Solution of the Last Open Four-Colored Rectangle-Free Grid: An Extremely Complex Multiple-Valued Problem

It is a challenge in the multi-valued domain to solve problems that depend on a large number of variables, as large as possible. We selected for this paper the problem of rectangle-free colorings using four colors which could not be solved so far for the grids of the sizes 12 × 21 and 21 × 12. This problem depends on 12*21 = 252 four-valued variables. It is the last of so far unsolved rectangle-free grid problems for four colors. This paper aims at the solution of a multi-valued problem with an exceptionally high complexity. The search space for this finite problem is 4252 which is approximately 5.2 * 10151. A similar coloring problem was solved for the grid of the size 18 × 18 that relies on the extreme search space of approximately 1.1 * 10195. The construction of a cyclically reusable solution for a single color simplifies this search space approximately to 3.4 * 1097. Unfortunately, such a restriction to a single color is not possible in the case of a grid of the size 12 × 21. Hence, the complexity which must be handled in maintainable time grows additionally by a factor of more than 1054. Based on a very deep analysis of the properties of the problem we have constructed a strongly restricted SAT-model. This final model depends on 504 Boolean variables and 85.344 clauses. Using this SAT-instance we could calculate not only one solution but 38, 926 representatives of different permutation classes of four-colored rectangle-free grids of the size 12 × 21.