Beyond The 2nd Dimension

Describing 3D Solutions

Just like with the two-dimensional solutions to this puzzle, we need a brief and consistent way of describing three-dimensional "cube" solutions. We actually have two options, each of which is related to the various 2D solutions that we fitted together to make up the 3D solution.

Using 2D Board Numbers

View of the chess-cube containing the first possible 3D solution
710 25 811 36 91 4
25 811 36 91 47 10
811 36 91 47 102 5
36 91 47 102 58 11
91 47 102 58 113 6
47 102 58 113 69 1
102 58 113 69 14 7
58 113 69 14 710 2
113 69 14 710 25 8
69 14 710 25 811 3
14 710 25 811 36 9

The diagram shows the 11³ cube containing its first possible 3D solution. In this plan view of the cube, the Queens on each Layer are represented by that Layer's number, counting upwards from the bottom of the cube. So all Queens in Layer 1, for example, are indicated by a number 1 in the appropriate square.

The 2D solutions (or boards) that we used to complete this 3D solution are as follows, numbered by the Layer in which each one sits:

  1. No.1
  2. No.976
  3. No.2,358
  4. No.98
  5. No.1,371
  6. No.2,366
  7. No.326
  8. No.1,781
  9. No.2,593
  10. No.579
  11. No.2,109

We can use these board numbers to quickly describe the solution by arranging them into a comma-separated list:

1, 976, 2358, 98, 1371, 2366, 326, 1781, 2593, 579, 2109

However, to interpret this description we need to have the full list of 2,680 solutions to the 2D 11² chessboard puzzle so we can use the board numbers to find the Queen positions. An alternative approach would be to use the full list of 121 Queen positions themselves, but to do this we first need to better understand how our view of the cube relates to the 2D solutions.

Using Queen Positions

View of the chess-cube containing the first possible 3D solution
810 13 57 911 24 6
46 810 13 57 911 2
112 46 810 13 57 9
79 112 46 810 13 5
35 79 112 46 810 1
101 35 79 112 46 8
68 101 35 79 112 4
24 68 101 35 79 11
911 24 68 101 35 7
57 911 24 68 101 3
13 57 911 24 68 10

This diagram shows the same 3D solution viewed from the front, rather than from above. The numbers now signify which Row a Queen occupies, numbered from front to back. But most importantly, each horizontal row on the diagram now contains the Queen positions of the 2D board on each Layer of the cube using our 2D notation. For example, the board on Layer 1 is No.1 (the first possible 2D solution), for which the Queen positions are 1, 3, 5, 7, 9, 11, 2, 4, 6, 8, 10.

To use these numbers to describe the 3D solution, we could arrange them either into a grid as in the diagram, or into a comma-separated list. And to make it clear which Queens belonged in which Layer, we could surround each Layer's Queens with parentheses:

(1, 3, 5, 7, 9, 11, 2, 4, 6, 8, 10), (5, 7, 9, 11, 2, 4, 6, 8, 10, 1, 3), etc...