The Knight's Tour instructions and game on this site have always been one of Grey Matters' most popular sections. To keep it fresh and helpful, it's time I made some updates.
I'm going to start by assuming you're familiar with the basic patterns, the systems, and the Danger Zone. Once you've learned about the Danger Zone, you might think that getting that out of the way is all you have to worry about.
However, you may sometimes notice that you can't continue on in the same system, as per the instructions. Let's say you've started in a left-handed square system, and made 8 moves in the following order:
From the 8th square, the only way to stay in a left-handed square pattern is to move onto a square you've already visited (#3). The only other possible moves will place you in one of the diamond patterns! How do we prevent this situation?
When you're working through a system, and you're entering a new quadrant, you need to base your decision to go clockwise or counterclockwise on which finishing square will allow you to move to the next quadrant. In the above example, the decision to go clockwise led back to a completed quadrant. If the decision had been made to go counterclockwise (effectively switching the two moves in blue), the knight would easily be able to move to the next quadrant, and continue with the left-handed square pattern.
There is one point in any tour where this situation will always develop. When you enter the last quadrant of your 3rd system (square 45 in the example below), going one direction will allow to proceed to another quadrant and begin a new system, and going the other direction will leave your knight without any possible moves.
In the above example, the knight began in the upper right quadrant, and the left-hand diamond system was completed, followed by the left-hand square system (starting at 17), and then the right-hand diamond system (starting at square 33). As soon as the last quadrant of the third system was entered (square 45), the decision to go clockwise created the problem. If the player here had stopped briefly to consider their options, it wouldn't be difficult to see that going counterclockwise (effective switching the two moves in blue), would allow the knight to begin working on the final system.
In the second of three parts of the Knight's Tour update, I'll discuss the approach to a problem that is faced by those who allow their audience to choose the beginning and ending squares. In the third and final version of the update, I'll introduce my improved version of Petri Kallberg's Knight's Tour Dashboard widget (Mac OS 10.4.3 or later required). If you want to get a sneak peek, you can download my 2.0 version here (Mac OS 10.4.3 or later required), or from the Downloads section in the rightmost column.