# 3D 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.