Evaluating SAT Solver Metrics as Predictors of Human-Perceived Nonogram Difficulty

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.

Neural solvers are built to deduce, branch, and revise intermediate states. The Lattice Deduction Transformer (LDT) appears to do exactly that. In clue-rich Sudoku, it does not: one forward pass commits essentially the entire grid (every blank cell on standard 6x6, 94-96% on augmented 9x9), turning the iterative solver into a one-shot predictor wrapped in an exact verifier. All hard-slice failures are decided before search begins, when the first pass confidently deletes a value required by the true solution. We call this first-pass poisoning. Adding learned branching, MRV, backtracking, value exclusion, and shared nogoods (CoLT) does not change which Sudoku instances are solved; it cuts repeated invalid derivations 1,497-fold. At the frozen training budget, constraint-graph attention alone matches full-CoLT accuracy, while positional tables recover only under substantially longer training, indicating an optimization and sample-efficiency advantage rather than an absolute capacity difference. The diagnosis predicts two effective interventions. Digit-permutation augmentation raises 9x9 accuracy from below 1% to 96.5 +/- 0.3 across three training seeds on a symmetry-disjoint split. Test-time union over symmetry-transformed passes raises all three hard-slice checkpoints from 72.8-78.9% to 100% without retraining. On from-scratch graph coloring, one-shot behavior disappears and search changes accuracy. In clue-rich completion, LDT-like systems are one-shot amortized predictors rather than learned search procedures: accuracy is determined by calibration and symmetry, while search primarily removes computational waste.

Arithmetic puzzle games provide a controlled setting for studying difficulty in mathematical reasoning tasks, a core challenge in adaptive learning systems. We investigate the structural determinants of difficulty in a class of integer arithmetic puzzles inspired by number games. We formalize the problem and develop an exact dynamic-programming solver that enumerates reachable targets, extracts minimal-operation witnesses, and enables large-scale labeling. Using this solver, we construct a dataset of over 3.4 million instances and define difficulty via the minimum number of operations required to reach a target. We analyze the relationship between difficulty and solver-derived features. While baseline machine learning models based on bag- and target-level statistics can partially predict solvability, they fail to reliably distinguish easy instances. In contrast, we show that difficulty is fully determined by a small set of interpretable structural attributes derived from exact witnesses. In particular, the number of input values used in a minimal construction serves as a minimal sufficient statistic for difficulty under this labeling. These results provide a transparent, computationally grounded account of puzzle difficulty that bridges symbolic reasoning and data-driven modeling. The framework supports explainable difficulty estimation and principled task sequencing, with direct implications for adaptive arithmetic learning and intelligent practice systems.