The Jealous Neighbors

Aug 07, 2021

RAY: This week's Puzzler was sent to me from Bruce Robinson, a professor of civil and environmental engineering at the University of Tennessee. I think he sent it sometime in 1980, so he’s probably quit or retired by now!

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.

The first row starts with 1, the second row starts with 6, the third row starts with 11, and so forth.

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: 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. 

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.

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.

My first guess was that it had to be one, or zero. But my brother guessed “Millions of moves!” 

And it turns out that he was right. Or at least, he was close! 

If you don't number the squares 1 through 25, but instead, letter them, alternating A and B (so the first one is 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, by some stroke of bad luck, you’ve got 13 A squares and only 12 B squares.

Someone's got to move out of the building. 

So there is no fewest number of moves. It is impossible for this to happen. I know, it was a little sneaky.

Impossible is a good answer.


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