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Lyndenlea3D Solutions So Far
This page lists the number of three-dimensional solutions to the n² Queens in an n³ "chess-cube" problem that we have found in each size of cube using our algorithm.
As we had been feeding that algorithm the full set of two-dimensional solutions (or "boards") for each cube – the magnitudes of which are given in the second column of the table below, and are taken from sequence A000170 in the On-Line Encyclopedia of Integer Sequences (OEIS) – our progress became increasingly hampered by available computer memory and processing power and reluctantly we stopped searching after finally exhausting the 16³ cube. (Had we sought to persevere with the 17³ cube, we'd have needed to shuffle almost 96 million boards!)
Aside from the theoretical solution for the 0³ cube (into which one can successfully place zero Queens) and the trivial solution for the 1³ cube (comprising a single Queen in the only available cell), we have only found a complete set of 3D solutions in two cubes so far, as shown in the third column of the table; however, I've already started to identify patterns in the results:
| Cube size | Candidate 2D boards | Number of 3D solutions | Date that we first completed the search |
|---|---|---|---|
| 0³ | 1 | 1 | Theoretical |
| 1³ | 1 | 1 | Trivial |
| 2³ | 0 | 0 | No 2D boards |
| 3³ | 0 | 0 | No 2D boards |
| 4³ | 2 | 0 | Insufficient 2D boards |
| 5³ | 10 | 0 | 7th February 2009 |
| 6³ | 4 | 0 | Insufficient 2D boards |
| 7³ | 40 | 0 | 7th February 2009 |
| 8³ | 92 | 0 | 7th February 2009 |
| 9³ | 352 | 0 | 7th February 2009 |
| 10³ | 724 | 0 | 7th February 2009 |
| 11³ | 2,680 | 264 | 7th February 2009 |
| 12³ | 14,200 | 0 | 7th February 2009 |
| 13³ | 73,712 | 624 | 7th February 2009 |
| 14³ | 365,596 | 0 | 8th February 2009 |
| 15³ | 2,279,184 | 0 | 12th December 2010 |
| 16³ | 14,772,512 | 0 | 29th November 2017 |
One obvious feature of these results is the lack of solutions for the 12³ cube and the 14³ cube, even though the 11³ cube and 13³ cube both yielded solutions. As they (along with the 8³, 10³ and 16³ cubes) 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, as I found whilst analysing our earliest generated solutions in November 2008. I was confident that this couldn't be a coincidence, and in fact it led me to devise a way of Taking a Shortcut to finding solutions in even larger odd-numbered cubes.
Another cube with no solutions is the 15³ cube, which is of course an odd-numbered cube. However, my analysis in November 2008 revealed that 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 could also explain the lack of solutions for the 9³ cube as it might otherwise be large enough to produce solutions.
As for the 7³ cube and its smaller cousins, they are simply too small to bear any solutions, just like the 2×2 and 3×3 chessboards in the original two-dimensional problem. Specifically, their respective sets of 2D boards are insufficient either in number or in variety to allow a 3D cube solution to be assembled.