Numbers in a cube

IBM Ponder This #294 · October 2022

The following puzzle was suggested by Evert van Dijken: Consider a $n \times n \times n$ cube. It has $3 n^{2}$ rows, $6 n$ diagonals from one edge to the opposite edge, and 4 corner-to-corner diagonals.

To each $1 \times 1 \times 1$ cube, we assign one of the numbers $1 \dots n$ . The big cube does not need to have the same number of 1's, 2's.., or n's. Given such a numbering, we count the number of rows and diagonals that contain all the numbers $1 \dots n$ at once. As an example, consider for $n = 3$ the numbering given by

There are 27 rows, each containing all the numbers 1, 2, and 3. To illustrate, we mark three such rows, each in a different axis, using different bracket styles:

[1] [3] [2]
 2   1   3
 3   2  {1}

(2)  1   3
(3)  2   1
(1)  3  {2}

 3   2   1
 1   3   2
 2   1  {3}

There are 4 corner-to-corner diagonals, three of which contain 1, 2, 3 and one containing only 2:

[1]  3   {2}
 2   1    3
 3   2    1

 2   1    3
 3 {[2]}  1
 1   3    2

 3   2    1
 1   3    2
{2}  1   [3]

And 18 edge diagonals, 9 of which contain 1, 2, and 3, while the other 9 do not:

[1] {3}   2
 2   1    3
 3   2    1

 2  [1]   3
 3  {2}   1
 1   3    2

(3)   2   [1]
 1  (3)    2
 2  {1}   (3)

The total count is 27+9+3=39. We can compactly represent this solution by the string

(Note the count in the final line).

Your goal: Find a solution of at least 110 for $n = 6$ . Supply your solution in the above format.

A bonus "*" will be given for getting at least 60 solving the case $n = 5$ when maximizing the count over diagonals (both edge and corner) first. A solution with more diagonals is always better, even if the final count is lower, but for the best diagonal count we still wish to maximize the total count. Supply your solution in the same format as above (no need to explicitly write down the diagonal count, only the total count).

Solution

Best opened after a real attempt

To be added.

IBM Ponder This #294 · October 2022 ​

IBM Ponder This #294 · October 2022