Knights Tour
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I can't see any relevant patterns in this thing besides the fact that it typically makes half a dozen or so moves on the queenside then half a dozen on the kingside. I wonder what a perfect knight tour would look like on a 9x9 or 10x10 board (if even possible)
When you fail the knight's tour so bad
I was actually just looking back to post something related to this in another thread, but apparently I forgot to explain here! 7 years late but here's the explanation:
A knights tour can very easily be completed using this method of "4 Quadrants 4 Systems". Each color represents a system that exists in each of the four 4x4 subsections of the board. All of the individual systems connect with each other in the other quadrants, so what you do is you complete 1 system in all 4 quadrants (always a way to do it where you won't box yourself in early), and then jump to the next, and repeat until all 4 are done.
The blue is the "right diamond system" since it looks like a rhombus tilted to the right. I color coded each of the unique systems once in different quandrants each so that it's easier to see, but all 4 systems exist in all 4 quadrants.
The green is the mirror image of the blue, the "left diamond system".
The red is the "right square system" since it forms a square that's tilted a little to the right.
And the Yellow is the mirror image of that, left square system (square tilted to the left).
Each system in each specific quadrant connects at least once to a square in that same system in the neighboring quadrant. So to do the knights tour, you complete 1 system at a time in all 4 quadrants, and then jump to a square within another system and do the same looping around the 4 quandrant, finishing all the 16 squares in that system, one system at a time. In this one I did the extra thing of making sure the 64 also connected back to 1, so I deliberately left the 64 square blank and just made sure the last system I did connected back to the right diamond system in the 1st quadrant, and did them in an order to make that convenient. If you follow the numbers you can see they go through each system one at a time.
Here is the completed color-coded view. The purple arrows indicate where I jumped from 1 system to another after completing each system throughout the 4 quadrants.
Doing it this way is the most reliable and easy to understand way to make a knights tour, and its just as easy regardless of what square you start and end on.
AntonioEsfandiari wrote:
I can't see any relevant patterns in this thing besides the fact that it typically makes half a dozen or so moves on the queenside then half a dozen on the kingside. I wonder what a perfect knight tour would look like on a 9x9 or 10x10 board (if even possible)
They are possible on all board greater than size 4x4, but only even dimension ones can be "re-entrant" (meaning the last square also connects to loop back to the first square in 1 continuous loop) and this can be proven quite simply:
Every knight move alternates color of the square it's on. Therefore all even numbers will be on the same color square and all odd numbers will be on the opposite colored square. But boards with odd number dimensions (5x5, 7x7..etc) have an odd number of squares, so for a tour to be re-entrant, the first and last squares would have to also be a knights move away, but they would both be odd! (1 and 25, 1 and 49...etc), so the first and last squares could never be connected by a knights move on a 5x5, 7x7, 9x9, or any other OddxOdd board, since a knights move always changes square color and thus " even or odd-ness"
The Knight's Tour is such an interesting challenge! It’s amazing how the knight can visit every square exactly once, following its unique movement. Solving it not only improves your chess skills but also sharpens problem-solving abilities. The diagram you've shared looks like a great example of one possible solution! Writing a term paper requires a lot of research, and I didn’t have time to do it properly. I found Academized’s professional term paper writers, and they exceeded my expectations! The writers from this https://academized.com/professional-term-paper-writers site was knowledgeable in my subject and delivered a well-structured, original paper. The formatting and references were done perfectly, and I received a high grade. I highly recommend their services!