The Neighbors Shuffle
Remember those little sliding number puzzles?
The ones with 15 tiles and one space, where no matter how smart you thought you were, you always ended up with 14 and 15 swapped—and no way to fix it?
Yeah, the ones that taught us early in life that the universe is unfair.
Well, this puzzler reminded me of that… only worse.
So here’s the deal:
Imagine a five-by-five grid—25 squares total. Think of it as a very cramped, very unfriendly apartment building. And in each apartment lives exactly one person.
Now, these are not your average neighbors. These are jealous people.
Each person looks only at their adjacent neighbors—that means up, down, left, or right. No diagonals. Diagonals don’t count, just like in real estate disputes.
And every single one of them wants to move, not across town, not to Florida—just into one of their adjacent neighbor’s apartments. Anybody else’s place will do, as long as it’s next door and not their own.
Let’s number the apartments just to keep things civilized:
Top row: 1 through 5
Next row: 6 through 10
All the way down to 25 in the bottom-right corner.
For example, the person in apartment 1 can only move to apartment 2 or apartment 6.
The question is:
What is the fewest total number of moves required so that every one of the 25 people successfully moves into one of their adjacent neighbors’ apartments?
And of course, this being a proper puzzler:
No teleporting.
No diagonals.
No staying put. Everyone must move.
And when you think you’ve got the answer…
Prove it.
