# High-dimensional Sudoku

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 *n*². 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, …, n _{d}* be natural numbers as well.
A basic block of a sudoku with these parameters has side length

*n*for each direction

_{i}*i*. Stack these blocks together to obtain an enormous hypercube, where each side has sidle length

*n₁ * … * n*.

_{d}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 *
… * n _{d}* 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.