needed? In the given puzzle, they go through what—a total of three statements before they both know the answer? (Okay, technically two, since Albert knows the date as soon as Bernard makes his statement, even before Albert makes his second statement.) What if you wanted an arbitrary number of statements to occur before they both knew the answer? Say you wanted 7 statements to occur back and forth that they didn't know the answer before they would both be able to deduce the answer. How could you choose a set of dates to present such that they would make 7 statements before they would know the answer?

Actually, this gets even trickier: It might take *them* 7 statements to know the date, but there could be another date in the list of possibilities which would also take 7 statements for them to know fully. Then A and B would be able to find the answer (given the month and day, respectively) but the outside observer (i.e. the reader) wouldn't be able to tell which of the dates was actually the solution. This gives me an idea for an evil a brilliant version of the puzzle, where you design it deliberately with two possible answers which would require 7 statements, you include an onlooker named Dan, and then after A and B both know the date, B comments that Dan can't possibly know the date, and then A comments that after Bernard's 2nd statement that he didn't know the date, A already knew that Dan would never know the date from A and B continuing their exchange.)

Also, the graph for month-day version only would be easier than I thought. You would just need a bipartite graph with, say, each node on the left representing a month and each node on the right representing a day. The solution is represented by a chosen edge, and A and B are each told one node of the edge. The month-day-year version is much trickier.

I'd like to get up to an arbitrary number of pieces of information (month-day-year = 3 pieces of information) and arbitrary number of "don't know" cycles before everyone knows.
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Re: Cheryl's Birthday

Postby Vytron » Sun May 03, 2015 9:06 pm UTC

August 17 is my favorite answer, because it allows for the following variant:

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 know that Bernard does not know the date, but I do!
Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.

So when is Cheryl’s birthday?

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Re: Cheryl's Birthday

Postby TheGrammarBolshevik » Sun May 03, 2015 9:25 pm UTC

Huh?
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Re: Cheryl's Birthday

Postby Moonbeam » Sun May 03, 2015 10:37 pm UTC

TheGrammarBolshevik wrote:Huh?


My sentiments exactly :)

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Re: Cheryl's Birthday

Postby Vytron » Sun May 03, 2015 10:44 pm UTC

TheGrammarBolshevik wrote:Huh?


Spoiler:
Albert knows Bernard didn't get a 18 or a 19, because if he did, Bernard would be announcing that he knows the date.

This means Albert was given June and Bernard was given 17, because:

Suppose Albert was given May:
Then, May 19 is ruled out (Bernard didn't announce)
Albert doesn't know if it's 15 or 16, so he can't say "but I do!"

Suppose Albert was given July:
Albert doesn't know if it's 14 or 16, so he can't say "but I do!"

Suppose Albert was given August:
Albert doesn't know if it's 14, 15 or 17, so he can't say "but I do!"

Albert then should have gotten June, and after he announces that he knows the answer, Bernard, which had 17, and initially doesn't know the answer (it could be June or August) then knows it must be June, as otherwise, Albert wouldn't know if his number was 14, 15 or 17.

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Re: Cheryl's Birthday

Postby Moonbeam » Sun May 03, 2015 11:18 pm UTC

Vytron wrote:
TheGrammarBolshevik wrote:Huh?


Spoiler:
Albert knows Bernard didn't get a 18 or a 19, because if he did, Bernard would be announcing that he knows the date.

This means Albert was given June and Bernard was given 17, because:

Suppose Albert was given May:
Then, May 19 is ruled out (Bernard didn't announce)
Albert doesn't know if it's 15 or 16, so he can't say "but I do!"

Suppose Albert was given July:
Albert doesn't know if it's 14 or 16, so he can't say "but I do!"

Suppose Albert was given August:
Albert doesn't know if it's 14, 15 or 17, so he can't say "but I do!"

Albert then should have gotten June, and after he announces that he knows the answer, Bernard, which had 17, and initially doesn't know the answer (it could be June or August) then knows it must be June, as otherwise, Albert wouldn't know if his number was 14, 15 or 17.


I understand the original question and answer, it's the variant of your's that I'm having problems with ??

Vytron wrote:August 17 is my favorite answer, because it allows for the following variant:

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 know that Bernard does not know the date, but I do!
Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.

So when is Cheryl’s birthday?


..... how can Albert know the date - he's only been told that the month is August ??

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Re: Cheryl's Birthday

Postby Vytron » Sun May 03, 2015 11:34 pm UTC

In the variant, the answer isn't August 17, it's

Spoiler:
June 17


However, the same reasoning that allows for that gives August 17 as the answer to the original Puzzle.

Spoiler:
Imagine this isn't a puzzle, but a real world scenario, and you are Albert.

You are given June, and face off with Bernard. You just wait. He doesn't do anything.

So, why isn't he doing anything? Because he doesn't know the date! If the date was June 18, on the face off Bernard would be saying "I know the date", but he isn't. The silence itself gives you enough information to figure out it must be June 17.

The same reasoning would allow you to figure out it was August 17 in the original puzzle, because you can rule out dates just because Bernard isn't claiming he knows the date, so you can rule out more dates that you'd do just by looking at the month you got.

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Re: Cheryl's Birthday

Postby TheGrammarBolshevik » Mon May 04, 2015 12:52 am UTC

Spoiler:
Couldn't you get the same puzzle, but without the implicit "Bernard would speak up if he knew anything" rule, if you just have Bernard first say that he doesn't know?
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Re: Cheryl's Birthday

Postby Vytron » Mon May 04, 2015 1:20 am UTC

Spoiler:
The point is Albert doesn't need Bernard to say he doesn't know to know what's the date of the birthday. Bernard saying he doesn't know is superfluous.

It helps to take a look at base scenarios:

Scenario 1:
The date is May 19.

Bernard gets a 19. He knows that the month must be May. He knows the full date.

Face off:
In the Faceoff Bernard says "I know that Albert does not know the date, but I do!"
Albert looks at his month, May. Now he thinks:

Suppose Bernard got a 15 or a 16. If he got the former, he wouldn't know the date because it could be May or August. If he got the latter, he wouldn't know the date because it could be May or July. Therefore, he got a 19.

Albert says: "At first I didn't know when Cheryl’s birthday is, but I know now."

Scenario 2:
The date is June 18.

Bernard gets a 18. He knows that the month must be June. He knows the full date.

Face off:
In the Faceoff Bernard says "I know that Albert does not know the date, but I do!"
Albert looks at his month, June. Now he thinks:

Suppose Bernard got a 17. He wouldn't know the date because it could be June or August. Therefore, he got a 18.

Albert says: "At first I didn't know when Cheryl’s birthday is, but I know now."

Scenario 3:

The date is June 17.

Bernard gets a 17. He doesn't know if the month is June or August.

Face off:
In the faceoff, Barnard stays quiet.
Albert knows that if this was Scenario 1 or 2, Barnard would have spoken, so he knows it's not May 19 or June 18. His month is June, so he knows it must be June 17.

Albert says: I know that Bernard does not know the date, but I do!

Bernard knows that if Albert had August, it'd be impossible for him to know the date, so it must be June 17.

Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.

--------

This is like the blue eyed puzzle: With two blue eyed islanders, if one doesn't see the other leave, they know their own eyes are blue, and the other islander didn't need to say anything. In this case, Bernard didn't need to say anything.

As you keep adding scenarios, you realize August 17 is the answer that is most consistent with a real world scenario for the original puzzle (ie. if you were Albert and it actually happened, after being given August the main OP would happen as described, because you would know Bernard doesn't have 18 or 19 or he'd be claiming that he knows the date on the faceoff).

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Re: Cheryl's Birthday

Postby TheGrammarBolshevik » Mon May 04, 2015 5:34 am UTC

Spoiler:
Vytron wrote:This is like the blue eyed puzzle: With two blue eyed islanders, if one doesn't see the other leave, they know their own eyes are blue, and the other islander didn't need to say anything. In this case, Bernard didn't need to say anything.

As you keep adding scenarios, you realize August 17 is the answer that is most consistent with a real world scenario for the original puzzle (ie. if you were Albert and it actually happened, after being given August the main OP would happen as described, because you would know Bernard doesn't have 18 or 19 or he'd be claiming that he knows the date on the faceoff).

I don't see why, in a real-world scenario (as if this would ever happen in the real world anyway), Albert would assume that Bernard would be saying he knows the date if he knew it. There is nothing in your description of the problem that says anything about a "face-off" or about Albert and Bernard sitting down and telling each other if they know when Cheryl's birthday is. And that interpretation doesn't really seem natural for problems like this, especially since it wasn't the intended interpretation of the original problem.

The comparison to the blue eyes puzzle seems out of place. In the blue eyes puzzle, it's specifically stipulated what each islander will do if she finds out her eye color. There's nothing in your puzzle that says "Once Albert or Bernard finds out Cheryl's birthday, he will say so.

Vytron wrote:The point is Albert doesn't need Bernard to say he doesn't know to know what's the date of the birthday. Bernard saying he doesn't know is superfluous.

I feel that my point isn't fully clear. You said that you like the interpretation that gives August 17th as the answer, because it allows for the alternative puzzle that you present. But my point is that it's just as easy to present that alternative puzzle while following the intended interpretation. You just have to have Bernard say that he doesn't know, first.

Sure, your way means that it isn't necessary for Bernard to say it out loud. But that's only because you've taken the information that Bernard would have conveyed by speaking, and instead stuffed it into a counterintuitive implicit premise.
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Re: Cheryl's Birthday

Postby Moonbeam » Mon May 04, 2015 6:57 am UTC

Vytron wrote:In the variant, the answer isn't August 17, it's

Spoiler:
June 17


However, the same reasoning that allows for that gives August 17 as the answer to the original Puzzle.

Spoiler:
Imagine this isn't a puzzle, but a real world scenario, and you are Albert.

You are given June, and face off with Bernard. You just wait. He doesn't do anything.

So, why isn't he doing anything? Because he doesn't know the date! If the date was June 18, on the face off Bernard would be saying "I know the date", but he isn't. The silence itself gives you enough information to figure out it must be June 17.

The same reasoning would allow you to figure out it was August 17 in the original puzzle, because you can rule out dates just because Bernard isn't claiming he knows the date, so you can rule out more dates that you'd do just by looking at the month you got.


That's why I was confused, 'cos you said:

Vytron wrote:August 17 is my favorite answer, because it allows for the following variant: .........

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Re: Cheryl's Birthday

Postby Vytron » Mon May 04, 2015 8:58 am UTC

Spoiler:
@TheGrammarBolshevik:

So, suppose there's an alternative puzzle, that works identically, except:

Albert and Bernard want to marry Cheryl, so she holds a competition where, the first person to guess her birthday gets to marry her.

The events happen identically as in the OP, so Bernard got to marry her because he guessed her birthday first. Albert may facepalm now because had he stayed quiet, Bernard wouldn't have figured it out.

Now, what was Cheryl's birthday? It had to be August 17.

If you disagree, let's go back to my variant.

Exactly like above, but now Albert gets to marry her, because he managed to figure it out first, because the fact that Bernard isn't claiming that he knows eliminated the only alternative to June 17.

And now, let's get it a step further:

You claim that if Bernard doesn't have to make any announcement at all, in the original puzzle, the answer must be July 16.

I have shown that if both Albert and Bernard are interested in announcing they know the birthday ASAP, the answer must be August 17.

So, suppose Cheryl hands Albert the month, and Bernard the day, then she flips a coin, if it's tails we get our original puzzle, and if it's heads we get the alternative one where they announce ASAP.

She gets some result, tells them about it, and this conversation takes place:

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.

Now, you can't say "if she flipped tails then the answer is July 16, and if she flipped heads it's August 17", because she told them the month and date before flipping the coin!

So the only answer that fits both scenarios is August 17, so August 17 is an answer that works for the original puzzle where she flips a coin to decide whether she goes with it or not, and she does go with it, which doesn't change the nature of the puzzle.

But my point is that it's just as easy to present that alternative puzzle while following the intended interpretation.


What makes the variation cool is that the logic used to solve the original puzzle doesn't work for the variation if Bernard stays quiet, so both Albert and Bernard never say anything and they never figure anything out.

Nobody expects Albert to jump and announce that he knows the birthday with the information given, because all the months have more than one option. So given the scenario someone using that logic would go "that's impossible, because Albert doesn't have enough information to have figured out Cheryl's birthday", and yet:

If a one million dollar prize was offered to the first person to figure out Cheryl's birthday, and there were a bunch of Alberts, and a bunch of Bernards, competing to get it, and I was given June, the very fact that no Bernard announces that they know the birthday immediately tells me it's June 17, for certain, and I can get the prize by announcing it before the other Alberts.

Albert's or Bernard's intentions don't change what Cheryl gave them, and that's why I like August 17 as a better answer.

(as if this would ever happen in the real world anyway)


We're meant to imagine these puzzles as hypothetical scenarios, wouldn't we? So we get to imagine what would happen if they actually happened.

Like, I show you a real life video of the events and tell you that Albert and Bernard were being truthful at all times, and ask you if you can figure out Cheryl's birthday.

You say it's July 16 but it turns out Cheryl's actual birthday is August 17, and then I show you how Albert knew Bernard didn't know because he had August, so he knows Bernard can't have a 19 or a 18, because there's no August 19 or August 18. He claims so, so Bernard knows it can't be June (if it was June 17 then Albert would have claimed he already knew it, as in the variant). There's no May 17 or July 17, so he now knows it's August 17. And after he claims he didn't know, but he now does, Albert does as well.

This is a plausible explanation of the conversation recorded on video.

The comparison to the blue eyes puzzle seems out of place. In the blue eyes puzzle, it's specifically stipulated what each islander will do if she finds out her eye color. There's nothing in your puzzle that says "Once Albert or Bernard finds out Cheryl's birthday, he will say so.


The thing is, we're not following what these agents are doing and analyzing whether they''re behaving as we expect or not. The events related are past events. They happened (in the hypothetical). So in the variant, if what already happened was that Albert managed to deduce Cheryl's date without Bernard speaking (or because of it) and there's an explanation, the only explanation is that it's June 17, and this possibility would enable Bernard to know it's August 17 in the original puzzle just because he has a 17 and knows if it was June Albert wouldn't be saying that he doesn't know.

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Re: Cheryl's Birthday

Postby Xanthir » Tue May 05, 2015 5:02 pm UTC

Back to the original puzzle, I finally understand how people can get the Aug 17 date. It's because they forget that Arnold and Bernard already know their pieces of information.

When they interpret Arnold's first statement as meaning "we can rule out May 19 and June 18, because then Bernard would know", they're thinking at their own level, where they don't know the month. Arnold, tho, does know the month - it's August! (According to their solution.) So Bernard saying "I dunno lol" at the beginning gives Arnold exactly zero information - he already knows it's August, and all of the August days show up in other months too, so of course Bernard doesn't know yet. The rest of the logic is similarly broken.

In other words, they're interpreting the statements as:

1. If you knew the month or the day, you still couldn't tell what the date was. (This rules out May 19 and June 18.)
2. Based on the remaining possible dates, if you knew the day you could tell what month it was, but not vice versa (this rules out June 17).
3. Based on the remaining possible dates, if you knew the month you could tell what day it was.

In other other words, they're thinking of the problem as a third person, Dora, who wasn't told either the month *or* the day, and is asking Albert and Bernard in turn whether they know it or not. Neither Albert nor Bernard is allowed to do any non-trivial logical deduction; they just have to look at Dora's chalkboard, where she's crossing out possible dates, and answer whether, if Dora knew their piece of information, there'd only be one uncrossed date left.

Dora has less information than either Albert or Bernard, so it's not too surprising he'd get a different answer when she tries to interpret their statements (which are based on their actual knowledge). What is somewhat surprising is that she ends up with a single possibility!
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