Enumerating Solutions to Grid-Based Puzzles with a Fixed Number of Rows

Abstract

In this paper we demonstrate a method for counting the number of solutions to various logic puzzles. Specifically, we remove all of the “clues” from the puzzle which help the solver to a unique solution, and instead start from an empty grid. We then count the number of ways to fill in this empty grid to a valid solution. We fix the number of rows k, vary the number of columns n, and then compute the sequence $$A_k(n)$$ , which gives the number of solutions on an empty grid of size $$k \times n$$ .