3D Knight's Tour
There is a famous puzzle called Knight's Tour. Knight is a piece in the game of chess. The puzzle is to move the Knight on an 8x8 chessboard and find a tour that goes through all 64 squares only once.
It seems difficult because there are 64 squares, but you can also go back to the first square through all the squares as shown in the following figure.
According to Wikipedia, there are 26534728821064 ways just for such a tour back to the first square. There are several algorithms and heuristics for finding tours.
I thought "Is there a 3D Knight's Tour?" Is it possible to pass through all 512 squares in an 8x8x8 cube only once, rather than on an 8x8 flat chessboard?
The 3D knight can move to 24 squares as shown in the figure.
After writing and running a program, I found a tour back to the first square. It looks good.
By the way, you can find various examples of 3D Knight's Tour on the net. For example, St. Olaf College is decorated with wooden works created by mathematician Loren Larson. There is also a site where you can register the longest route of a non-intersecting 3D Knight's Tour. It is an attempt to find the longest route, as it is not possible to go through all the squares without crossing. It seems difficult to break the record.
Loren Larson? I noticed I have known his name for a long time! I loved the Japanese translation of his book "Problem-Solving Through Problems" when I was a student!
Is there a tour that starts from a corner and finishes at a corner on the other side? It would have a pleasant appearance if a part of one straight line is a Knight's Tour.
I was able to find it. I made it "Knight in Night" together with a night view. I listed some NFTs of my artworks.
This road sign pole is with a 4x4x4 tour.
I want to try a larger cube. Doubled one side and solve 16x16x16 = 4096 squares. It feels better to be denser than when it was 8x8x8.
Next, tripled one side to 48x48x48 = 110592 squares. It exceeded 100,000 squares. It's tight. Something seems to be inside. I want to jump into the gap and go through to the other side.
you can find a tour with 100x100x100 = 1 million squares, but as the volume increases in proportion to the cube of the length of one side, it becomes hard er and harder to make a solid.
いいなと思ったら応援しよう！
