Explanation III

by zenquaker

I did promise an explanation of how I do the knight’s tour, so here it is. To be clear, the knight’s tour is where you take a Chess knight and move it around a Chess board so that it goes to every square once without going to any square twice. I use it as a visualization exercise to help see possible knight moves when playing Chess. I find it is also good for practicing thinking ahead a few moves, because at certain points you have to be careful about where you will be three moves later.

Note that this method is not something I came up with. I got it off some other website, which I can no longer find.

The basic plan is the four patterns shown in the diagram below. Each quarter of the board is one of the patterns. I think the two with the white knights are called the diamonds, and the two with the black pieces are called the windmills. If not, those are good enough names for them. Note that they don’t overlap: none of them use the same squares as any of the others. That means if you lay them on top of each other, they completely fill one quarter of the board.

Diagram of Patterns in the Knight's Tour

That is basically what you do. Take the square you start on, it can be any square on the board. Figure out which of the four patterns that square is a part of. Finish that pattern for that quarter of the board. You want to do the pattern so you finish it near the center of the board. Then you should be able to move to a square in the same pattern, but in a different quarter of the board. Then do the pattern in the second quarter of the board, the third, fourth.

Once you have done the first pattern in all four quarters of the board, you move on to the next pattern. You are going to have to switch from diamond to windmill or vice versa. You may also have to turn around. If you did the first pattern going clockwise from quarter to quarter, you may have to go counter clockwise when you switch patterns. Keep in mind that you can switch to the next pattern in the same quarter you finish one pattern in.

Switching patterns is the easiest place to get yourself into a trap. For that reason it is good to plan ahead when you start the fourth quarter of a pattern. You generally don’t have a choice where to start the pattern when shifting from quarter to quarter, but once you are in a quarter, you will have to choices of which of two squares to end on. Think ahead to those two choices, and make sure you are going to one that will allow you to switch to the next pattern. Typically this will be the choice that is closest to the center of the board, but it doesn’t hurt to check.

That’s all there is to it. Do each pattern one quarter of the board at a time, switch from windmill to diamond to windmill. When you’ve done all four patterns in all four quarters of the board, you will have finished a knight’s tour. Not only that, if you do it right you can end up one knight’s move away from the square you started on, making an infinite loop.

The images for the diagram were made by Paul Gorman, and released under a Creative Commons Attribution 2.5 license.