Beyond The 2nd Dimension
3D Solutions So Far
The first table on this page lists the number of three-dimensional solutions we have found in each size of "chess-cube" using our algorithm. As we have been feeding that algorithm the full set of two-dimensional solutions for each cube our progress is beginning to be hampered by available computer memory and processing time. However, although we have only managed to generate results for two cubes, I've already started to find patterns in the results:
| Cube Size | Solutions |
|---|---|
| 7 | 0 |
| 8 | 0 |
| 9 | 0 |
| 10 | 0 |
| 11 | 264 |
| 12 | 0 |
| 13 | 624 |
| 14 | 0 |
| 15 | 0 |
One obvious feature of these results is the lack of solutions for the 12³ cube and 14³ cube, even though the 11³ cube and 13³ cube both produced solutions. As they (along with the 10³ cube and 8³ cube) are even-numbered cubes, their sets of 2D solutions lack the 1, 3, 5, 7... boards that form the basis of the 11³ cube and 13³ cube solutions. I suspect this isn't a coincidence and look forward to seeing the results of the 16³ cube that we're currently processing.
Another cube with no solutions is the 15³ cube, which is of course an odd-numbered cube. However, it too lacks the 1, 3, 5, 7... boards as this arrangement would result in two Queens trying to share the same diagonal line. The same is true of any chessboard size that's divisible by three, which probably also explains the lack of solutions for the 9³ cube.
But by far the most interesting pattern I've seen in these results might actually allow me to calculate the number of solutions for any odd-numbered cube whose size is not divisible by three. This would be an enormous benefit considering the computer memory constraints we are currently experiencing with just the 16³ cube and 17³ cube. Following a detailed analysis of the solutions generated by the 11³ cube and 13³ cube I came up with a formula - however, it assumes that these bigger cubes only contain solutions that fit the patterns I've seen so far, so it may yet prove to be incomplete. Nevertheless, here are the results of that formula for the next few odd-numbered cubes whose size is not divisible by three:
| Cube Size | Solutions |
|---|---|
| 17 | 2,040 |
| 19 | 3,192 |
| 23 | 6,624 |
| 25 | 9,000 |
I am indebted to my friend and colleague Clare Willis, and her father John Willis, for helping me to present my analysis in mathematical terminology.
So why are there no solutions for the 7³ cube? And what about the 5³ cube?
Just like the 2 by 2 and 3 by 3 chessboards in the original 2D puzzle, they're simply too small. Specifically, there isn't enough variety within their respective sets of 2D boards to allow a 3D cube solution to be built.
