The knight's tour in chess: covering the whole board

Have you ever wondered if a knight can travel across all 64 squares of the board without landing on the same square twice? That’s exactly the knight’s tour problem, also known as Euler’s knight. It’s one of the most fascinating mathematical challenges surrounding chess. And the good news: it has a solution, and I’m going to show it to you step by step.

The idea is simple: move the knight across all 64 squares of the board without repeating any. It sounds easy, but as soon as you try it you realize it isn’t so simple. Let’s solve it systematically.

Euler’s knight solved step by step

The trick is to break the problem into smaller parts. That way you don’t lose track.

1. Divide the board into four quadrants of 16 squares each. As you can see in the image, this helps you keep the route under control at all times. In each quadrant you’ll make 4 distinct moves, without repeating ranks or files each turn.

2. Pay special attention to the corners. They’re the most dangerous squares: the knight has few options from there and it’s easy to get stuck. Once you finish the tour of one quadrant, move to the next and repeat the process. A full lap of the board: 16 moves.

3. Reverse the direction of rotation when starting the second lap. That adds another 16 moves, bringing you to 32 in total.

4. Reverse the rotation again on the third lap: 48 moves. And one more lap gets you to the 64 moves needed to cover the whole board. Keep reading to see how the full solution looks!

Euler’s knight: solution

Does it look hard? Here’s a practical example starting from square d4. Notice how the knight chains the quadrants together without getting trapped:

With this method you solve the knight’s tour across the whole board without repeating a single square. Systematic, clean and elegant. Once you master it, you’ll see the knight’s movement with different eyes.

If you want to better understand how the pieces move in general or dig deeper into the knight, here are some guides that will come in handy:

1. How the knight moves in chess
2. How to set up a chess board
3. How much is each chess piece worth?
4. Origin of the chess knight