Imagine there is a hat sitting on the table. And there are two contestants. You, Tommy, will be one of the contestants, and we'll call the other one Vinnie.
You reach into the hat, and pull out a number. Then, Vinnie does the same. Now, the reason this hat is magical is that, no matter what number you pull out, Vinnie will always pull out a number that is either one above or one below your number. For example, if you pull out a two, you know Vinnie has pulled out either a one or a three.
To make it simple, we'll limit the numbers to between one… and infinity.
So, each of you pulls out a number. Let's say you pick three and Vinnie picks two. I'm the moderator, and I ask Tommy, "Do you know what number Vinnie has?" Tommy looks at his number, which is 3, and says, "No, I don't."
I then ask Vinnie, "Do you know what number Tommy has?" He looks at his number two and says, "Yes." He knows Tommy has to have a 3.
Regardless of the numbers that are picked, and assuming that both contestants answer truthfully, if I keep asking the question of both of them, eventually one contestant will know what number the other contestant has.
In other words, if I ask Tommy, then ask Vinnie, then Tommy again… then Vinnie again… eventually one of them will know the other's number.
The question is this:
How can this be proven in a mathematically rigorous fashion?
This question was taken from the Car Talk Puzzler Archive and is located at http://www.cartalk.com/content/magic-hat-0?question
Best Answer
This is known as Conway's Paradox, and you can find a mathematically rigorous solution here.