For Bernard to be unable to determine Cheryl’s birthday upon being told the day, the day must be insufficient to specify the month. In other words, the day has to be one of the numbers that appears more than once in the list. This immediately rules out 18 and 19, which both only appear once. Moreover, for Albert to know that Bernard doesn’t know Cheryl’s birthday, the month he was given must not contain either 18 or 19 as a possible day; otherwise, it would have been possible for Bernard to figure out the month from the date, and Albert could not know that Bernard did not know Cheryl’s birthday. This rules out May (which contains 19) and June (which contains 18).
Upon hearing that Albert knew he did not know Cheryl’s birthday, Bernard would gain the above information, and know that Cheryl’s birthday falls in either July or August. This means that the information he was given must be sufficient to discriminate between these two months, i.e. whatever the day Cheryl gave him, it cannot appear in both months. This rules out 14. The remaining possibilities are July 16, August 15, and August 17.
This is where I got stuck. There doesn’t seem to be any more information in the problem that would allow further discrimination between these three possibilities. Moreover, this makes Albert’s assertion that he now knows Cheryl’s birthday after hearing Bernard absurd; how could he possibly know which month it is?
I’m still unsure how to proceed right now. I’ll give it ten or so more minutes of thought, and if I fail to come up with anything after that, I’ll look at the answer.
EDIT: Man, I feel stupid. The answer came to me right after I commented, and it turns that my mistake was that I had unconsciously conflated Albert with the reader. The reader doesn’t know the month, and therefore without further information, it’s impossible to determine which of the three possibilities Cheryl’s birthday actually falls on. However, Albert does know the month, and whichever one it is, it cannot contain more than a single possible day for Cheryl’s birthday (because if it did, Albert wouldn’t be able to tell which day it was). Of the set of possibilities I had originally {July 16, August 15, August 17}, the month of August contains two possible days—so if Cheryl’s birthday were in August, Albert would not know whether it was August 15 or August 17. Therefore, Cheryl’s birthday must fall in July, and so the answer is:
July 16
(META: That was pretty fun! We should do this more often.)
(Posted without looking at the replies.)
For Bernard to be unable to determine Cheryl’s birthday upon being told the day, the day must be insufficient to specify the month. In other words, the day has to be one of the numbers that appears more than once in the list. This immediately rules out 18 and 19, which both only appear once. Moreover, for Albert to know that Bernard doesn’t know Cheryl’s birthday, the month he was given must not contain either 18 or 19 as a possible day; otherwise, it would have been possible for Bernard to figure out the month from the date, and Albert could not know that Bernard did not know Cheryl’s birthday. This rules out May (which contains 19) and June (which contains 18).
Upon hearing that Albert knew he did not know Cheryl’s birthday, Bernard would gain the above information, and know that Cheryl’s birthday falls in either July or August. This means that the information he was given must be sufficient to discriminate between these two months, i.e. whatever the day Cheryl gave him, it cannot appear in both months. This rules out 14. The remaining possibilities are July 16, August 15, and August 17.
This is where I got stuck. There doesn’t seem to be any more information in the problem that would allow further discrimination between these three possibilities. Moreover, this makes Albert’s assertion that he now knows Cheryl’s birthday after hearing Bernard absurd; how could he possibly know which month it is?
I’m still unsure how to proceed right now. I’ll give it ten or so more minutes of thought, and if I fail to come up with anything after that, I’ll look at the answer.
EDIT: Man, I feel stupid. The answer came to me right after I commented, and it turns that my mistake was that I had unconsciously conflated Albert with the reader. The reader doesn’t know the month, and therefore without further information, it’s impossible to determine which of the three possibilities Cheryl’s birthday actually falls on. However, Albert does know the month, and whichever one it is, it cannot contain more than a single possible day for Cheryl’s birthday (because if it did, Albert wouldn’t be able to tell which day it was). Of the set of possibilities I had originally {July 16, August 15, August 17}, the month of August contains two possible days—so if Cheryl’s birthday were in August, Albert would not know whether it was August 15 or August 17. Therefore, Cheryl’s birthday must fall in July, and so the answer is:
July 16
(META: That was pretty fun! We should do this more often.)
Perhaps a monthly puzzle thread.