As Prince Geoffrey says in The Lion in Winter, “I know. You know I know. I know you know I know. We know Henry knows, and Henry knows we know it. We’re a knowledgeable family.” Or, as Princess Leia and Han Solo say, “I love you.” “I know.” (Or, on problems of knowledge, see also here.) But really, this is about the Cheryl’s birthday problem from Singapore that the Internet has been talking about recently. Here’s the original problem (including the infelicitous English usage):

Albert and Bernard just become friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates:

May 15, May 16, May 19

June 17, June 18

July 14, July 16

August 14, August 15, August 17

Cheryl then tells Albert and Bernard separately the month and 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 don’t know when Cheryl’s birthday is, but I know now.

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

So when is Cheryl’s birthday.

I won’t bother solving this one — it’s all over the Internet by now — or answering the obvious questions like why would Cheryl even do such a thing, and why don’t Albert and Bernard just pool their information??? Maybe it’s for the best that they “just become friends” and don’t go any further. But if your problem with this problem was that it was too easy, that’s a problem I’ve solved for you.

Here’s one of the same type (i.e., people revealing their information sets) but a lot harder, which came to me indirectly from the UVa math department about a year ago:

Two numbers a and b are between 2 and 99. [Note: They’re not constrained to be different from each other. Also, I interpret this as permitting a or b to be 2 or 99, i.e., “2 or above" and “99 or below", or, if you will, 2 ≤ a,b ≤ 99.]

Peter is given the product of the numbers, ab (and knows he is given the product).

Sarah is given the sum a+b (and knows she is given the sum).

They also know the numbers are between 2 and 99.

They are UVa math majors, so they are great at math and completely honorable!

Peter says, “I don’t know the numbers.”

Sarah says, “I knew you didn’t know the numbers.”

Peter then says, “I know the numbers now.”

Sarah then says, “Ah ha! I know the numbers now.”

What are the numbers?

This problem took a little more brute force to solve than seems elegant to me, but maybe there’s a more elegant solution that I missed. Feel free to chime in in the comments.