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REVIEW 5 major objections 6 minor 26 references

Project Patti: Why can You Solve Diabolical Puzzles on one Sudoku Website but not Easy Puzzles on another Sudoku Website?

T0 review · 5 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A randomized trial-and-error Sudoku solver's average cycle count matches website-assigned difficulty rankings for four of five puzzle sites, and the paper builds a universal Easy/Medium/Hard scale from it.

desk verdict A genuinely useful empirical study of Sudoku difficulty, but the main metric lacks a stability check and the paper overclaims support for its second metric. read the letter →

arxiv 2507.21137 v1 pith:64JRWJTK submitted 2025-07-22 cs.AI

Arman Eisenkolb-Vaithyanathan This is my paper
classification cs.AI
keywords SudokudifficultyNishioHumanCyclesSATencodingClauseLengthDistributionSpearmancorrelationclassificationrandomizedsolverpuzzlerating
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that Sudoku difficulty labels can be reduced to a single procedural number: the mean number of human-strategy cycles a randomized trial-and-error solver needs, averaged over 50 random starts. On a dataset of 1,320 puzzles from five websites, that number ranks difficulty in the same order as each site's own labels for four of the five sites, with Spearman correlations of 0.84, 0.84, 0.89, and 0.70, while a SAT-derived structural metric correlates less consistently. The paper interprets this as evidence that website-assigned difficulty is mostly about how many logical steps a human-like solver must take, not about static puzzle structure. It then builds a three-bin universal scale — Universal Easy, Universal Medium, Universal Hard — from the two metrics and maps every website's difficulty levels onto it, with one website, Extreme Sudoku, failing to align with either metric. The motivation is practical: a common scale would let a player know whether a 'Diabolical' puzzle on one site is genuinely harder than an 'Easy' puzzle on another.

What carries the argument

The central object is a randomized Nishio solver with human strategies. Nishio is trial-and-error backtracking: the solver guesses a candidate digit in a cell, applies the four strategies repeatedly, and backtracks when a contradiction appears. The key metric, Nishio Human Cycles, counts how many times the four-strategy cycle is applied while solving a puzzle, averaged over 50 random starting guesses; this number is meant to represent procedural difficulty as a human would experience it. The secondary object is the Clause Length Distribution, computed from a minimum-encoding SAT (boolean satisfiability) representation of the same puzzle, specifically the percentage of 'short' clauses; this is meant to capture structural complexity such as the number of givens and their positions. The argument works by correlating both metrics against each website's own difficulty labels, then putting the 1,320 puzzles into three equal-count bins to make the categories comparable.

What would settle it

Recompute Nishio Human Cycles for all 1,320 puzzles using a much larger number of random starts, such as 1,000 per puzzle, and check whether puzzle-level means stabilize; if the means shift enough to change Spearman rankings or bin assignments, the 50-start proxy is not stable. A complementary check is a user study in which people solve a sample of the same puzzles and their solve times and strategy choices are compared with cycle counts.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that procedural difficulty, measured as Nishio Human Cycles, tracks the difficulty ordering that websites assign to their puzzles, while structural complexity, measured by the clause-length distribution of a SAT encoding, does not. Nishio Human Cycles is the mean, over 50 randomized restarts, of the number of times the solver applies a fixed cycle of four human strategies (Naked Singles, Hidden Singles, Naked Twins, and X-wing) inside a trial-and-error backtracking procedure called Nishio. The paper reports that this single number correlates strongly with labeled difficulty for the New York Times, Sudoku.org.uk, Sudoku of the Day, and Sudoku of the Day UK, but is essentially uncorrelated for Extreme Sudoku, whose five labels all behave as hard puzzles under the metric. The paper also proposes that a simple univariate binning of the 1,320 puzzles into three equal-count bins yields a universal rating system that classifies both individual puzzles and entire website difficulty levels consistently across sites.

Load-bearing premise

The load-bearing premise is that the average number of strategy cycles used by a randomized Nishio solver, over 50 random starts, mirrors how hard a real human finds a puzzle, an assumption the paper states but does not validate with variance data, convergence checks, or a user study.

Editorial extensions

If this is right

  • A puzzle labeled 'Easy' on one website and a puzzle labeled 'Diabolical' on another can be compared on the same numeric scale, so a player can see which labels are inflated.
  • The four websites whose labels correlate with Nishio Human Cycles have their difficulty levels ordered by the metric, so those sites' labels are measuring something the solver captures.
  • For Extreme Sudoku, the near-zero correlation means either the website's labels are arbitrary with respect to solving effort or the metric misses something that site does; the paper leaves this as an open question.
  • An unlabeled puzzle can be rated directly by computing its mean cycle count and applying the bin cutoffs, without needing a website's own difficulty label.
  • Applied to an unrated dataset, the same bins place more than half of its puzzles in Universal Easy, showing that the scale separates datasets, not just levels within one website.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • As an extension, the metric predicts that any Sudoku website whose labels do not correlate with cycle counts is labeling by something other than solving effort; a straightforward check is to compare that site's labels to human solve times.
  • The universal cutoffs are sample-dependent: re-running the binning on a different corpus would move the boundaries, so the durable contribution is the calibration procedure rather than the specific numeric cutoffs.
  • The cycle count depends on the chosen four-strategy vocabulary; adding or removing strategies would shift the counts, implying that 'universal' difficulty is relative to a defined strategy set, not an absolute property of a puzzle.
  • Recording the full distribution of cycle counts across the 50 random starts, instead of only the mean, could expose a variance-based difficulty signal that the paper does not explore.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

Summary. Project Patti proposes two difficulty metrics for Sudoku puzzles: (1) Clause Length Distribution (CLD), specifically the percentage of short clauses in a minimum SAT encoding, and (2) Nishio Human Cycles (NHC), the mean number of strategy-application cycles used by a randomized Nishio solver interleaved with four human strategies. Using a corpus of 1320 puzzles from five websites, the paper reports Spearman correlations between these metrics and website-provided difficulty levels (Table 2), finding strong NHC correlations for four of five websites, with Extreme Sudoku anomalous. It then builds an unsupervised three-bin universal classifier on each metric and maps all 22 difficulty levels into Universal Easy, Universal Medium, and Universal Hard categories. The paper also reports the percentage of puzzles solvable by human strategies alone and introduces a heuristic-based Nishio solver for human practitioners.

Significance. If the correlations in Table 2 are robust, the paper provides a novel and useful procedural difficulty metric for Sudoku and a practical cross-website rating scheme. Strengths include a large multi-website dataset, a clearly specified random-solver simulation, a comparison with the escape-rate hardness measure of Ercsey-Ravasz and Toroczkai, and an application to the nikoli_100 benchmark from SudokuBench. The main claims are empirically grounded rather than assumed. However, the absence of any variance or convergence analysis for the random solver, the in-sample estimation of the universal bins, and several internal inconsistencies currently prevent the paper from fully establishing its central claims.

major comments (5)
  1. [§7.2 and Table 2] The Nishio Human Cycles metric is a mean over 50 random starts of a randomized solver, but the paper reports no variance, standard error, or convergence check for these per-puzzle means. Because step 2 of Figure 3 chooses a random cell and a random candidate, the cycle count for a fixed puzzle is a random variable; with only 50 samples per puzzle and 60 puzzles per difficulty level, the Spearman correlations in Table 2 and the bin boundaries in §8.2.1 could change with the random seed. Please report the distribution of per-start cycle counts for representative puzzles (e.g., SD and IQR across the 50 starts) and a convergence plot of the mean as a function of the number of starts, for at least one easy, one medium, and one hard puzzle per website. This is necessary to establish that the headline correlations are stable.
  2. [§8.2.2 and §8.3, Observation 3] Observation 3 states that '19 of the 22 difficulty levels fall into or near the Universal Medium difficulty level' when classified by Clause Length Distribution. This count is not supported by the data in Table 1 and the bin boundaries given in §8.2.2 ([100%, 22.6), [22.6, 17.6), [17.6, 0]). Using the table means, five levels (NYT Easy, Sudoku.org.uk Gentle, Sudoku of the Day Beginner, Sudoku of the Day Easy, Sudoku of the Day UK Easy) fall into the Universal Easy bin, twelve fall into the Universal Medium bin, and five fall into the Universal Hard bin; 'near' is never defined. Please provide the exact classification of all 22 levels and correct the count or define 'near' quantitatively.
  3. [§8.2.1 and §8.3] The universal classification is built by fitting equal-bin-count thresholds on the full 1320-puzzle dataset and then evaluating the alignment of the same 1320 puzzles' difficulty levels in Observations 1–4. This is an in-sample evaluation, so the reported alignment counts are not evidence for generalization to new puzzles or websites. To support the 'universal' claim, the binning should be fit on a training subset and evaluated on held-out puzzles, or a cross-validation procedure should be reported. At minimum, the in-sample nature of the alignment should be stated explicitly.
  4. [§8, 'Percentage of Puzzles solved by only Human Strategies Assertions'] The first assertion for the 'Percentage of Puzzles solved by only Human Strategies' analysis states that 'the percentage of puzzles solved should increase monotonically as difficulty level increases.' This is contrary to both intuition and the data in Tables 3–7, where, for example, New York Times Easy is 100% and Medium and Hard are 0% for two strategies. The later statements that 'all assertions hold true' are therefore inconsistent with the stated assertion. The assertion should be corrected (presumably to 'decrease monotonically'), and the corresponding analyses in §8.1.2–8.1.6 should be re-worded to match the corrected direction.
  5. [Abstract and §11 Conclusion] The abstract and conclusion state that the proposed metrics correlate strongly with website-labeled difficulty for 4 out of 5 websites, and the conclusion adds that 'the labeled difficulty levels align with the two difficulty metrics.' Table 2, however, shows strong correlations for Clause Length Distribution only for NYT (-0.80) and Sudoku of the Day (-0.75); for Sudoku.org.uk (-0.47), Sudoku of the Day UK (-0.28), and Extreme Sudoku (-0.02) the correlation is weak. The 4-out-of-5 claim is supported only for Nishio Human Cycles with four strategies. Please qualify the claim so it does not imply that both metrics perform equally.
minor comments (6)
  1. [§7.1 text vs Table 1] The text says 'the mean percentage of short clauses for New York Times Easy is 29%', but Table 1 reports 67% for that entry; this typo should be corrected.
  2. [Table 2] Spearman correlations are reported without p-values or confidence intervals; given that each correlation is based on 180–360 puzzles, significance tests or bootstrap intervals would strengthen the empirical claims.
  3. [§4.1.1] The term '1-in-k SAT clause' is used to denote a clause of length k, but in the SAT literature 1-in-k SAT refers to exactly-one semantics. Since the clauses here are ordinary at-least-one disjunctions, using this term may confuse readers; a different term such as 'k-clause' would be clearer.
  4. [§4.1.2] The sentence 'At-most-one clauses creates pairwise relationships...' has an agreement error ('creates' should be 'create'), and the total clause count for Figure 1a (12026) should explicitly state that this includes the 38 given-digit clauses.
  5. [§6] The dataset is not yet public; since the analysis is empirical, releasing puzzle strings and per-puzzle metric values would greatly aid reproducibility and allow other researchers to verify the correlations.
  6. [§9.2] The sentence 'Additionally, the Hsolver should always apply the human strategies before using Nishio, as described in Step 1.5' refers to Step 1.5 that appears later; consider moving the description before this sentence or adding a forward reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proposed metrics and correlations are defined independently of website labels, and the in-sample universal binning is a validation limitation rather than a circular reduction.

full rationale

The paper's two metrics are defined without reference to website difficulty labels. Clause Length Distribution is computed from the SAT minimum-encoding of each puzzle (Section 7.1), and Nishio Human Cycles is the mean cycle count over 50 randomized Nishio solves (Section 7.2). The Spearman correlations in Table 2 are empirical associations between these independently computed metrics and ordinal website labels; no parameter is fitted to the labels. The universal classifier (Section 8.2.1) uses equal-count binning on the pooled 1320-puzzle metric values, explicitly as an 'unlabeled dataset', so the bin boundaries are not derived from the target labels. The subsequent mapping of difficulty levels (Section 8.2.2) and the 'aligns well' claim are in-sample descriptions rather than out-of-sample predictions, which is a methodological limitation but not a definitional or fitted-input circularity. There is no self-citation chain, no uniqueness theorem imported from the authors, and no ansatz smuggled in via citation; the related-work comparison to Ercsey-Ravasz and Toroczkai [7] is external. The Extreme Sudoku near-zero correlation and the stated need for future validation are acknowledged in the paper. Therefore no specific circular step can be quoted; the robustness concern about 50 random starts is real but is not a circularity argument.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claims rest mainly on the pooled 1,320-puzzle dataset and on design choices: the 50-run averaging, the fixed strategy order, the 3-bin equal-count binning, and the short-clause definition. No external benchmark calibrates the universal categories, so all thresholds are dataset-defined.

free parameters (6)
  • Number of universal bins = 3
    Hand-chosen because New York Times has three levels and because Ercsey-Ravasz and Toroczkai used three categories, as stated in Section 8.2.1.
  • Short-clause threshold for Clause Length Distribution = k = 1, 2
    Clause lengths 1 and 2 are defined as short, and this definition directly determines the percentage of short clauses used as the CLD metric in Section 7.1.
  • Universal bin thresholds for Nishio Human Cycles = [1.30, 3.48), [3.48, 6.52), [6.52, 98.14)
    Computed by equal-bin-count binning on the pooled 1,320-puzzle dataset, so the universal categories are defined by that dataset rather than by an external standard, as shown in Section 8.2.2.
  • Universal bin thresholds for percentage of short clauses = [100, 22.6), [22.6, 17.6), [17.6, 0)
    The same equal-bin-count binning applied to the CLD metric produces these ranges in Section 8.2.2.
  • Number of random starts for Nishio = 50
    Each puzzle is solved 50 times with different random choices, and the mean is taken as the puzzle's Nishio Human Cycles value, as described in Section 7.2.
  • Strategy set and ordering = Naked Singles, Hidden Singles, Naked Twins, X-wing
    Section 5.2 assumes Hsolvers know between two and four strategies and apply them in this fixed order, with no user study to support the ordering.
assumptions (5)
  • standard math The maximum and minimum SAT encodings correctly capture all Sudoku constraints
    Sections 4.1 and 4.2 rely on standard CNF encoding for Sudoku and cite Lynce and Ouaknine for the maximum and minimum encoding forms.
  • domain assumption The randomized Nishio solver of Figure 3 terminates and solves every valid Sudoku puzzle
    The paper asserts in Section 5 that the method can solve every Sudoku puzzle, but it gives no proof of termination or completeness.
  • domain assumption Website difficulty levels are a meaningful ordinal benchmark for evaluating the proposed metrics
    Section 8.1.1 assigns integer ranks to website labels and uses them as ground truth, even though the paper's motivation is that these labels are inconsistent across sites.
  • ad hoc to paper Three universal categories with equal-bin-count binning are appropriate for cross-site comparison
    Section 8.2.1 chooses 3 bins because one site has 3 levels and because reference [7] used 3 categories, but no external validation of the bin count is provided.
  • ad hoc to paper Human solvers behave like the stated strategy ordering and random trial-and-error
    Assumptions in Section 5.2 are based on the author's own solving experience rather than on a controlled user study, which weakens the claim that the simulation captures human solving effort.

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Cite this review

Pith. "Pith review of Project Patti: Why can You Solve Diabolical Puzzles on one Sudoku Website but not Easy Puzzles on another Sudoku Website?." pith.science (2026). https://pith.science/paper/64JRWJTK

@misc{pith2026250721137,
  author       = {Pith},
  title        = {Pith review of: Project Patti: Why can You Solve Diabolical Puzzles on one Sudoku Website but not Easy Puzzles on another Sudoku Website?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/64JRWJTK}},
  note         = {Machine review of arXiv:2507.21137}
}
read the original 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.

Figures

Figures reproduced from arXiv: 2507.21137 by the authors.

Figure 1
Figure 1. Two New York Times Sudoku Puzzles from July 23rd 2024 with candidates filled in Numbered Rows and Columns In Figure 1a, there is a digit on top of each column and to the left of each row. These digits, ranging from 1 to 9, represent the row and column numbers. A cell in any given Sudoku is represented by (x, y), where x is the row number and y is the column number. Candidates Candidates are the possible digits that … view at source ↗
Figure 2
Figure 2. A rectangle is drawn between the cells where X-wing should be used 5.2 Simulating Human Solver via Randomized Nishio with Human Strategies Armed with the above knowledge, I now state the assumptions on how a human solver, knowing these strategies, will solve Sudoku puzzles. I have made these assumptions loosely based on how I, a beginner to Sudoku, solved Sudoku puzzles. Going forward, human solvers will be referred… view at source ↗
Figure 3
Figure 3. Randomized Nishio Interleaving Human Strategies 6 Datasets Used in this Study For this study, puzzles from five popular Sudoku websites have been collected, along with the difficulty level of each puzzle in accordance with that of the website. Most websites have an archive, including Sudoku.org.uk, Extreme 8 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Nishio Human Cycles distribution over the 1320 Sudoku puzzle dataset. In this histogram, the width of each bin is is one Nishio Human Cycle [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Clause Length Distribution over the 1320 Sudoku puzzle dataset. In this histogram, the width of each bin is one percentage (short clauses). 16 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: shows the result: each of the 22 difficulty levels (across the 5 websites used in the Sudoku puzzle dataset) is mapped onto one of three universal categories using both its mean and median Nishio Human Cycles using 4 Human Strategies [PITH_FULL_IMAGE:figures/full_fig_…
Figure 7
Figure 7. Figure 7: Difficulty levels from the 5 websites classified into one of three universal categories based on Mean percentage of short clauses (from [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Nishio using Heuristics The solver in [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.