What is a natural generalisation of Sudoku to (arbitrarily) high dimensions?

Prior work considered: nearly none. Only the Wikipedia page.

The usual two-dimensional Sudoku is a 9 by 9 grid where in each of the following areas every digit from 1 to 9 has to appear at least once:

  • in each row,
  • in each column,
  • and in each of the marked 3 by 3 blocks

If we substitute 9 by 3² and then each 3 by n, the rules generalize to Sudoku boards of arbitrary size.

Variant 1: Multi-Planar

Let n be a positive integer and d be an integer and at least 2. The sudoku board shall now consist of a d-dimensional hypercube with sides of length n². Now consider each axis-aligned 2-dimensional subset of the hypercube as a “standard” two-dimensional sudoku of size n. A solved sudoku for this variant satisfies all the constraints of its two-dimensional “subsudokus”.

Variant 2: Hyperblocks

Let d be a natural number (0 included) be the number of dimensions and let n₁, …, nd be natural numbers as well. A basic block of a sudoku with these parameters has side length ni for each direction i. Stack these blocks together to obtain an enormous hypercube, where each side has sidle length n₁ * … * nd.

As in the initial set of rules, each axis-aligned one-dimensional subset of the hypercube and each basic block shall contain each number from 1 to n₁ * … * nd at most once.

Questions

How do the symmetry groups look like now? Sudoku is NP-complete, so the above variants are as well (for a fixed dimension and variable side-lengths), by embedding the two-dimensional sudokus into higher dimensional ones.

Example

For a high dimensional sudoku of the first variant, with parameters n = 2 and d = 4, the solutions are very constrained. If the coordinates are indexed starting at 1, and writing (i, j, k, l) for the entry of the sudoku with these coordinates, we can deduce the following equations:

  • (1, 1, 1, 1) = (2, 1, 2, 2)
  • (1, 1, 1, 1) = (1, 2, 2, 2)
  • (1, 1, 1, 1) = (2, 2, 1, 1)
  • (1, 1, 1, 2) = (2, 1, 2, 1) etc.

I did not solve a whole such sudoku.