Combining Hopfield neural networks, with applications to grid-based mathematics puzzles

Hopfield neural networks are useful for solving certain constrained set-selection problems. We establish that the vector fields associated with general networks of this type can be combined to produce a new network that solves the corresponding combination of set-selection/constraint problems, provided a relatively simple condition is satisfied. That is, we establish that just this one condition needs to be verified in order to be able to combine such networks. We introduce some generalizations of networks that exist in the literature, and, to demonstrate the usefulness of the work, we combine these networks to solve two well-known grid-based math puzzles (i.e. constraint problems): Kakuro and Akari (called Cross Sums and Light Up in North America). We present examples to illustrate the evolution of the solution process. We find that the difficulty rating of a Kakuro puzzle is strongly connected to the number of iterations used by the neural network solver.