Technique frequency by difficulty

Every number is computed from the stored puzzle files by a committed script, and the sample sizes and method are shown so you can judge what they do and do not prove.

What was measured, and on what

The site keeps a pool of verified classic puzzles for each of its five difficulty levels: Easy, Medium, Hard, Expert and Evil. Each pool holds 1,000 puzzles, so the full sample is 5,000 puzzles. For every puzzle the generator recorded which techniques the site’s own technique engine used while solving it, and the script scripts/technique-frequency.ts adds those records up. This page reports the totals for the classic 9×9 game only. The variants have their own pools and are not covered here.

The engine solves cheapest technique first. It looks for a naked or hidden single, and only when none exists does it try the next technique in its list, then the next. A technique is counted for a puzzle if the solve used it at least once. That makes the numbers a record of what the engine reached for, which is a proxy for what a human solver would need, not a direct measurement of human behavior. The full ordering and the tier each technique belongs to are described in how Sudoku difficulty is graded, and how the puzzles were built and checked is in how puzzles are generated and verified unique.

The table: how many puzzles needed each technique?

Each cell shows the number of puzzles out of 1,000 that used the technique at least once, with the percentage in brackets. The second column gives the tier in which the site’s grading first requires the technique.

Source: data/stats/technique-frequency.classic.json, generated by scripts/technique-frequency.ts. The killer-only cage techniques are also in the engine, and they are zero in every classic row.

How should the zeros be read?

Several of the zeros are built in. A puzzle is placed in the tier of the hardest technique it needs, so a puzzle that required an XY-Wing is by definition not an Easy, Medium or Hard puzzle. That is why XY-Wing, Swordfish and simple coloring show 0 in the first three columns, and why Easy shows 0 for every technique above the singles. These zeros follow from how the tiers are defined and are not independent findings. The same goes for the Evil column of unique rectangles: 1,000 of 1,000 is the definition of the tier showing up in the data, since an Evil puzzle is one that needs a unique rectangle or something beyond the implemented techniques.

The numbers that are not forced by definition are the interesting ones, and they are the ones worth reading closely.

What does the table show for each level?

Easy

All 1,000 Easy puzzles used naked singles, and 979 also used hidden singles. The other 21 were solved by naked singles alone. Nothing else appears, which is the definition of the level. If you can read a cell’s candidates and spot a digit that has only one possible cell in a row, column or box, you can finish every puzzle at this level. Both techniques have their own lessons: naked singles and hidden singles.

Medium

Medium adds the four tier-2 techniques, and the spread between them is wide. Pointing pairs were used by 952 puzzles (95%). Box/line reduction was used by 301 (30%), naked pairs by 285 (28%) and hidden pairs by 114 (11%). If you are moving up from Easy, pointing pairs is the technique that the data says you will meet in almost every puzzle. Hidden pairs, the hardest of the four to spot, are needed in barely one puzzle in nine.

Hard

Hard is the first level with tier-3 techniques. Naked triples were used by 578 Hard puzzles (58%) and X-Wing by 462 (46%). The two overlap, since together they add to more than 1,000, so many Hard puzzles used both. The tier-2 techniques stay common: pointing pairs 88%, box/line reduction 51%, naked pairs 46%. A Hard puzzle is therefore mostly Medium technique with a small number of heavier steps. Naked triples and X-Wing are the new skills to learn at this stage.

Expert

The Expert column is where the pattern shifts. XY-Wing was used by 607 puzzles (61%) and simple coloring by 570 (57%), so these two carry most of the load. Swordfish appears in only 43 Expert puzzles (4%). Naked triples drop to 76 (8%) and X-Wing to 168 (17%), well below their Hard figures. The data does not explain why. One possibility is that the engine’s cheapest-first order means a puzzle that has an XY-Wing available often never gets as far as a triple, but that is a guess and this page does not test it. What the table supports is the practical point: XY-Wing and simple coloring are the techniques to practice before Expert, and Swordfish can wait.

Evil

Evil puzzles all need a unique rectangle, and they also lean on the earlier techniques more than Expert does. Naked pairs appear in 67% of Evil puzzles against 49% of Expert, and box/line reduction in 56% against 41%. Simple coloring (34%) and XY-Wing (30%) are used less than in Expert. In these pools, then, Evil is not simply Expert with harder chains. It is defined by the unique rectangle and reached through a different mix of steps. Unique rectangles rely on the puzzle having exactly one solution, which is why the lesson spells out when the shortcut is safe.

Does clue count predict difficulty?

Not in these pools. The second table gives the number of starting clues for the same 5,000 puzzles.

The mean runs from 24.4 on Easy to 24.8 on Evil, a difference of 0.4 clues, and the ranges overlap almost completely. A puzzle with 21 clues can be Easy, and one with 29 can be Evil. The generator removes clues while keeping the puzzle uniquely solvable and then grades what is left by technique, which is why this site grades by technique and not by clue count. These puzzles all sit near the low end of what classic Sudoku allows, so the result applies to this pool and should not be read as a general law about Sudoku.

What this data cannot tell you

• It is not human solving data. The engine uses a fixed order and never gets tired or misses a pattern. A person may solve a puzzle with different steps, or miss a technique the engine would find.
• It covers only this site’s puzzles. The pools are the puzzles the generator produced and the grader accepted for each tier. They are not a random sample of all Sudoku puzzles, and a different generator or grader would give different numbers.
• “Used at least once” hides how often. A puzzle that needed a single pointing pair and one that needed ten both count as one. The table says nothing about how many times a technique is applied.
• The grading is this site’s own. The tiers are the site’s method, not an official or industry standard, and other sites label difficulty differently.
• Only the classic game is covered. Variant pools are smaller, some have fewer tiers, and none is reported here.

How to use the table when you practice

1. Pick the level you currently finish comfortably and look at its column. Any technique with a high percentage there is one you are already using or should be.
2. Look one column to the right and find the techniques whose percentages rise from zero or from a small number. Those are the new skills for the next level.
3. Learn the high-frequency ones first. Pointing pairs for Medium, naked triples and X-Wing for Hard, XY-Wing and simple coloring for Expert.
4. Practice on the matching level on the home page, where you can choose the difficulty. The strategy guide for each difficulty turns this into a step-by-step plan.

Where the numbers come from and when they change

Nothing on this page is typed by hand. Running node scripts/technique-frequency.ts reads every puzzle file in data/puzzles/classic, takes the technique list stored with each puzzle, and writes the totals to data/stats/technique-frequency.classic.json. This page was written from that file after the script was re-run and produced the same counts as the committed copy. The only difference was a new all-zero column for a killer-cage technique that the engine includes.

If the pools change, for example when more puzzles are added, the script is run again and this page is updated from the new file. Because every stored puzzle is also re-checked in continuous integration for exactly one solution, the counts describe puzzles that are verified unique. For the grid construction, the clue-removal loop and the solution counter, read the generator guide. For the technique series in learning order, start at the techniques hub, and for other tutorials see the guides hub.