Knight's Tour Update (2/3)

Published on Sunday, April 27, 2008 in , , , ,

Click here to go to Grey MattersHopefully you've been practicing the Knight's Tour starting from any square, and learned how to make sure that you leave yourself openings, as we previously discussed.

After learning to do the Knight's Tour starting from any square, many people find that it almost seems too simple. All you really have to remember is the simple alternation of diamond and square systems, and to keep your paths open. Naturally, many move on to the advanced version of the Knight's Tour, where you let someone else choose the starting and ending squares.

It's great when one chosen space is in a diamond system, and the other is in a square system. As long as you choose your path carefully, this isn't much different from the simpler version.

The challenge comes when they're both square or both diamond systems. On this page, I briefly mention to taking care of a few spaces in one pattern (one system in one quadrant). However, I do feel that I need to go into more detail on this point.

For our first example, we'll start with two diamond patterns, with the starting point (S) part of a left-hand diamond system, and the finishing point (F) as part of a right-hand diamond pattern.

Since the systems are two different orientation, you can complete the first one without conflict (as in Start-16 below), then move onto an opposite system (17-32 cover the right hand square below). As the next step in our example, we need to go to the right-hand diamond system, but since the final square is in this system, we need to minimize the steps in it.

Besides minimizing steps, you'll also need to consider your ability to enter and exit that particular quadrant. You want to go into a pattern, but make sure you leave enough squares that will allow you to enter and exit the quadrant. In the current example, I chose to only move into one square in the right-hand diamond system (#33 below) before moving into a left-hand square system (#34-#37 below), as the squares marked A and B in blue allow for easy entry and exit for that quadrant (not forgetting to move into upper rightmost square for completeness).

From here, you would finish the left-hand square system (#38-#49 below). This leaves us the right-hand diamonds, the location of the closing square, as the only remaining system! It's not over yet, though, as we need to proceed through this final system in a manner than allows finishing on the right square.

In the example below, I began the right-hand diamond system in the lower right quadrant (#50-#53), finished the partially filled upper right quadrant (#54-#56), partially completed the upper left quadrant (#57-#59, skipping the finishing square, of course), then completing the lower left quadrant (#60-#63) so that I finished on the chosen final space (F).

As you can see, a large part of the challenge of the advanced Knight's Tour is getting used to properly breaking up the systems. In the previous example, we not only had to break up the upper right quadrant, but later, the upper left quadrant to allow for ending in the chosen square, as well. Looking ahead to check can seem challenging, but since this is mainly done as your working through the final system, there are fewer spaces, and thus fewer paths with which to deal.

When you're dealing with two chosen spaces that are both in the same system and in the same orientation, such as the starting (S) and final (F) squares in left-hand square systems below, you'll need to get the partial pattern out of the way as the first step.

Moving from the starting point to space #2 below is an easy choice, as it gets a space from the Danger Zone out of the way. From there, you could certainly move to the space marked A (in blue), as it would still leave one space in the upper left quadrant's left-hand square pattern (space B in blue) that would allow you to move in from one quadrant, and exit to another. You'll note that I chose to start on a right-hand diamond system instead (#3-#6).

Yes, the first move could be from S directly into a left-hand diamond system, but since moving to space #2 eliminates a space with limited options (a Danger Zone space), it's preferable to get rid of it as soon as possible.

Leaving two spaces (A and B) open, instead of just one (B), will allow for more flexibility later on. Since you won't be coming back to the left-hand square system until the end of the game, allowing for more options later is a good idea.

From here, the moves in the middle of the game are fairly straightforward. The right-hand diamond system is completed (#7-#18), then the right-hand square system (#19-#34), and the left-hand diamond system (#35-#50).

Now that we're back to the final left-hand square system, we have to take that partially completed upper left quadrant into consideration, and work out how to use it to get to the final square (F). In this particular example, I chose to finish off the lower left quadrant (#51-54), then break up the patterns of the remaining 3 quadrants. The bottom right quadrant was first (#55-#57), then along the outside of the upper right quadrant (#58-#60), and the two spaces we left open at the beginning (#61-#62). The remaining space in the upper right quadrant (#63) is then used to get back to the chosen final space (F).

The real challenge in learning the advanced Knight's Tour is getting used various ways of seeing paths and breaking the patterns that were used so rigidly in the simpler version. As an aid to getting used to pattern-breaking, I created a second level of my Knight's Tour game where you have to wind up just one knight's move away from your starting point. This allows you to use the simpler system alternation strategy you learned initially, while still frequently requiring the pattern-breaking needed for the more advanced version.

I hope you've been enjoying this article, and part 1 of the Knight's Tour series, and found it useful. On the original instruction pages here and here, I've added links to these blog posts, so they're immediately available as you need them.

The final third of this series will be dedicated to my newly-improved OS X Dashboard widget for the Knight's Tour. As I mentioned in the first part, you can already download the improved Knight's Tour widget here (or in the Downloads column to the right) and get a sneak peek before my next post.

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