The Covetous Neighbors

Nov 17, 2012

RAY: This week's Puzzler was sent to me from Bruce Robinson, a professor of civil and environmental engineering at the University of Tennessee.

There are 25 jealous people who live in the squares of a five-by-five grid. We're gonna number the squares, starting in the upper left-hand corner, 1 through 25.

TOM: So the first row starts with 1, the second row starts with 6, the third row starts with 11, and so forth.

RAY: Right. Remember, each person is jealous of his adjacent neighbor. Not his diagonal neighbor, but the person up or down or left or right of him. Each aspires to move into the apartment of his adjacent neighbor.

The question is very simple: What is the fewest number of total moves that can accomplish this?
RAY: Here’s how you solve it. 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.

Let's letter the first one A. The next one B, the next one A, the next one B, et cetera, et cetera. Then, everyone who's on an A square must, by definition, move to a B square. And everyone who's on a B square must move to an A square.

Now, if you add them up, you got 13 A squares and only 12 B squares. So, there are no fewest number of moves. It’s impossible for this to happen.

Do we have a winner?

TOM: The winner is Martha Lozzanno from Richmond, Virginia. Congratulations, Martha! 

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