Latin Square

Latin Square

Abstract

Quoting common sources, “A Latin square is a k×k array filled with k different symbols, each occurring exactly once in each row and in each column.”
That definition reminds of a Sudoku grid without subdivision (3×3 boxes) constraints. In order to create puzzles, however, we need such regions; albeit arbitrarily shaped/positioned ones, because there won't be nicely tiling square boxes in grids other than sizes 4×4, 9×9, 16×16, et cetera, and our goal will be to construct 10×10 grids (using digits [0…9]) anyway.
On top of that, additional hints may be provided, just like in traditional/variant Sudoku.

Construction

The core algorithm for filling a k×k grid is relatively simple and compact, built on 3 helper arrays, 3 vars, and a function for array shuffling (here: amx()) only. Here the resulting grid is stored in array g:

//--- Declarations
const a=[],r=[],c=[];let i=k,o,j;
//--- Initialisation
while(i--)a.push(i);r.push(...amx(a));c.push(...amx(a));
//--- Calculation
for(i=k;i--;)for(o=r[i],j=k;j--;)g[c[++o%k]+i*k]=a[j];

Basic demo

This generator uses base36 for symbols ([0…9,a…v] or [1…9,a…w] respectively) to create a grid up to size 32×32 cells:

					332
Start value	0		Size	0

Regions

For the placement of 10 regions (in a 10×10 grid, as mentioned above, each containing all digits [0…9]), the following strategy is used:

1: Populate grid with digits.
2: Choose 10 seed (starting) cells, one per region.
3: Grow each incomplete region by one cell.
4: Repeat step 3 until all regions are complete.

For the seeding in step 2, there's at least two methods:

Symmetry
Four
seeds can immediately be placed in the corners.

The other 6 seed positions are each randomly chosen
from a 4×4 area.

Four of these areas are also located in the corners, and
the remaining two sit closer to the center, which seems
to be the best compromise as of symmetry.
Symbol X
Every region starts at symbol X (in our case: digit 0).

After all 10 seeds have been placed, the algorithm for step 3 (growth) loops through each region and does the following:

If the region is complete, we’re already done; check out the next.
Collect all adjacent neighbouring cells, each being put in a dedicated basket:
- If the digit is free and…
  … needed, put cell into basket 1
  … not needed, put cell into basket 2
- Else, if the digit is needed, put cell into basket 3
If basket 1 (free and needed) isn't empty, pull a random cell to extend the region.
Else, if basket 2 (free but not needed) isn't empty, pull a random cell to exchange digits and reshape the region.

Here’s a step-by-step simulation of the process: