# Explaining Cheryl's birthday

I understand the solution. But it feels wrong to me, because Albert cannot possibly make the third statement with the information he has at the moment he makes it.

Hear me right : if he does, and if it is true, then we know for sure the answer is what it is. But if I understand correctly, he can't.

The first statement eliminates May and June for everyone. Bernard, had he been told "14th", could still not deduce the correct date. But he was not told the 14th. So he can make the second statement.

Then, the only thing Albert can deduce is "Bernard has been told 15th, 16th or 17th"

These three dates are enough, with the first statement, to fully deduce the date.

Bernard is able to make the second statement because he as a piece of information Albert does not, and the second statement only provides little information to Albert.

I say again, if Albert actually makes the third statement just for the sake of the problem, and if the statement is true, I do understand why the solution is July 16th. But Albert, with the information he has after the second statement, cannot make the third statement.

Let's take an example : let the date be August 15.

Albert can still make the first statement. Bernard can still make the second statement, because "15" (initial information he has) and "not May/June" (which he can deduce from first statement) are enough to deduce August 15 at this point for him. Albert cannot make the third statement.

Now, the date is of very little importance. let the date be July 16th again. Albert can make the first statement. Bernard can still make the second statement, because "16" (initial information) and "not may/june" are enough for him to deduce the date. But the second statement only tells Albert "Bernard was not told 14th".

What would be a good answer to this question ? There are two possible good answer for my question. The first one would be "you're just supposed to accept the clues and not question them". The second would be a detailed explanation of why Albert can make the third statement righteously.

How can Albert, with the information he has after the second statement, make the third one?

• look at lorisN's answer! puzzling.stackexchange.com/q/12022/9420 Apr 16 '15 at 9:09
• I did, and when I tried to solve the problem, I drew the exact same table. But lorisN says "Since Albert states that now he knows"... I want to know How can Albert know with the information he has, not "how can we deduce from the third statement which date it is" Apr 16 '15 at 9:11
• as 14 excluded and August has two value so it is also excluded ! finally only july 16 is remaining Apr 16 '15 at 9:15
• Let's say Cheryl's birthday is July 16. At the start of the game, then, Albert was told that the correct month is July. There are only two possible dates in July: the 14th and the 16th. So, once Albert deduces that 14 cannot be correct, he is left with only the 16th. Apr 16 '15 at 11:04
• @Kalissar "Albert can deduce it's not 14. Why does that leave only 16?" Because Albert knows it is July. That's what he was given at the start. There are only 2 possibilities for July: 14th and 16th. If you can eliminate 14th, that leaves 16th. Apr 16 '15 at 16:09

How can Albert, with the information he has after the second statement, make the third one?

In your post, you have already accepted that Bernard was not told the 14th. This leaves the following numbers: 15, 16, 17, 18, 19. However, since Albert was told "July", before Bernard has even said anything, Albert has eliminated 15, 17, 18, and 19. This leaves only July 16.

Note: What follows is a complete description of the logic for the remainder of the puzzle. Cheryl's birthday is July 16. At the beginning of the puzzle, this is unknown to Albert and Bernard (and us as spectators).

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

Albert knows: Cheryl's birthday is in July. It is either 14 or 16.

Bernard knows: Cheryl's birthday is the 16. It is either July or May.

Albert knows that it is not the 18 or 19, and therefore also knows that Bernard does not know the birthday without Albert's help. This is how he comes to the conclusion he states first:

Albert: I don't know when Cheryl's birthday is [it is either July 14 or July 16], but I know that Bernard does not know, too [because it is certainly not May 19 or June 18].

Bernard knows that the month could only be May or July (since he was told 16). Had Albert been told "May", Albert's initial reasoning would have instead produced the statement:

Albert: I don't know when Cheryl's birthday is [it is one of May 15, May 16, or May 19], and I also don't know if Bernard knows [because it might be May 19].

Because this was not Albert's initial statement, Bernard knows that he must have been told "July", leading him to join together with his own information and conclude that it is July 16.

Bernard: At first, I didn't know when Cheryl's birthday is [because it could have been in May or July], but I know now [because Albert's information allowed me to eliminate May as a possibility].

With Bernard's conclusion, Albert is given the information he needs to rule out July 14. Let's consider for a second that Bernard might have been told "14". Had this been the case, Albert's initial statement wouldn't have been enough to let Bernard choose between July and August. This makes Bernard's second statement:

Bernard: At first, I didn't know when Cheryl's birthday is [because it could have been July or August], but I still don't know [because Albert's information has not narrowed my selection, it has only eliminated the two months that I had already ruled out myself].

Because he didn't say this, Albert knows it must be 16 of the month he was told.

Albert: Then I also know when Cheryl's birthday is [and that is July 16].

I think we can verify which ones are the incorrect answers by using reductio-ad-absurdum on all the possible dates.

The date which will not produce a contradiction or logic leap (eg: "Why can we exclude May/June dates?" or from above "Albert: I don't know when Cheryl's birthday is [it is either July 14 or July 16], but I know that Bernard does not know, too [because it is certainly not May 19 or June 18]." --- why not [because Bernard thinks May 16 or August 14]) is August 17.

Using August 17 as the DOB.

Albert knows August. Bernard knows 17.

Albert can't figure out the DOB because there're possible 3 options (all August dates).

Albert can figure out that Bernard doesn't know because: 1. June 18 and May 19 are out of questions (they're not in August). 2. Bernard has to choose between June 17 and August 17.

That's why he could make his first statement about himself and Bernard. Bernard or Cheryl didn't have to tell Albert about Bernard's not knowing Cheryl's DOB.

After Albert made his first statement, Bernard knows: 1. June 17 is not correct because if this the DOB, then Albert must have figured it out and said it.

Bernard now only has one option August 17.

That's why Bernard could say what he said.

Based on what Bernard said, Albert knows that Bernard must have discarded June 17 (because if this the DOB, Albert would have said it in his first statement) and chosen August 17.

So as far as I can see, your thinking is along the lines of:

• If the date is July 16th, A and B can make statements 1 and 2.
• However, if the date is August 15, A and B can also make statements 1 and 2.
• Therefore, A, having heard statements 1 and 2, can't distinguish between the possibility of July 16th and the possibility of August 15th, since these lead to identical statements.

The first two points are true. However, you're forgetting that A has an additional piece of information: he still knows the month.

So if the date was August 15, A would have heard statements 1 and 2 and know the month was August, whereas if the date was July 16, A would have heard statements 1 and 2 and know the month was July. Because of this, he's able to distinguish between the two situations.

In fact, of all the remaining dates after hearing statements 1 and 2, only one has month July, which is how A is able to pinpoint the exact date and make the third statement. If the date HAD been August 15th, A would have to admit that he still doesn't know what the correct date is (as it could be August 15th or 17th).

Albert: "I don’t know when your birthday is, but I know Bernard doesn’t know, either."

The first half of the sentence is obvious — Albert only knows the month, but not the day — but the second half is the first critical clue.

The initial reaction is, how could Bernard know? Cheryl only whispered the day, so how could he have more information than Albert? But if Cheryl had whispered “19,” then Bernard would indeed know the exact date — May 19 — because there is only one date with 19 in it. Similarly, if Cheryl had told Bernard, “18,” then Bernard would know Cheryl’s birthday was June 18.

Thus, for this statement by Albert to be true means that Cheryl did not say to Albert, “May” or “June.” (Again, for logic puzzles, the possibility that Albert is lying or confused is off the table.) Then Bernard replies:

Bernard: "I didn’t know originally, but now I do."

So from Albert’s statement, Bernard now also knows that Cheryl’s birthday is not in May or June, eliminating half of the possibilities, leaving July 14, July 16, Aug. 14, Aug. 15 and Aug. 17. But Bernard now knows. If Cheryl had told him “14,” he would not know, because there would still be two possibilities: July 14 and Aug. 14. Thus we know the day is not the 14th.

Now there are only three possibilities left: July 16, Aug. 15 and Aug. 17. Albert again:

Albert: "Well, now I know, too!"

The same logical process again: For Albert to know, the month has to be July, because if Cheryl had told him, “August,” then he would still have two possibilities: Aug. 15 and Aug. 17.

The final possibilities are July 16, August 15,August 17. I hope everyone is clear about that.

If Albert was told August .He would be left with 2 options August 15 and August 17. So he could not have made the final statement.

In that case he would would have said something like "It must be Aug15 or Aug 17"

And Lets imagine he was told July

Then He would know Its July 16.

He would say "Now I know it".

Since this is exactly what he said.We can infer that the B'day is July 16 provided
Albert is not "stoned" or has lost his mind.

The same question I also got, and in the event of searching for an answer I routed here. Finally I figured out myself.

Albert knows the month (July/August)

Since Albert knows the month that is July but he does not know the date, after 2nd statement he was 100% sure that he knows the answer,becuase July having 2 dates(14, 16), if it is 14 then Bernard could not make the second statement so it should be 16.

If it is in Auguest then it is hard for Albert to make the third statement as there are two dates 15 and 17.

Hope this makes clear.. :)