An attempt to devise a transfinite version of the Cheryl birthday puzzle.

I won't describe the original puzzle here -- just Google it if, by some miracle, you haven't seen it already. Let us define a logic puzzle of this sort to be of level N if it requires you to consider a chain of length N of the type "I know that you know that I know that you know ... etc." Then a level-omega puzzle would be one that requires you to consider chains of length N for arbitrarily large N. That's what I've tried to devise below. I do not guarantee that I have succeeded -- I may have made a silly mistake.

Cheryl presents Albert and Bernard with the following set of pairs of positive integers. For even n, let r=n^2, s=(n+1)^2 and let A(n) be the set that consists of the pairs (1,r), (r,r), (r,r+1), (r+1,r+1), (r+1,r+2), and so on up a staircase until you reach the point (s-1,s-1). For odd n, take instead the pairs (r,1), (r,r), (r+1,r), (r+1,r+1), (r+2,r+1), ... , (s-1,s-1). So, for example, A(1) consists of the points (1,1), (2,1), (2,2), (3,2), (3,3), while A(2) consists of the points (1,4), (4,4), (4,5), (5,5), ... , (8,8).

Cheryl then tells Albert the x-coordinate of a special point and tells Bernard the y-coordinate. Albert and Bernard then have the following conversation.

Albert: I don't know which point Cheryl has chosen.

Bernard: Even now you've told me that, I don't know which point she has chosen.

Albert: Even now you've told me that, I don't know which point she has chosen.

Bernard: Even now you've told me that, I don't know which point she has chosen.

and so on and so on. Eventually Cheryl gets bored and says, "Hey, can we talk about something else now? You're never going to work out the answer."

At that point Albert and Bernard both say, "Ah, now I know which point you've chosen." What was the point?
Shared publiclyView activity