# Jealous Neighbors

RAY: So, if you draw this grid, the square in the upper left-hand corner we could say is one, and the one next to it is two, three, four, five, and then the line below that is six, seven, eight, nine, 10, 11, 12, right? All the way to 25. Got it?

TOM: I got it.

RAY: Now, each person who lives on the floor aspires to move into the apartment of one of his adjacent neighbors. So number one can move to square number two, or number six, for example.

TOM: But not diagonally.

RAY: Not diagonally. Number two can move to number one.

TOM: Or number three, or number seven.

RAY: There you go.

TOM: I can visualize all that.

RAY: So, here's the question. Why would anyone live in such a stupid building? No, the question is, what is the fewest number of total moves that will allow every person to move to an adjacent square.

TOM: All right. I see --

RAY: I know Doug Mayer has been, did you get the answer, Mayer? He says, no, I didn't get the answer.

TOM: Well, unencumbered by the thought process, I concluded right away when you gave this problem --

RAY: It had to be one or zero, right?

TOM: -- that the answer was, it was either going to be 26 --

RAY: Ah! Very good!

TOM: -- or millions.

RAY: Well, millions is close.

TOM: Millions is closer, huh?

RAY: If you don't number them one through 25, but instead, letter them.

TOM: Yeah.

RAY: And not A, B, C, D, E. Let's letter the first one A.

TOM: Yeah.

RAY: The next one B, the next one A, the next one B, et cetera, et cetera.

TOM: Oh, oh, oh.

RAY: Then, everyone who's on an A square must, by definition, move to what?

TOM: A B square.

RAY: Right.

TOM: And then, vice versa.

RAY: And everyone who's on a B square must move to an A square. It's pretty obvious if you draw it out.

TOM: Yeah. That's true.

RAY: Yeah. Now, if you add them up, by some stroke of bad luck, you got 13 A squares and only 12 B squares.

TOM: Someone's got to move out of the building.

RAY: They've got them mixed up, like. So, there are no fewest number of moves. It is impossible for this to happen. I know, it was a little sneaky.

TOM: No, no. Impossible is a good answer.

RAY: Impossible is a good number of moves. So, who's our winner, anyway?

TOM: That's very good, you know? I like that question.

RAY: Well, you're going to love this week's if you --

TOM: Yeah.

RAY: Poor Mayer just threw a waste basket at me! He was up to 26,215 moves, and he almost had it!

TOM: Ah, the winner is Martha Lozzanno from Richmond, Virginia.

Support for Car Talk is provided by: