# Chessboard distances

I saw this visualization posted on the internet a couple of weeks ago. I love it. It shows the minimum number of moves required for a chess piece to achieve any given square on the board. It assumes White pieces, and an empty board (Other than to allow castling for the King). If anyone knows the original creator of this excellent graphic, please let me know as I’d like to give them full attribution/credit.

At first glance it’s easy to think, “hang on, I think there’s a mistake”, then you realize that when you promote a pawn it can be to any piece, or that pawns are allowed to move two spaces on a first move.

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## Follow up

Thinking about this infographic lead to another interesting, related, puzzle. Imagine each of the numbers 1–64 are placed on the squares of the board so that each number appears exactly once.

Prove that, no matter how the numbers are arranged, there must be two squares sharing an edge whose numbers differ by more than four.

Prove that, no matter how the numbers are arranged, there must be two squares sharing an edge whose numbers differ by more than four.

## Solution

Imagine the worst case that could happen. The maximum Manhattan distance between any two squares on the board is fourteen. The biggest number on the board is 64, and smallest is 1. This is a delta of 63.

63÷14 = 4.5 which is greater than four, so there has to be

*at least*one step where the jump is greater than four.## More

For more chess related puzzles:

- The classic eight Queens problem.
- The chess tournament puzzle.
- How many squares are there on a chessboard?
- The impossible escape (the Devil’s Chessboard).