Are You Smarter than a Singaporean Math Olympiad Student?

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.

 

I wrote the dates this way- it makes it easier to see them laid out like this

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.

 

With May and June out of the way

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.

 

Only three possibilities left!

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.

 

Voila!

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