Latin squares generator? (Sudoku-like constraint problem)

Question

Purpose

We're designing a Latin square (sudoku-like sequence) for an experimental design that needs to follow these constraints:

• Values cannot be repeated in a row
• Values cannot be repeated in a column
• Values cannot be repeated pairwise in any two rows

Example for the first 3 constraints:

2     3    5    7    11   13
7     2    11   3    13   5
11    5    2    13   7    3
3     7    13   2    5    11
5     13   3    11   2    7
13    11   7    5    3    2 

Here, we chose primes, but the values are arbitrary (as long as there are 6 distinct values). Notice that it's the same as sudoku in a 6 x 6 grid, with the additional constraint that there are no repeated pairs (aka bigrams) across rows. That is, 2 3 only appears on the first row, but in no others, and so on for every other pair.

• Values should be pair with another 6 values that fit these constraints:
  • The 2nd-set values cannot be repeated in a row
  • The 2nd-set values cannot be repeated in a column
  • When paired with the 1st-set values, the 2nd-set values cannot be repeated.

That is, we need to have another six values (again, arbitrary -- they could be "a, b, c, d, e, f") that pair with the first six. The last constraint means that if you use ( 2 , a) anywhere , you cannot use it again.

The last 2nd-set constraint is the problem. There is no solution for nxn grids where n = 6. That's the monkey wrench in the works. We want to minimize the number of repeats, to make a "sorta-kinda" orthogonal pair of Latin squares.

Question

Have you encountered this problem before? (It'd be great if there were a tool that will generate solutions.) We only need one such example for our purposes. We've tried several different construction strategies already.

We're flirting with the idea of writing a solver for it, but we'd like to avoid doing so if something already exists.

Answer 1

Take a look: Dancing Links

In computer science, Dancing Links, also known as DLX, is the technique suggested by Donald Knuth to efficiently implement his Algorithm X. Algorithm X is a recursive, nondeterministic, depth-first, backtracking algorithm that finds all solutions to the exact cover problem. Some of the better-known exact cover problems include tiling, the N queens problem, and Sudoku.

Answer 2

We wrote a solver that randomly permuted solutions and took the best balanced solution available after running it overnight. It might not be the best approach, but since a perfect solution doesn't exist for the constraints, we thought it should be better than other methods of finding a "not-so-bad" solution.