# When is Cheryl's Birthday? [duplicate]

I saw this question on Facebook:

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 16 19

June 17 18

July 14 16

August 14 15 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 didn't know when Cheryl's birthday is, but I know now.

Albert: Then I also know when Cheryl's birthday is.

When is Cheryl's birthday?

There is an official solution which is explained in the link below, but I don't understand the logic.

July 16, as explained here. https://www.facebook.com/kennethjianwenz/posts/386479228197631

What is wrong with the logic in the answer I posted below (I put it in spoilers if others want to try)?

## marked as duplicate by leoll2, Community♦Apr 14 '15 at 11:28

• – Joe Z. Apr 14 '15 at 5:37
• @JoeZ. there's one which is exactly the same but with different numbers, it was about 2 students guessing the birthday of their professor with about the same dialogue. It's convoluted logic but fun none the less. I'll see if i can find the question – Vincent Apr 14 '15 at 9:33
• Just wondering how this duplicate question got 13k views in 12 hours... – leoll2 Apr 14 '15 at 17:50
• This question is viral on social media. thats why .. – Himanshu Apr 14 '15 at 18:34
• Can someone suggest a website or book that has bunch of such questions? EDIT: puzzling.stackexchange.com/questions/tagged/logic-puzzle is a good source. More suggestions are welcome. – Mustafa Apr 15 '15 at 3:16

I will try to help you understand the answer. It is important to note that Albert and Bernard both know the answer before we do.

With Albert's first statement, it is clear to us that Albert was told either July or August. Had he been told May or June, he would not be able to state definitively that he knew Bernard didn't know. (May 19 and June 18 both could be uniquely identified immediately by Bernard without Albert's help, so in order for Albert to know that Bernard doesn't know, the month he was told must not be May or June.)

Because Bernard has been able to identify Cheryl's birthday after Albert's statement, he must not have been told 14. Because Albert's statement revealed July and August, had Bernard been told 14, he would still be unclear on the date. (If he said he didn't know, then Albert would know the birthdate, but Bernard would never be able to deduce it.)

We still don't know whether it is July 16, August 15, or August 17.

However, since Albert (who was only told the month) is able to state that he also knows the birthday, he must not have been told August. Had he been told August, he would be unable to decide if Bernard was told the 15th and 17th.

It is only after the third statement that we (as outsiders) can deduce the birthday. Bernard knew after the first statement and Albert knew after the second.

• Even though this isn't the highest rated answer, and a lot of them were similar, I picked this one because after reading it, I can finally understand why July 16 is the answer. – Fodder Apr 14 '15 at 11:40
• Thanks @RichardKennethNiescior, I don't know why I thought his name was Bertrand. How odd. – Ian MacDonald Apr 14 '15 at 13:56
• Here is an easy to follow picture without so many words bit.ly/1ys1Hea – D.Deriso Apr 14 '15 at 21:04
• @IanMacDonald I am sorry but I still having problem understanding your answer,How are we so clear that Albert was told July or August Had he been told May or June, he would not be able to state definitively that he knew Bernard didn't know. (May 19 and June 18 both could be uniquely identified immediately by Bernard without Albert's help 18,19 are unique but we also have 15 and 16 of May and 17 of June. – Suraj Singh Apr 15 '15 at 7:45
• @SurajSingh the trick is not that there are dates available, but rather that he stated that he knew that Albert didn't know. Since Bernard was only told the month, suppose that he was told "May". Looking at the set of dates that it could be, Albert could have been told 15, 16, or 19. Bernard doesn't know which number Albert was told, but he is able to confidently say that it wasn't 19. The only way he can confidently say this is if he were not told May as the month. A similar argument is made for June. – Ian MacDonald Apr 15 '15 at 12:11

Albert: I know Bernard doesn't know when Cheryl's birthday is.

Since May and June contain unique dates 18 and 19, then Albert cannot have been given one of those months, since if Bernard had received 18 or 19, then he would have immediately known (and therefore Albert wouldn't know if Bernard already knew Cheryl's birthday or not). So Cheryl's birthday must be in July or August.

Bernard: Now I know when Cheryl's birthday is.

Since Bernard didn't know before and now he does, he must have deduced some information about the date, which means the date can't be 14 because then Bernard would still have been stuck with the month being either July or August.

Albert: And now I know too.

Since Albert needs this information to discover which date it is, there must only have been one possibility remaining from eliminating 14 as a possible day, which is only applicable if the month is July. The remaining possibility is July 16, and so that's Cheryl's birthday.

• then Albert cannot have been given one of those months I don't get this. What if the date was May 15? – dwjohnston Apr 14 '15 at 5:33
• 18/19 only eliminates May 19 and June 18, it doesn't eliminate the month as a whole/ – dwjohnston Apr 14 '15 at 5:33
• If it were May 15, then Albert would know May, and in his perspective, there would be a 1/3 possibility that Bernard knew the full date already if it were 19. So he couldn't know for sure that Bernard didn't know the full date given just the number. – Joe Z. Apr 14 '15 at 5:33
• Yip, I get it now. – dwjohnston Apr 14 '15 at 5:37

Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know, too.

No matter what month is given to A he can not know the birthday. If A was given May: B was given 15 or 16, 19. But if B was given 19(not ambiguous), he could deduce the birthday so A was not given May.

If A was given June, B was given 17 or 18, again 18 is not ambiguous.

Conclusion: A was given July or Aug, in which all dates are ambiguous

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

So the date give to B may be 14, 15, 16 or 17. 14: Still ambigous as present in both months. So he had 15, 16 or 17 and now he knows the birthday

Albert: Then I also know when Cheryl's birthday is.

Now A can rule out 14, otherwise B could not know the date. Eliminating 14, 16 is the only date remaining in July but 2 more dates remains in Aug leading to ambiguity. As A knows the answer now, it must be July and can not be 14.

July 16 is Cheryl's birthday

• Your third step doesn't account for why the month can't be August either. – Joe Z. Apr 14 '15 at 5:39
• @JoeZ. OK, thanks. Let me add. Because Aug is left with 2 more dates leading to ambiguity – Mohit Jain Apr 14 '15 at 5:41

The simplest way is by putting the data into a table and crossing out months and days step by step

May -- 15 16 -- -- 19

Jun -- -- -- 17 18 --

Jul 14 -- 16 -- -- --

Aug 14 15 -- 17 -- --


Edited first paragraph for clarification due to comment:

Albert can only be 100% sure that Bernard does not know the date if and only if Albert was told either July or August. If he was given one of the other two months, there would have been a chance that Bernard knew the correct date (i.e. if he was told 18 or 19, since these two values are unambiguous as easily seen in the table) and Albert could not have made his statement. But, since Albert made his statement, Bernard can deduce (in the same way we just did), that Albert was told either July or August.

Both Albert and Bernard now know that it must be either July or August. Bernard now tells that he knows and, hence, the 14th can be excluded, since it is found in both months. If it was the 14th, he could not know.

That leaves three possibilities: Jul 16, Aug 15 and Aug 17. Since Albert states that now he knows, too, it only can be Jul 16, since it is the only unambigous combination of month/day left.

• But how does Albert know is the question? It makes sense how we can get to the answer from knowing that 'Albert now knows'. But how does Albert know is my question. – pnizzle Apr 17 '15 at 4:41
• Albert can only be 100% sure that Bernard does not know the date if and only if Albert was told either July or August. If he was given one of the other two months, there would have been a chance that Bernard knew the correct date (i.e. if he was told 18 or 19) and Albert could not have made his statement. – symphonic Apr 17 '15 at 7:46

The date I thought of was:

August 17

This is because:

Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know, too.

If Bernard was told 18 or 19, he would know right away, as they are unique dates. But he doesn't, so Albert can eliminate May 19 and June 18. If he was told June, then he would know it's June 17, but as he does not know at this stage, we know he wasn't told June. So we've eliminated three dates: May 19, June 17, and June 18.

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

Now that Bernard knows Albert doesn't know, Bernard knows that Albert was not told June. He can also eliminate those three dates. The only way for him to know based on this information which date is correct is if he was told 17, which would make August 17 the answer.

Albert: Then I also know when Cheryl's birthday is.

Now that Albert knows Bernard knows, Albert can also work out that Bernard was told 17, so he comes to the same conclusion.

• Since Albert knows the month, he can be certain that May and June are wrong. This is because if it was one of those months, Albert could not be absolutely sure that Bernard doesn't know the date due to the unique days. From that we can deduce that the month is neither May nor June and eliminate 5 dates - not 3. – Beno Apr 14 '15 at 5:17

July 16, because the first statement Albert makes immediately eliminates May and June.

May and June both contains unique dates - May 19 and June 18 respectively - and for Bernard to DEFINITELY not have a chance to know the answer, it couldn't be those 2 months.

Then Bernard confirms the birthday. Which means that July 14 and August 14 is eliminated, because otherwise, Bernard who only knows the date, wouldnt know.

After which Albert claims that he knows it too, meaning the only answer would be July 16, as August has 2 dates remaining and Albert still wouldn't know for sure because he only knows the month.

Behind your first spoiler: as Beno pointed out in a comment, Albert's first statement let's you eliminate even more. After he speaks, Bernard (and we) know that the month cannot be May or June.

Behind your second spoiler, you say "The only way for him to know based on this information would be if he was told 15". 15 is not the only possibility. Since Albert's first statement let's you eliminate May and June, there are three dates Bernard could have heard that would allow him to have certainty, namely, the 15, 16 and 17 (it couldn't be 14, because then Bernard would think both June 14 and August 14 were possible).

The logic in the question is nonsensical.

The statement by Albert: "Then I also know when Cheryl's birthday is" cannot be true based on Bernard saying that he now knows.

Cheryl could have said the 17th to Bernard. Since Bernard has deduced the only possible months are July and August he can now be certain he knows. BUT Albert cannot be certain - Cheryl could have told Bernard either the 15th, 16th, or 17th. So the logic in the question is wrong.

The question would work as written if given the following change:

July 14 15 16
August 14 15


Then 14th and 15th can be ruled out by Bernard's certainty, leaving only July 16

• Albert can be certain because he knows the correct month. Given that he remembers "July", the 17th is already ruled out. It is only we who don't know until after Albert speaks the second time. – Julia Hayward Apr 14 '15 at 10:24
• The logic of the question is sound. Please read the correct answer to understand it. – Ian MacDonald Apr 14 '15 at 10:34
• OK, I agree, I have it wrong – Kent S Apr 14 '15 at 14:59