I wonder if the thing that is going viral here is the fact that Singapore 5th graders are being challenged by the puzzle — rather than the puzzle itself, which is cute but not difficult.

There is something fun about a race to solve a puzzle, though… I always give my students a logic puzzle as a bonus question on in-class exams. At the beginning of the semester it’s usually a variant of the Zebra Puzzle — which did the 1960s equivalent of going viral in NYC social circles when Life International published first the puzzle, then the names of people who managed to solve it. On the 2nd exam I typically choose one of Raymond Smullyan’s inimitable problems — and then on the 3rd exam a variant of the so-called Impossible Puzzle.

The interesting thing is that I actually have students every year who manage to solve each of the three types of puzzles — under exam conditions, without any advance notification of what the type of puzzle will be. It gives me hope for the future to think that our collective global capacity for this kind of analytic reasoning might be increasing very significantly.

The Impossible Puzzle, for those who want a better morning coffee challenge, is harder than the Singapore 5th graders’ puzzle:

X and Y are two different integers, greater than 1, with sum less than 100. S and P are two logicians. S knows the sum X+Y, P knows the product X*Y, and both know the information in these two sentences. The following conversation occurs.

P says “I cannot find these numbers.”
S says “I was sure that you could not find them. I cannot find them either.”
P says “Then, I found these numbers.”
S says “If you could find them, then I also found them.”

What are X and Y?

(For an even more interesting challenge: can you come up with variants of this? And then can you come up with a technique or calculus for producing variants?)

I had actually just come across this puzzle elsewhere, worked through it, and sent an email to a colleague who’s smarter than I am to get his insight before I saw it posted here. I was able to arrive at the correct answer, and apparently I did it in the right way (according to published solutions), but it seems to me that there is something wrong with the design of the question. Given that Albert knows the answer by the end of the story, I know it, too. But my question is how he knows. It seems to me that at the last step Albert doesn’t actually have enough information to deduce the answer. Am I missing something here?

“Are there other such problems that could gain such traction so quickly in social media?”

This puzzle seemed to get a lot of public attention:
http://io9.com/can-you-solve-the-worlds-other-hardest-logic-puzzle-1645422530

Here’s the chart I made:

“When’s your birthday?” Albert asked Cheryl. She wrote down a list of 10 dates:
May 15 — May 16 — May 19
June 17 — June 18
July 14 — July 16
August 14 — August 15 — August 17

Then Cheryl whispered in Albert’s ear the month — and only the month — of her birthday. Month:
May 15 —16 —19
June 17 — 18
July 14 — 16
August 14 — 15 — 17

To Bernard, she whispered the day, and only the day.
14 – July August
15 — May August
16 — May July
17 — June August
18 – June
19 – May

“Can you figure it out now?” she asked Albert.

Albert: I don’t know when your birthday is, but I know Bernard doesn’t know, either.

July 14 — 16
August 14 — 15 — 17

Bernard: I didn’t know originally, but now I do.
15 — August
16 — July
17 — August

Albert: Well, now I know, too!
July 16
August 15 — 17

When is Cheryl’s birthday?
July 16

It might help.

Dynamic epistemic logic to the rescue! Logicians Audrey Yap and Barteld Kooi both have solutions to the puzzle:
http://richardzach.org/2015/04/15/logicians-yap-kooi-explain-viral-birthday-logic-puzzle/
http://m-phi.blogspot.nl/2015/04/dynamic-epistemic-logic-solves-birthday.html

I follow the logic all the way to Bernard figuring out the date. But it’s unclear to me how Albert knows the date? If Bernard now knows the date, then it must be July 16, August 15, or August 17. The reason Bernard knows is because it must be July or August AND he has been given the actual date, so if that’s right, then he eliminates the 14th dates and selects one of the three remaining answers ‘in virtue of knowing the number already.’

Albert doesn’t have that number, though. So from his perspective, he cannot determine between the three remaining dates because he is not an epistemic peer with respect to Bernard regarding the number. The ONLY way Bernard selects the date is by Cheryl telling him the date. Albert has not been told that, so how can he know that it is the 16th?

Bernard does not arrive at the answer by reasoning that there is only one month with a single date remaining. So how can Albert arrive at the answer that way?

Right, I see how the 14th dates are eliminated, but how (from Albert’s perspective) does he know it’s not August 15 or August 17? If Bernard had been told 15 or 17 as the number, then all the previous logic would still lead him to know Cheryl’s birthday. Regardless of whether Bernard had been told 15, 16, or 17, he would still know the date of the birthday given Albert’s elimination of May and June as possibilities. So Albert, at the end, knows that Bernard was told one of those three numbers, because otherwise Bernard would not know the answer. But Albert has no way of knowing WHICH of those three Bernard was told. So how can he be confident the birthday is in July? Nothing Bernard or Cheryl has said (that Albert is also aware of) indicates July 16 over against the August dates.

Now, if we assume that Albert KNOWS the date, then it has to be July 16 because that’s the only unique possibility left (i.e., both the August dates are not unique because they are both in August). So the only way he could know is if he’s learned something that eliminates August. Since we know that’s the only way he could know, we can confidently say that the date is July 16. But here’s my question: HOW does Albert know that? Not how do WE know that, but how does HE know that? What did he learn that eliminated the August dates?

FYI: All CAPS are for emphasis only.

AH! Yes! Thank you, guys! I suspected there was something fairly obvious that I was overlooking, and now you’ve shown me what it was! Much obliged.

The world has gone mad for epistemic logic! I issued my own transfinite epistemic logic puzzle challenge at http://jdh.hamkins.org/transfinite-epistemic-logic-puzzle-challenge/. Can you solve it? Please post your solution there.

No