What actually makes a Sudoku puzzle hard?

1 September 2026 · 7 min read

Ask most people what makes a Sudoku hard and they will say “fewer numbers to start with”. It sounds obvious and it is mostly wrong. Clue count and difficulty are only loosely related, and understanding why explains what difficulty actually is.

Why counting clues does not work

What makes a puzzle hard is not how much is missing but which techniques you need to fill it in. Two puzzles with the same number of clues can be minutes apart, because difficulty lives in the structure of what is given, not the quantity.

A concrete example from building this site: the best-known Sudoku on the internet — the one on its Wikipedia article — has 30 clues. By clue count that lands it squarely in most sites’ Hard band. Run it through a solver that knows only the simplest technique in the game and it finishes anyway, with naked singles alone, never needing anything else. It is an easy puzzle wearing a hard puzzle’s clue count.

The reverse happens too. A 32-clue puzzle can require pattern-hunting across two units at once. The clues are there; what is missing is a cheap route through them.

What difficulty actually measures

A useful grade answers: what is the hardest thing you are forced to do? That is settled by solving the puzzle with a restricted solver — one that knows only certain techniques — and seeing which restrictions still succeed.

That is exactly how the four difficulties here are defined:

• Easy — naked singles alone are enough, start to finish.
• Medium — hidden singles are also needed somewhere.
• Hard — a naked pair or a pointing pair is genuinely required.
• Expert — those four are not enough; something beyond them, such as an X-wing, is needed.

The wording matters. Not “an Easy puzzle has 40 clues” but “an Easy puzzle can be finished with the simplest technique there is”. That is a promise about your experience rather than a statistic about the grid — and it is checkable, which is why every puzzle here is graded by solving it before you see it.

The puzzle below is graded Easy on precisely that basis: naked singles will carry it all the way, despite the 36 empty cells.

The 17-clue floor

There is a real lower limit, and it is one of the few genuinely settled facts about Sudoku: no puzzle with 16 or fewer clues has a unique solution. Many 17-clue puzzles do.

This was not proved by clever argument but by exhaustive computer search — a team at University College Dublin checked every possible arrangement and published the result in 2012, after roughly a year of computation. It had been suspected for years; nobody could rule out a 16 until the search finished.

And here is the part that matters for difficulty: 17-clue puzzles are not automatically the hardest. Some are, some are unremarkable. Minimal clues make a puzzle interesting to mathematicians, not necessarily hard to solve.

What “unique solution” buys you

A properly made puzzle has exactly one solution. That is not a stylistic preference — it is what makes the puzzle solvable by reasoning at all.

If a grid had two solutions, then somewhere there would be a set of cells that could be filled either of two ways with no way to distinguish them. Every deduction would run out and you would have to guess, with no logical reason to prefer one guess. A puzzle with two solutions is not a hard puzzle; it is a broken one.

Uniqueness is why you can trust “this digit must go here”. It is the guarantee behind every technique on this site, and it is checked for every generated puzzle — a candidate grid is only accepted if a solver confirms it has exactly one answer.

How puzzles get made

Backwards, which surprises people. You do not design a puzzle and then solve it; you start from a full solution and take digits away.

1. Generate a complete, valid 9×9 grid.
2. Remove a clue.
3. Check the puzzle still has exactly one solution. If not, put the clue back and try a different one.
4. Repeat until enough clues are gone.

Difficulty grading needs one more step, and it is the one that separates a good generator from a lazy one: after each removal, also check that the puzzle is still solvable within the intended difficulty band. Without that check, a generator aiming for Medium produces something that happens to have Medium’s clue count but may need an X-wing — which is precisely the bad experience of a puzzle labelled Medium that you cannot finish.

Why some puzzles feel harder than they are

Two puzzles at the same grade can feel very different, for reasons that have nothing to do with technique:

• How many moves are available at once. A puzzle that offers one forced move at a time feels much harder than one offering five, even when the techniques needed are identical. You spend the whole solve hunting rather than placing.
• Where the hard step sits. A single tricky deduction on move 4 is discouraging; the same deduction on move 40, after momentum has built, barely registers.
• Whether the clues are clustered. Clues bunched into a few boxes leave large empty regions where scanning finds nothing, which feels like being stuck even while the puzzle is technically easy.
• How pretty the pattern is. Symmetrical clue layouts look better and are slightly harder on average, because symmetry constrains where clues can go and rules out some of the helpful placements.

The practical version

If you want to get better, stop reading clue counts and use difficulty labels as a statement about technique. Stay at a level until its hardest required technique is one you find quickly, then move up. Jumping to Expert because 24 clues “looks doable” means meeting patterns you have no method for, which teaches nothing except that Sudoku is frustrating.

And when a puzzle at your level does stop you, that is a specific, fixable gap rather than a wall — there is a named technique you have not met yet. The order to work through when stuck is the fastest way to find out which one.

Every diagram on this page is a real position from a generated puzzle, and every elimination shown is checked against that puzzle’s unique solution as part of the test suite. Want a move explained on your own board? Play a puzzle.