Sudoku metrics
I’ve been experimenting with metrics for sudokus. For example, in the image above, each color occurs once as the center of a 3-by-3 block, four times as a corner, and four times as a side. There are also no instances of squares of the same color being adjacent diagonally, which is allowed by the sudoku rules is the squares are in different 3-by-3 blocks, but which visually makes the distribution of that color seem less random.
My thinking is that images with certain properties are more esthetically pleasing than other images. I have a computer program that generates sudokus, and I can plug in different metrics. The metrics also restrict the number of sudokus generated to more manageable levels. The image above is one of only 36, for example. Runs with less restrictive metrics go into the tens of millions.
July 26th, 2008 at 3:37 pm
Metrics? What about a hollow cube where each side of each plane was shared with its right angle neighbor?
July 26th, 2008 at 4:23 pm
I’m not sure what you mean. Cubes have six faces, but sudokus have nine blocks.
July 27th, 2008 at 7:37 am
Say you have 81 cubes instead of 81 squares for each of 6 sudokus and you put them together like a cube, but the nine cubes on the right most column of the sudoku on the yz plane were the same cubes as on the left most side of the sudoku on the xy plane. Like the intersection of two walls in a room would share the same cubes at the corner, at all corners leaving the center empty. I am interested in how it would look from the center. Would the pattern repeat and reverse?
July 27th, 2008 at 10:26 am
Reverse-and-repeat works for the 4 walls, but it messes up the ceiling and floor. Suppose the cubes in the top row of your south wall are numbered 1 through 9. If you reverse for the east and west walls, then the top of the east wall reads 9 through 1, and the top of the west wall reads 9 through 1 (reading left to right in all cases). The 5 in the top of the east wall lines up with the 5 in the top of the west wall, which invalidates the sudoku in the ceiling.
So reverse-and-repeat doesn’t work, but I think it would be more interesting anyway if the 6 sudokus had completely different patterns, except for matching along the seams.
July 28th, 2008 at 6:18 pm
That would be interesting. So by putting in the cubes along the seams first you could create 6 different sudokus? Could each of the four outling strings of cubes on one face create more than one pattern inside that face? When it came time to join the last face wouldn’t it also misalign?
July 28th, 2008 at 6:49 pm
Good questions. Suppose we take a standard 9×9 grid and fill in the edges (32 squares) in a way that doesn’t violate the sudoku conditions. My guess is that there are many, many ways to fill in the center.
These problems typically have many solutions, OR there is some gimmick that allows a simple counterexample, like the fact that reversing 1 through 9 leaves the 5 in same position.
If I am right that there are many ways to fill in the centers, we could specify all the edges of a cube first, and then fill in the center of each of the 6 faces. Everything would be pre-joined, and there would be no alignment problem at the end.
Of course I don’t know any of this for a fact without actually constructing a solution.
July 29th, 2008 at 6:43 am
The multi-colored cubes constructed out of paper (which is a perfect cat toy) could be created in real life. 9 colors each with {[81 x 6 - (9x8)/2 ]/9 }cubes of that color.
August 13th, 2008 at 8:15 am
I forgot to account for the eight corners. Actually, couldn’t a rubic’s cube be adapted for the six sides of nine squares each? It would have to be bigger of course to allow for the 81 cells per face.
Maybe some sort of cubic metal box and colored squares with magnets to get close. The cubes on the edges would be represented by two of the same colored squares.
August 13th, 2008 at 12:51 pm
1. One could take a color sudoku and print it on 9 stickers, one 3-by-3 block per sticker. Do this for 6 sudokus and put the stickers on a Rubik’s cube.
2. I have an acrylic photo cube that I use to help myself visualize. I draw diagrams and put them in the photo cube.
3. I have not set up the computer to solve for the faces yet.
4. Consider a solid 3-D sudoku, not just the outside faces of a cube. Slice it anywhere, along the x-, y- or z-plane, and you always get sudokus along the cut. Possible or not?