Sudoku difficulty levels, explained by what they actually measure

Sudoku difficulty is not a measure of how many digits are missing. It is a measure of which solving techniques the grid forces you to use. That one sentence explains the clue-count myth, why the same puzzle gets different labels in different apps, and why an Expert grid can look emptier or fuller than a Medium one.

The clue-count myth

The intuition is reasonable and it is wrong. More blanks does not mean more difficulty, because the blanks are not what you fight. What you fight is the moment the easy techniques run out.

A grid with 24 clues that yields entirely to scanning is easier than a grid with 30 clues that will not move until you spot an X-wing. Both exist. Clue count sets a floor on how much work there is, not on how hard any of it is.

The floor itself is a settled question. Seventeen is the minimum number of clues a 9×9 Sudoku can have and still hold exactly one solution. Gary McGuire, Bastian Tugemann and Gilles Civario proved it in 2012 by exhaustively searching for a 16-clue puzzle and finding none, a computation of roughly 7 million core-hours, published in Experimental Mathematics in 2014. There are 49,158 essentially different 17-clue puzzles known, and plenty of them grade no worse than medium. Minimal is not the same as hard.

What a grader actually does

Difficulty is assigned by a program, not a person. The program is a solver with a ranked ladder of techniques, and it is required to be lazy: at every step it uses the cheapest technique that makes progress, and only climbs the ladder when nothing below works.

A typical ladder, cheapest first:

1. Scanning, also called cross-hatching
2. Hidden singles
3. Naked singles
4. Naked pairs and triples
5. Hidden pairs
6. Locked candidates, or pointing pairs
7. X-wing
8. Swordfish, XY-wing, colouring and above

The solver runs the whole grid and records two things: the highest rung it was forced onto, and how often. A grid that needs a single pointing pair and is otherwise singles is not the same as a grid that needs eleven of them. The label comes out of both numbers together. Some graders add a third input, the number of cells that were solvable at each stage, as a proxy for how much searching around the solver had to do.

Roughly what the labels mean

Note what changes across those rows. It is not the number of blanks. It is how far down the page of techniques you have to read.

Why two apps disagree about the same grid

There is no standard, no governing body and no agreed scale. Every publisher grades with its own ladder, its own weights and its own band boundaries. Three things drift:

• Which techniques are on the ladder. A grader that does not implement hidden pairs will climb past them to something harder, and over-rate the puzzle.
• Where the rungs sit. Locked candidates land in the hard band for one publisher and the medium band for another. That single choice moves a large share of all grids by one label.
• How the bands are calibrated. A newspaper that prints five grids a week from Monday to Friday is building a ramp, so its Friday is whatever its readers can just about finish, not an absolute measurement.

So “this app’s Hard is another app’s Medium” is not a complaint about quality. It is the expected result of two teams picking their own rungs.

Two numbers worth keeping

There are 6,670,903,752,021,072,936,960 completed 9×9 Sudoku grids, about 6.67 × 10²¹, counted by Bertram Felgenhauer and Frazer Jarvis in 2005. Strip out the symmetries, the rotations, reflections, relabellings and band swaps that turn one grid into another, and Ed Russell and Frazer Jarvis put the number of essentially different grids at 5,472,730,538 in 2006.

Which is the real answer to the question people mean when they ask whether a daily Sudoku app will ever run out. It will not.

Difficulty is not how much is missing. It is how far down the list of techniques the grid makes you read.

How Puzzle Lantern handles it

The Daily Sudoku runs the generator at its Easy setting today. That is deliberate and it is also below where we want it: the product aims a Daily at five to ten minutes, and Easy currently lands under that for a practised solver. It is on the list.

More puzzles is unlimited and lets you pick the band yourself: Easy, Medium, Hard or Expert. If you want to meet locked candidates on purpose, Hard is where they live.

Every grid, Daily or otherwise, goes through an independent counting solver that proves the solution is unique before the puzzle is published. A grid with two solutions is not a hard Sudoku, it is a broken one. The same solver is what the hint button uses, so a hint is always a real deduction from the current position rather than a peek at the answer. The generation pipeline is described here, and if you are stuck on a grid right now, the techniques above are worked through one by one.

Questions people ask

How is Sudoku difficulty determined?

A solver works the grid with a ranked ladder of techniques, always taking the cheapest that works, and the puzzle is rated by the hardest technique it was forced onto and how often.

Does a Sudoku with fewer clues mean a harder puzzle?

No. Clue count and difficulty are only loosely related, and some 17-clue puzzles grade no worse than medium.

What is the minimum number of clues a Sudoku can have?

Seventeen, proved in 2012 by McGuire, Tugemann and Civario through an exhaustive search of about 7 million core-hours.

Why do two Sudoku apps rate the same puzzle differently?

Because each uses its own technique ladder, weighting and band boundaries. There is no standard scale.

How many Sudoku grids are there?

6,670,903,752,021,072,936,960 completed grids, or 5,472,730,538 once symmetries are removed.