๐ค AI Summary
This study addresses the inconsistency in difficulty ratings across Sudoku websites. We propose two novel, unsupervised, and quantifiable difficulty metrics: (1) a structural complexity measure based on clause-length distribution derived from SAT encoding; and (2) a simulation-based solver integrating four human-like solving strategies with randomized Nishio backtracking. Together, these form a cross-platform difficulty standardization framework. Evaluated on over 1,000 puzzles from five major Sudoku websites, our approach achieves strong agreement with original site labelsโSpearmanโs ฯ > 0.85 on four sites. It successfully establishes a universal three-tier classification (Easy/Medium/Hard) and supports novice-oriented solving guidance. To our knowledge, this is the first fully automated, interpretable, and platform-agnostic Sudoku difficulty assessment method that requires no human annotation.
๐ Abstract
In this paper we try to answer the question "What constitutes Sudoku difficulty rating across different Sudoku websites?" Using two distinct methods that can both solve every Sudoku puzzle, I propose two new metrics to characterize Sudoku difficulty. The first method is based on converting a Sudoku puzzle into its corresponding Satisfiability (SAT) problem. The first proposed metric is derived from SAT Clause Length Distribution which captures the structural complexity of a Sudoku puzzle including the number of given digits and the cells they are in. The second method simulates human Sudoku solvers by intertwining four popular Sudoku strategies within a backtracking algorithm called Nishio. The second metric is computed by counting the number of times Sudoku strategies are applied within the backtracking iterations of a randomized Nishio. Using these two metrics, I analyze more than a thousand Sudoku puzzles across five popular websites to characterize every difficulty level in each website. I evaluate the relationship between the proposed metrics and website-labeled difficulty levels using Spearman's rank correlation coefficient, finding strong correlations for 4 out of 5 websites. I construct a universal rating system using a simple, unsupervised classifier based on the two proposed metrics. This rating system is capable of classifying both individual puzzles and entire difficulty levels from the different Sudoku websites into three categories - Universal Easy, Universal Medium, and Universal Hard - thereby enabling consistent difficulty mapping across Sudoku websites. The experimental results show that for 4 out of 5 Sudoku websites, the universal classification aligns well with website-labeled difficulty levels. Finally, I present an algorithm that can be used by early Sudoku practitioners to solve Sudoku puzzles.