Datasets:

twistshan
/

Sudoku_DLM_Reasoning

This document defines the difficulty fields used by Sudoku DLM Reasoning release 1.1.0. Difficulty is an operational, solver-relative measurement; it is not a human Sudoku grade and is not defined by the number of clues.

1. Provenance of the rating

For the Sudoku Extreme rows, the upstream rating column is copied into official_rating and labeled rating_type = tdoku_backtracks. The converter does not recompute the rating:

Sudoku Extreme card: https://huggingface.co/datasets/sapientinc/sudoku-extreme
Conversion code: https://github.com/hengyuf/diffusion-vs-ar/blob/ba0445f4d0808113bf08a9a9d8c7041086bfdf10/tools/prepare_sudoku_datasets.py

The upstream card describes the value as the number of backtracks required by Tdoku. At public Tdoku commit af426180dc53aef89b82868e7b3fdfcf42165654, the counter is named num_guesses_ and is incremented whenever the DPLL solver expands a binary configuration decision:

Branch implementation: https://github.com/t-dillon/tdoku/blob/af426180dc53aef89b82868e7b3fdfcf42165654/src/solver_dpll_triad_simd.cc#L539-L559
Branch-variable selection: https://github.com/t-dillon/tdoku/blob/af426180dc53aef89b82868e7b3fdfcf42165654/src/solver_dpll_triad_simd.cc#L473-L536

Operationally, official_rating is best interpreted as the number of search decision nodes visited before the first solution, rather than literal failed undo operations.

2. Constraint propagation and the DFS root

Tdoku does not begin search from an empty board or from the first empty cell. For a standard 81-character puzzle it:

loads every given digit;
eliminates incompatible row, column, box, band, and stack configurations;
propagates those eliminations until the initialized state is consistent; and
calls depth-first search on that propagated partial state.

The initialization and call path are visible in that pinned public source:

Propagation applies only logical consequences of the current givens or branch assumption. It makes no new hypothesis and is not counted by the rating. It can end in a complete solution, a contradiction, or a fixed point that still has multiple configurations.

This matters because two puzzles can both have rating zero while requiring very different amounts of propagation.

3. What one branch means

For one digit in a three-row band (or a three-column stack), there are six possible global placements across the three boxes. Tdoku represents these as band configurations. When propagation leaves multiple configurations, the branching heuristic selects:

the unresolved band or stack with the fewest configurations; then
the digit in that band or stack with the fewest configurations.

It then creates a binary decision:

left branch: force the first remaining configuration;
right branch: exclude that configuration and retain the others.

The counter increases once for this binary split. If the left branch fails and the right branch immediately solves the puzzle, the count for that decision is still one. A later split of the remaining configurations increases it again. Therefore the rating is not the recursion depth, the number of filled cells, the number of contradictions, or a human solving-step count.

4. Bucket definitions

The source converter applies these thresholds:

release label	operational rule	interpretation
`original`	no Tdoku rating	separate easy collection; not numerically calibrated to Extreme
`r0`	rating = 0	propagation resolves the puzzle without a search decision
`r1_4`	1 <= rating < 5	1--4 visited decision nodes
`r5_19`	5 <= rating < 20	5--19 visited decision nodes

The upstream converter also defines r20_49, r50_99, and r100_plus, but those buckets are outside this v1 candidate pool. Thus r5_19 is an adjacent held-out difficulty band, not the hardest part of the upstream corpus.

5. Quantitative audit of the raw candidate pool

The following statistics were computed over all 1,415,420 rows included under raw/. Rating and clue statistics use the full pool.

bucket	rows	rating mean / median	clue mean / median	clue range
`original`	101,000	not rated	33.813 / 34	29--37
`r0`	614,136	0 / 0	26.282 / 26	17--37
`r1_4`	588,453	1.976 / 2	25.604 / 26	17--31
`r5_19`	111,831	11.776 / 12	24.932 / 25	17--30

Clue count is not a substitute for this rating. Within r1_4, the Pearson correlation between rating and clues is -0.035; within r5_19 it is 0.002.

As an independent diagnostic, standard row/column/box naked-single propagation was run over all 101,000 original rows and deterministic 20,000-row reservoir samples of each Extreme bucket. The reservoir seeds were 20260816 for r0, 20260817 for r1_4, and 20260818 for r5_19:

bucket	rows checked	solved by naked singles
`original`	101,000	100,959 (99.959%)
`r0`	20,000	3,691 (18.455%)
`r1_4`	20,000	0
`r5_19`	20,000	0

The 41 original rows that stalled under naked singles were all solved after adding hidden-single propagation. This independent audit describes basic Sudoku logic, not Tdoku's own internal propagation, which is configuration-based and stronger.

6. Released train and evaluation distributions

The three training buckets are equal in size, but exact ratings inside r1_4 retain the upstream proportions:

exact rating	train rows
1	29,584
2	16,809
3	10,244
4	8,899

The main r1_4 test allocation instead uses 250 rows at each exact rating 1, 2, 3, and 4. The r5_19 test allocation uses 250 rows from each of [5,8), [8,12), [12,16), and [16,20); test_ood_confirm uses 1,000 per range. This supports per-rating or per-range reporting rather than only a bucket-wide average.

7. Source confounding

Difficulty and source collection are not independent in the raw pool:

r0: 77.40% puzzles1_unbiased, 16.28% puzzles0_kaggle, and 6.30% puzzles2_17_clue;
r1_4: 85.02% puzzles1_unbiased, 9.70% puzzles4_forum_hardest_1905, and 3.40% 01_file1;
r5_19: 68.19% puzzles4_forum_hardest_1905, 28.15% 01_file1, and only 2.18% puzzles1_unbiased.

Consequently, an r5_19 result combines a Tdoku-rating shift, a held-out training-exposure shift, and a source-distribution shift. It must not be interpreted as a pure causal effect of search count.

Recommended reporting includes exact rating or subrange, source collection, and clue count. A source- and clue-matched diagnostic set is appropriate when the goal is to isolate the rating axis.

8. Reproducibility limitation

The public Sudoku Extreme card does not pin the Tdoku commit, build flags, command, permutation policy, or seed used to create its rating column. This release therefore preserves official_rating as an upstream measurement rather than claiming to reproduce it.

For a fully controlled future rating, pin a Tdoku commit and configuration, record the branch count for the exact stored puzzle, and optionally measure its mean and variance over Sudoku-preserving isomorphic transformations.