Mathematical Definition and Systematization of Puzzle Rules

While logic puzzles have engaged individuals through problem-solving and critical thinking, the creation of new puzzle rules has largely relied on ad-hoc processes. Pencil puzzles, such as Slitherlink and Sudoku, represent a prominent subset of these games, celebrated for their intellectual challenges rooted in combinatorial logic and spatial reasoning. Despite extensive research into solving techniques and automated problem generation, a unified framework for systematic and scalable rule design has been lacking. Here, we introduce a mathematical framework for defining and systematizing pencil puzzle rules. This framework formalizes grid elements, their positional relationships, and iterative composition operations, allowing for the incremental construction of structures that form the basis of puzzle rules. Furthermore, we establish a formal method to describe constraints and domains for each structure, ensuring solvability and coherence. Applying this framework, we successfully formalized the rules of well-known Nikoli puzzles, including Slitherlink and Sudoku, demonstrating the formal representation of a significant portion (approximately one-fourth) of existing puzzles. These results validate the potential of the framework to systematize and innovate puzzle rule design, establishing a pathway to automated rule generation. By providing a mathematical foundation for puzzle rule creation, this framework opens avenues for computers, potentially enhanced by AI, to design novel puzzle rules tailored to player preferences, expanding the scope of puzzle diversity. Beyond its direct application to pencil puzzles, this work illustrates how mathematical frameworks can bridge recreational mathematics and algorithmic design, offering tools for broader exploration in logic-based systems, with potential applications in educational game design, personalized learning, and computational creativity.