Birthday Logic Question

Factual Questions
81

That is correct. He cannot say that. The most he could say is “I don’t know when Cheryl’s birthday is, but I don’t know (for certain) that Bernard doesn’t know” if Albert were given May or June as the month.

This is incorrect. If Albert gets May or June he cannot factually say for certain that Bernard doesn’t know the date. There is the chance that Bernard has been given the number 17 or 18 for the date. If Bernard has, then Bernard knows the date.

Work it out. You are Albert. Cheryl tells you “May.” Can you say that Bernard doesn’t know the date? You cannot make that statement if you are given May. You are given “June.” Can you factually make that statement? Once again, no you cannot. Only July and August satisfy the conditions where you can make that a true statement always.

82

Nice try, but nope. Those two statements are, strictly speaking, opposites, but it doesn’t follow that Albert has to say one or the other. For one thing, there’s nothing compelling him to speak to this question. But if he did wish to, and he’d been told one of the months in which he can’t make his deduction, he could say:
“I don’t know when Cheryl’s birthday is yet, and I don’t know if Bernard knows or not.”

That would actually provide significant information to Bernard, assuming that he knows Albert well enough that he would have followed through a deduction if he had sufficient information. Given that Albert does not have sufficient information to answer the question, Bernard would be able to deduce that Albert had been told–umm, either May or June, right?

83

I’ve had my third morning cup of tea. To work!

Let’s Brute force this. A for Albert B for Bernard.

The first A statement AS1 is I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.

Options:

A is told May. May has dates 15,16,19 so AS1 is true.
A is told June. June has dates 17,18. so AS1 is true.
A is told July. July has dates 14,16 so AS1 is true
A is told August. August has dates 15,17 so AS1 is true.

There are no other options.

B already knows all of the above, so as far as B is concerned no new information has been provided by AS1.

The first B statement BS1 is *At first I don’t know when Cheryl’s birthday is, but I know now. *

Now B simply can’t say that because the no new information has been provided by AS1.

Therefore the problem is flawed. Q.E.D.

84

How is this true if B is told 19 or 18?

ETA: Read through nolongerlurking’s thorough brute force walkthrough. I can’t quite figure out what you’re missing, but that brute forces all the statements that A can make about B’s knowledge at each point in time (and vice versa) and leaves only one possible date to satisfy all the conditions.

85

Ok, let’s try a different tack. Let’s reduce this to near-triviality.

There are three cards:

A of Hearts, A of Clubs
K of Hearts

If Albert says “I know the card.” Then Bernard can say “I didn’t originally know the card, but now I do.” We don’t even need to go that far, though. As soon as Albert says “I know the card,” we, the puzzle solvers, know it’s the King of Hearts.

Do you follow that?

If Albert says “I don’t know the card” and then Bernard says “I didn’t originally know the card, but now I do,” then we, the puzzle solvers, know the card is the Ace of Hearts. If Bernard said “I knew the card all along,” then we, the puzzle solvers, know the card is the Ace of Clubs.

Can you follow that?

86

And another fool leaps into the fray !!!

Ref OP’s post 83, you seem to be assuming there are only two possibilities for A’s opening statement. That’s wrong. There are three distinct non-overlapping possibilities.

When A makes his statement, there are three possibilities:
[ul][li]A can correctly work out that B positively has enough information to know for sure the birthday.[/li][li]A can correctly work out that B positively lacks enough information to know for sure the birthday.[/li][li]A can correctly work out that he, A, can’t say anything for sure about B’s knowledge of the birthday. Maybe B knows enough, and maybe B doesn’t, but A can’t tell which.[/li][/ul]
And when A comes to his conclusion and shares it with B, B then gains info. And when B replies, A in turn gains a tidbit of info.

87

Assume B for Bernard was told one of 14, 18, 19. He would know the Birthday at once, and save for his impassive Chinese face all around would know it.

A for Albert knows that 14,18,19 are dates that uniquely identify a month, so he can’t have been told any month where this is possible - i.e for him to then definitely say B doesn’t know.

So A must have been told August and B told 15 or 17.

By deduction B now knows the date (15 or 17 August) but his statement that he now knows the date doesn’t convey any information to A to differentiate between 15 and 17

88

14 is not a unique day. It occurs in both July and August as a possibility. Read your OP again.

ETA: Here are the dates, once again:

May 15 May 16 May 19
June 17 June 18
July 14 July 16
August 14 August 15 August 17

Or has the question changed somewhere and I didn’t notice?

89

May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17.

AS1: *I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too. *

-> A has a month with no singleton days (i.e a day number that only occurs once in the entire problem set). If a month did have a singleton day then A couldn’t be sure B hadn’t been given it.

May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17

BS1: At first I don’t know when Cheryl’s birthday is, but I know now.

-> B has a singleton date in either July August. i.e. not 14.

May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14,** August 15, August 17**

AS2: Then I also know when Cheryl’s birthday is.

-> A has a month with only one possible singleton date (not two)

May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17

90

By George, I think 'es got it.

91

Next:

Three logicians walk into a bar…

The bartender asks, “Do all of you want a drink?”

The first logician replies, “I don’t know.”

The second logician says, “I don’t know.”

The third logician says, “Yes.”

How did he know? Spooky. :eek:

92

Should we now move on to an island full of blue eyed and brown eyed people?

93

nitpick -
English is one of four official languages of Singapore

94

Common enough - I know an Albert, Bernard and two Cheryls in Singapore.

But for bonus points

Who would like to rewrite the question with decent grammar and better clarity?
(yes - it’s being slammed here for it’s grammar)

95

*its grammar. :stuck_out_tongue:

96

Anyhow, I might write it as such, if I want to be very clear and pedantic:

Albert and Bernard are perfect logicians. They just became friends with Cheryl and want to know when her birthday is. Cheryl gives them a list of ten possible dates:

[dates]

Cheryl then separately/privately tells Albert the month of her birthday and Bernard the date of her birthday.

Albert: I don’t know when Cheryl’s birthday is, but I also know that Bernard does not know.
Bernard: At first I didn’t know Cheryl’s birthday, but I know now.
Albert: Then I also know when Cheryl’s birthday is.

Assume that everyone is truthful. When is Cheryl’s birthday?

97

You might also state that they each know what type of information the other has been given.

98

For extreme overweening precision (something I’m infamous for) I’d make the following small adjustment:

Italics mine.

Otherwise, especially with Richard Pearse’s addition your restatement is a thing of unambiguous beauty.

99

Now that jezzaOZ got it, can we revisit my question: How are these types of questions/problems devised? Are they logical/deductive-reasoning scenarios that are dressed up a bit, or does someone encounter this scenario in real life and realize it makes for a good puzzle? How does someone concoct such a scenario?

100

I’d seen a version of this problem done more mathematically before, where a mathematician, M, thinks up two whole numbers, a and b, between 3 and 90-ish. To one other mathematician, P, he tells the product ab, and to the other, S, he tells the sum a+b.

P says, “I don’t know what the numbers are”.
S says, “I already knew that you didn’t know. I also don’t know”.
P says, “now I know what they are”
S says, “now I know, too”

What are the numbers?

I suspect that the current problem is a version of that one adjusted to have less math.