In the previous post I re-worded this logic puzzle that’s doing the rounds online at the moment. My goal in doing so was to hopefully make it easier to understand. So here is the problem as I presented it:
The Strange Saga of the Unknown Birthday
by SASMO
with editing and dramaturgy from Steve the Math Tutor
Albert & Bernard have just become friends with Cheryl and they want to know when her birthday is. Cheryl, being a huge puzzle-fan, decides to have a little fun. She gives them a list of 10 dates, one of which is her birthday.
| May 15 | May 16 | May 19 |
| June 17 | June 18 | |
| July 14 | July 16 | |
| August 14 | August 15 | August 17 |
Then, separately, she tells Albert the correct month and Bernard the correct day.
Cheryl: Ok, without actually telling each other, figure out which one is my birthday.
Albert: Well, knowing the month hasn’t helped me. I don’t know when your birthday is. But I do know that Bernard doesn’t know either.
Bernard: Aha! I didn’t know her birthday at first but, now that you’ve said that, I do.
Albert: Aha! Now that you’ve said that, I know as well!
So when is Cheryl’s birthday?
Ok, so lets have a look at a solution.
There are only those 10 possible dates that could be Cheryl’s birthday. We’re going to go through the given information, step by step, eliminating dates from the list until we only have one possibility remaining.
Albert: Well, knowing the month hasn’t helped me. I don’t know when your birthday is. This makes sense because every month in the list has multiple possibilities. Simply knowing the month doesn’t give Albert enough information to say exactly when Cheryl’s birthday is.
Albert: But I do know that Bernard doesn’t know either. The same is not true for the days. If Bernard had been told 18 or 19 then he would know the answer straight away since, out of the 10 possibilities, there is only one each with 18 or 19. Since Bernard doesn’t currently know, he can’t have been told 18 or 19. We can eliminate those possibilities from the list.
| May 15 | May 16 | |
| June 17 | ||
| July 14 | July 16 | |
| August 14 | August 15 | August 17 |
But remember Albert has only been told the month. So the only way for him to know that Bernard was not told 18 or 19, is for Albert to have been told neither May nor June. So we can eliminate all the remaining possibilities from those months.
| July 14 | July 16 | |
| August 14 | August 15 | August 17 |
Bernard: Aha! I didn’t know her birthday at first but, now that you’ve said that, I do. Bernard now knows the exact date. This could only be possible if Bernard was not told 14, because 14 appears in more than one of the remaining possibilities. All the others appear only once. So we can eliminate the two 14’s.
| July 16 | ||
| August 15 | August 17 |
Albert: Aha! Now that you’ve said that, I know as well! Now Albert knows the exact date as well. This is only possible if Albert was not told August because, again, August appears in more than one of the remaining possibilities. So we can eliminate the August dates.
| July 16 | ||
Only one possibility remains. Cheryl’s birthday is July 16th.