The original problem, as seen on Facebook |

We follow as many math sites as we can keep up with here at
the Center of Math. So when this image began circling the internet yesterday,
we were intrigued. I tried working the problem when I got back to the office
from lunch yesterday (kinda half-heartedly, I admit) and thought through the
logic, and originally got the problem wrong. My logic was there for a step or
two, but I didn’t take it far enough.

I didn’t like the explanations I found on various sites
early yesterday afternoon, so I decided to write up my own and post it here for
the Center of Math followers.

First, Albert says something along the lines of “I don’t
know the answer, but I

*do know*that Bernard also doesn’t have the answer.”
With this sentence, we know that we (and Bernard) can cross
off the first two months, May and June. This is because Albert knows that
Bernard doesn’t have the answer. May and June each have a unique day, the 18

^{th}or the 19^{th}, which doesn’t occur in any other month, so Bernard would know the full date if Cheryl gave him the date 18 or 19. If Albert had been given the month May or June, there is a possibility that Bernard would know the full answer already, and Albert’s sentence would have read “I don’t know the answer, and I don’t know if Bernard knows the answer.” So we can rule out the entire months of May and June.
Next, Bernard says something like “At first I didn’t know
Cheryl’s birthday, but

*now*I do know.”
We can figure that Bernard is smart kid, and he was able to
follow Albert’s logic like we did and cross off the months of May and June.
Bernard knows the answer now, so we can cross off the only repeated date left,
the 14

^{th}, because if Bernard had been given a date of 14 he wouldn’t know if it was July or August 14^{th}.
The final piece of the puzzle is that Albert then pipes up
again, and says something like, “

*Because**I know that Bernard knows*, I also know when Cheryl’s birthday is.”
Albert was only given the month, but he followed the same
logic that we did when Bernard said he determined the answer, so he could
narrow the choices down to the same three days. So we can figure out which
month Albert was given because of this: two of the dates lie in August, and one
lies in July. If Albert had been given August, he still wouldn’t know which of
the dates is correct. So Albert must have been given the month July.

After we cross out the two dates in August, we’re left with
only July 16

^{th}!
We're not so sure about Cheryl at this point- why the tricks? Will Albert and Bernard even want to go to her birthday party anymore? I can't solve that question for you guys, but I hope you enjoyed the problem anyways.

-Tori

meh

ReplyDeletenot correct

ReplyDeleteyou can't eliminate all of May based on Albert's first statement.

#aug17

Aug 17 is what I came up with...

DeleteI too understand that from the first statement, it cant be June...but its too soon to rule out May

DeleteWe actually CAN eliminate all of May and June right off the bat, and the trick is in Albert's first statement. Let's look at a couple scenarios:

DeleteLet's say Cheryl gave Albert the month May. Albert would say, "I do not know the full date, and I do not know if Bernard knows the answer." This is because Bernard could have been given the date 15, 16, OR 19. If Bernard was given the 15 or 16, he would not be able to deduce the answer. If Bernard was given the 19th, he would know immediately what the full birthday was.

If Cheryl gave Albert the month of June, the same situation occurs. Albert would have said "I do not know the date, and I do not know if Bernard knows the date" because it is possible that Cheryl gave Bernard the date 18 which would mean Bernard solved the problem immediately.

If Cheryl instead gave Albert the month August, Albert would say (like he did in the problem) "I do not know the full date, but I know that Bernard does not know either." This is because Bernard would have been given the date 14, 15, or 17, and all of these dates occur more than once. Therefore Bernard cannot possibly know the answer immediately. However, because he knows that Albert thinks there is no way Bernard can deduce the answer, Bernard can cross off the entire months of May and June.

I hope this makes a little more sense! -Tori