Birthday Logic Question

Factual Questions
1

Singaporean high-school students were presented with this logics problem in their Singapore and Asian Schools Math Olympiad test paper last week.

**The question **

Albert and Bernard just become 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?

(I’ve seen the ‘solution’ and don’t agree it is correct as it assumes more knowledge by the protagonists than stated, or false logic by them)

2

16th July.

  1. Albert knows that Bernard does not know.

This eliminates May and June as they are the two months with unique dates.

  1. Bernard now knows the birthday.

It can’t be the 14th, or Bernard would still not know. It must be the 15th, 16th, or 17th.

  1. Albert now also knows the birthday.

It must be the 16th of July. If Albert had been told August then he would still not know if it was the 15th or 17th, seeing as eliminating the 14th allows Albert to know, it must be July, therefore must be the 16th.

3

Reading the ‘result’ on the internet doesn’t change the fact that the answer assumes that the protagonists know more than they are told and are not acting logically.

How can Albert know what Bernard knows or doesn’t know? All Albert knows is Bernard was told one of several days. It doesn’t matter if they were unique or not. It’s a logic error to dismiss unique dates as per the answer.

4

Albert knows which month it is. He also knows the possibility for the days of the month that is Cheryl’s birth month, are not unique, meaning that there are other months in the list that share the same day as the day that Bernard has been told. So Albert assumes that Bernard must not know which month it is, since the day he knows is not unique.

5

Albert never said that Bernard didn’t know to the month. He just said he didn’t know the date (month and day) - which is always true so conveys no information. Hence the rest of the ‘proof’ fails on the first hurdle.

In fact Albert could never have said anything stronger because that would imply he know some of what Bernard knew privately.

6

Bernard doesn’t know the date.
But, Bernard does know the day.
Therefore, the part of the date that he doesn’t know must be the month.

7

And how does that advance knowledge? In particular to Bernard for the next part?

All you have stated is public knowledge and in no way makes a sudden change in Bernard’s appreciation of the problem.

8

Yes it does. Albert is not magically deciding what Bernard knows or doesn’t know. Albert is using the pieces of information that he has to deduce what Bernard knows. If Albert had gotten a different piece of information, then it might not have been possible for him to say what he did.

For example, say Albert had heard the month “May.” Then Albert knows that Bernard has “15,” “16,” or “19.”* And because it is possible, in this case, for Bernard to have “19,” it is possible for Bernard to know the whole date because there are no other dates that have a “19” in them. That is, if there is a “19,” there must also be a “May.” As a result, if Albert hears “May” then he cannot be sure that Bernard does not know the whole date.

So, by this logic, “May” is out of the running. And so is “19.”

*If he doesn’t have any of these days, then the problem is broken.

9

There are two days Bernard could have been told that would have identified the birthday without having to be told the month. If he had been told “the 19th” he would know the birthday was May 19th and if he had been told “the 18th” then he would know the birthday was June 18th. Albert, of course, is smart enough to realize this.

If Albert had been told “May” or “June” then he could not assert that Bernard does not know the birthday as for all he knows Bernard has just been told “the 18th” or “the 19th”. However, once Albert has been told “July” or “August” he knows Bernard has not been given one of those days and so he knows that the day that Bernard was given is not enough information by itself. That is how we (and Bernard) know that the month is July or August.

10

Once we get rid of “May” and “June” in the way I’ve described earlier, we now have a different problem.

Instead of choosing among: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, and August 17,

we now get to choose among: July 14, July 16, August 14, August 15, and August 17.

Bernard knows this for the same reason Albert does. Bernard is following the the same reasoning we did to eliminate “May” and “June.”

So, now Bernard knows that the 10 dates have been reduced to 5. And, with that information combined with the day that he was told at the beginning, he can figure out the whole date. Because he knows that with possible 10 dates, there are at least 2 months with the day he was given, but with 5 dates, there aren’t. This is the only way he can know the date for sure. What we are to do here is figure out when this can happen.

We proceed one day at a time.

If he has “14,” then he can’t know the full date since it could be “July 14” and “August 14.” So, this can’t be it (or Bernard is lying).

If he has “15,” then the date must be “August 15.”

If he has “16,” then the date must be “July 16.”

If he has “17,” then the date must be “August 17.”

In any of these last three cases, Bernard knows the date.
And so now, the 10 possibilities have been reduced to 3. The date must be July 16, August 15, or August 17.

Now we use the last clue, which is that Albert knows the date when there are only 3 possibilities left, even though he was only told the month. This is not possible if Albert was told “August,” because of the three possibilities left, two of them are in August.

Therefore, if Albert is to know the date at this point, it must be in July.

11

But Bernard has a range of days and whether or not they are specific to a month doesn’t alter the fact he doesn’t know which month or day.

Albert saying Bernard doesn’t know the exact date adds nothing to Bernard’s understanding. The statement would be true whether or not Albert knew any months at all.

12

How about fill out what Albert says more clearly…
Albert said " Bernard definitely cannot know the month as he does not have enough information ! " … Not enough information until he hears this from Albert, that is.

… How can ALbert know this ? If it was a month with a unique date, then Bernard MIGHT or MIGHT NOT KNOW the month… So what Albert said means it cannot be a month with a Unique date. He may as well have said “Its a month without a unique date”, its logically the same information.
Bernard knows the day number of course, and there is only one month that it can be then !. Since Albert knows that is the only way Bernard can deduce the month correctly, Albert deduces the date… It can’t be the date that leaves ambiguity, so it must be the date that is unique when considering the months specified by Albert.; Bernard might have said" good thing the day number is unique NOW!!"

13

Okay, let’s do it the long way.

If Cheryl had whispered “19” to Bernard, then he would instantly know that her birthday is May 19th, because May 19th is the only “19” on the list. Similarly, if she had whispered “18,” Bernard would know that her birthday is June 18th, because June 18th is the only “18” on the list. These are the only unique numbers in the list - if she tells Bernard “14,” “15,” “16,” or “17,” then he can’t know which month she was born in, because each of those numbers appears in at least two months.

Now for Albert. If Cheryl whispers “May” to Albert, it raises the possibility that she whispered “19” to Bernard - in which case he would know her birthday is May 19th. Similarly, if she whispers “June,” it raises the possibility that she whispered “18” to Bernard - in which case he would know that her birthday is June 18th. So if Albert knows that Bernard doesn’t know her birthday, it can only be because those possibilities were eliminated entirely just by knowing the month - in other words, she didn’t tell Albert “May” or “June.” That leaves July and August.

Back to Bernard. He, being almost as clever as a Doper, has worked out all of the above from what Albert said. So now let’s game out the other dates:

If Cheryl told Bernard “14,” he still can’t tell what month she was born in, because July 14th and August 14th are both on the list.

If Cheryl told Bernard “15,” then he now knows she was born on August 15th, because that’s the only 15 on the list.

If Cheryl told Bernard “16,” then he now knows she was born on July 16th, because that’s the only 16 on the list.

If Cheryl told Bernard “17,” then he now knows she was born on August 17th, because that’s the only 17 on the list.

So if Bernard declares confidently that he now knows Cheryl’s birthday, it’s because it’s either August 15th, July 16th, or August 17th.

Back to Albert. He, being almost as clever as Bernard, has also worked out all of the above possibilities, and confidently declares that he, too, now knows Cheryl’s birthday. If Cheryl had told him “August,” he wouldn’t be able to do that, because there are two possible dates in August, the 15th and the 17th. So if Albert knows Cheryl’s birthday, it can only be because she told him “July,” thereby eliminating the dates in August. This leaves the one remaining date - which, huzzah, is in July - July 16th.

14

Wrong. The possible day numbers usually map to several different months.

15

The question doesn’t mention whispers…

16

Except when they don’t. The entire point of the exercise is to figure out when that doesn’t happen. Or at the very least, when that isn’t consistent with Albert’s and Bernard’s statements.

You appear to believe that it is impossible for anyone to figure out what people know and don’t know, simply based on what they said. But we do know what they know, because they are telling us. Now the way they are telling us is a bit vague (and it does have a certain logic-puzzle-ness to it), but they are nonetheless telling us something.

Here’s a simpler problem in much the same vein:

Take the ten numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Write each of them on a ball, and drop each of the ten balls in a hat. I pick one and then you pick one. If I show a “2,” then I know, even before you’ve said anything, that your number is odd. If I don’t show a “2,” then I don’t know this, but if I do show a “2” then I do.

Moreover, if I say, “I know that you have an odd number,” then you know something, too. You know that I have “proof” that you have an odd number. And the only way that can happen is if I have a “2.”

17

2 is even…

18

2 is even, and all the other numbers are odd.

So, if I have the 2, then I know that you don’t. And if I say that you have an odd number, then you know that I have the 2.

19

I think the straightforward example people are used to dealing with is cards. If I’m playing Poker, and I have 4 aces (lucky me), then I can be sure nobody else has an ace.

In fact, if an ace was played last turn (assuming no reshuffling between hands), we can be sure that nobody has 4 aces, and that the ace of that particular suit is out of play.

20

rehash of this problem posted earlier: