pencilmark

pencilmark rates sudoku puzzles the way a human solver experiences them. It solves a puzzle step by step with human techniques, easiest first, and reports how hard it is, how long it takes, which band it belongs to and which techniques you need. It works for classic sudoku of any standard size and for many variants (killer, thermo, arrow, kropki, XV, sandwich, multi-grid boards and more). It also solves puzzles, checks uniqueness, imports puzzles from SudokuPad, f-puzzles, Penpa+ and Logic Masters links, and generates new puzzles to a target band or difficulty.

The engine is written in C++ (pybind11). Results are deterministic: the same puzzle always gets the same card.

Written with AI. Almost all of pencilmark's code, data analysis and documentation was written by an AI model (Claude), directed and reviewed by its maintainer. If you object to AI-written software, take that into account before you use it or contribute.

Install

pip install pencilmark
pip install "pencilmark[verify]"      # optional: OR-Tools CP-SAT as a fallback uniqueness checker

Python 3.10 or newer. Building from source needs a C++17 compiler.

Quick start

import pencilmark

card = pencilmark.rate("....4.8.3.6...8..29.86.5..1235.........7..2.........96..2......8.12..7...9.......")
card["band"], card["difficulty"], card["time"]          # ('Hard', 5.9, 10.7)

card  = pencilmark.rate(puzzle, steps=True)             # add the step-by-step solve with explanations
cards = pencilmark.rate_many(puzzles, jobs=4)           # several puzzles in parallel, input order kept
sol   = pencilmark.solve(puzzle)                        # list of digits; raises NotUnique / NoSolution
exp   = pencilmark.to_export(puzzle, title="My puzzle") # the sudoku-puzzle format

A puzzle can be a classic string (. or 0 for blanks), a sudoku-puzzle dict, a puzzle JSON dict or string, or a path to a .json file.

Command line:

$ echo '....4.8.3.6...8..29.86.5..1235.........7..2.........96..2......8.12..7...9.......' | pencilmark rate - --report
#1  Hard  -  difficulty 5.9  -  time 10.7 min
   contains: Naked Pair (or Hidden Pair), X-Wing (or 2-String Kite / Simple Colouring / Short chain in the head)

pencilmark rate puzzles.txt --json        # full card per line (python -m pencilmark works too)
pencilmark solve puzzles.txt              # the solution, or 'not unique' / 'no solution'

More in Usage.

The card

rate returns a "sudoku-grade" card: by default the band (Very Easy to Extreme: what you need to know), the difficulty (how hard the steps are to find, no ceiling; a newspaper medium is about 5), the predicted median solve time in minutes, and the techniques the puzzle contains, easiest first. On request it adds publishing checks (a difficulty spike, a rule nobody needs, too many givens), the steps with explanations, time percentiles and more; only requested fields are computed. A puzzle without exactly one solution has no card: rate raises NotUnique / NoSolution. Every field is in Grade card.

Puzzle format

to_export writes, and every entry point reads, the "sudoku-puzzle" format, format_version 1: every constraint is a rule, houses included, each with one exact meaning and a default drawing, and an unknown key makes the reader refuse the file instead of dropping a rule.

{"format": "sudoku-puzzle", "format_version": 1,
 "meta": {"title": "Heartstring", "author": "...", "source": "example.org", "type": "Killer Sudoku"},
 "board": {"rows": 9, "cols": 9},
 "digits": 9,
 "givens":   [0, 0, 5, 4, 0, ...],
 "solution": [6, 1, 5, 4, 2, ...],
 "rules": [
  {"type": "box_grid", "origin": "r1c1", "box": [3, 3]},
  {"type": "cage", "sum": 15, "cells": ["r1c4", "r1c5", "r2c4"]},
  {"type": "edge", "kind": "kropki", "color": "white", "cells": ["r5c5", "r5c6"]}]}

The specification is Puzzle format (JSON Schema: docs/sudoku-puzzle.schema.json).

Other formats

For convenience pencilmark also reads and writes sudokuwiki.org packed strings, text grids and ipuz, reads pencil-mark grids, and loads SudokuPad, f-puzzles, Penpa+ and Logic Masters links. None of these is needed to use the sudoku-puzzle format; see docs/formats/other-formats.md. How rules map onto the engine is in docs/library/engine-mapping.md; how to add one, in Add a rule kind.

Supported rules

Classic sudoku from 4x4 to 25x25, irregular (jigsaw) regions, diagonals and extra regions, multi-grid layouts (samurai, twodoku, butterfly, windmill and more), killer cages, little killers, sandwich, X-sums, skyscrapers, frame sums, arrows, thermometers (normal and slow), German and Dutch whispers, renban, region-sum, palindrome, zipper, between, entropic, modular and parity lines, kropki, XV, greater-than, difference / ratio dots, anti-knight, anti-king, non-consecutive, odd / even, low / high, Rossini, Vudoku, equal sums, counting rules, and given candidates (Sukaku). Rules that cannot be translated exactly are refused, never dropped silently.

Generator

r = pencilmark.generate({"format": "sudoku-spec", "format_version": 1,
                         "board": {"rows": 9, "cols": 9}, "digits": 9,
                         "rules": [{"type": "box_grid", "origin": "r1c1", "box": [3, 3]}],
                         "where": {"band": "Hard"}})
r["puzzle"], r["card"], r["info"]       # sudoku-puzzle, its card, run details

The request is a sudoku-spec (docs/formats/generator-spec.md).

Targets: band, a difficulty range or the hardest / easiest found, required techniques, symmetry, given counts (per grid on a multi-grid layout), and rule elements with counts and shape limits (cages, thermos, whispers, renban, between lines, arrows, kropki, XV, parity). Boards: one grid, or a named multi-grid layout such as samurai, with locked rules of different digits. Every result is checked for uniqueness.

Puzzle sources

card = pencilmark.rate("https://sudokupad.app/...")  # SudokuPad, f-puzzles, Penpa+, Logic Masters, short links
exp = pencilmark.to_export("https://sudokupad.app/...")  # the same puzzle as a sudoku-puzzle file

The command line takes links too: pencilmark rate links.txt, one link per line.

Loaders that fetch pages identify themselves, obey robots.txt and pause between requests. Override the user agent with the PENCILMARK_UA environment variable.

Limitations

• Bands and difficulty measure different things. The band is the hardest technique a puzzle needs; difficulty is how hard it is overall. A puzzle that needs one X-Wing (Hard) is often easier than a long Medium one, and solvers agree (philosophy). This is by design, not an error.
• Difficulty and time are models fitted to human solving data. Differences under one point of difficulty are not meaningful, and a single solve can vary a lot from the predicted time.
• Bands depend on the technique catalogue; a puzzle that needs a technique the engine does not have is solved with Trial and error at that point and rated Extreme.
• Very hard puzzles with many-cell rules (killers, arrows, palindromes, thermos) can take minutes while the technique list (contains) is made minimal; use max_seconds to cap it (the card then says completed: false).
• The Sudoku Explainer-style ratings (se=True) are approximate and meaningful for classic sudoku only.

Benchmark

On a frozen held-out set of 7,623 puzzles with real human solve data from seven sources, pencilmark's difficulty ranks puzzles by human solve time at Spearman 0.61 to 0.80 within each group, closer than four established graders on the same classics. Details: docs/grades/benchmark.mdx.

Known issues

Being worked on after 1.0:

• Speed. Grading takes a median 0.49 s per puzzle on the benchmark (classics 0.04 s; 90th percentile 3.8 s), and the slowest few percent take 10 s to minutes. Sudoku Explainer and HoDoKu answer in under a millisecond (they do not model human solving), Rangsk's solver in 0.07-0.13 s. Use max_seconds to bound a call.
• Generator speed and range. About 20-70 s per puzzle that meets a spec (HoDoKu: 0.05-1 s), and classic Extreme puzzles are not reached within the default 120 s.
• Variants on one scale. On classics, difficulty matches established ratings of the same puzzles; across variant types the same difficulty does not yet mean the same experience (puzzle types differ by about 2 points on average, and variants often score above the classic range of their band). Time is most off for the same reason (about 1.4x per puzzle type).
• The ends of the scale. Between about 2 and 8, each point of difficulty means about 20-40% more solving time per cell. Above about 8 a point is worth less time (5-15% on three sources). Below 2 the scale is gentler (about the square root of effort, so it never goes negative); there it predicts solve times about 1.5% worse than a pure log scale would.
• Uniqueness on some killers. A few large killers can raise Undecided without OR-Tools (pip install "pencilmark[verify]") although human logic solves them.

License

MIT, see LICENSE.