Magic Hat Puzzler

RAY: Here's what I want you to do: Take a pencil and paper and draw five boxes, starting at the top. In the first box, in the upper left corner, put the number one.

In the upper right of the first box, put the number two. Then, right below that, in the lower left and right corners, reverse the digits, so they're labeled two and one.

The second box, below the first, is going to have the numbers two and three in the upper right and upper left corners. Then in the lower left corner three, and two in the lower right.

RAY: The next box is three and four, left to right.

TOM: And then four, three.

RAY: Then four, five, five, four. You see the trend.

Now, let's say you have the number three. In other words, you're on the left. Vinnie, meanwhile, has four.

TOM: I have three and Vinnie has four. So I'm in the third box.

RAY: Correct, Tommy. I ask you, "Do you know what number Vinnie has?"

TOM: No.

RAY: Right--because Vinnie could have either a two or a four.

Now I ask Vinnie, "Do you know what number Tommy has?" He also answers, "No." Then I come back to you. I ask you again, "Do you know what number Vinnie has?"

And you answer, "Yes." How do you know? Well, let's go to the previous example, which is just above.

You have three and Vinnie has two. I ask you, "Do you know what number Vinnie has? As in the previous example that I gave, you have to say, "No."

TOM: Right, because Vinnie could have a two or a four since I have a three.

RAY: That's right. But when I ask Vinnie if he knows your number, he answers, "Yes." Because if he has two, you must have either one or three. And if you had one, you'd have known right off what number he had.

TOM: Right. He'd have had a two.

Okay. Now, let's go back to the first example, where you have 3 and Vinnie has 4.

When you answered, "No," the first time, and Vinnie answered, "No," you knew he couldn't have a two--because if he had a two he would have said, "Yes."

Because Vinnie didn't answer, "Yes," he must have four. So, when I ask you the second time, "Do you know what number Vinnie has?" you answer, "Yes."

Now go to the next example, where you have four and Vinnie has five. Believe it or not, the sequence of answers is, "No," "No," "No," and "No," and it comes back to you...

TOM: And I, Tommy, answer, "Yes."

RAY: Tommy, you got it man! Congratulations.

 Here's another way of thinking about it--a way that, some might say, is a little bit less confusing than Ray's explanation. (Sorry, Ray.) Thanks to John McCoy for this graphic representation.

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