Enumerating solutions to grid based puzzles with a fixed number of rows

TL;DR

This paper introduces a method to count solutions to grid-based puzzles by analyzing empty grids with fixed row counts and varying column sizes, providing a sequence that enumerates solutions.

Contribution

It presents a novel approach for enumerating solutions to logic puzzles by removing clues and systematically counting valid configurations for fixed row counts.

Findings

Derived sequences for solution counts across different grid sizes

Demonstrated the method on various puzzle types

Provided insights into the combinatorial structure of grid puzzles

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 $r$, vary the number of columns $k$, and then compute the sequence $A_r(k)$, which gives the number of solutions on an empty grid of size $r \times k$.