Ask most people what makes a sudoku hard and they will say: fewer given numbers. It is the obvious answer, it is what most generators use as their difficulty slider, and it is mostly wrong.
The count is a poor proxy
Two puzzles with an identical number of givens can be worlds apart. What matters is not how many numbers are on the board but where they are and what techniques the gaps force you to use.
A puzzle that can be finished entirely by scanning — looking along a row and asking what is missing — is easy no matter how sparse it looks. A puzzle that requires you to hold two candidate placements in mind and eliminate one by contradiction is hard even with plenty of numbers showing.
The real difficulty ladder looks roughly like this:
- Scanning — the answer is visible by elimination in a row, column or box.
- Pencil marks — you have to write candidates down before anything resolves.
- Pairs and triples — two cells that can only hold the same two numbers, which removes those numbers from everything else in the unit.
- Chains — following a hypothesis several steps until it contradicts itself.
A generator that only counts blanks cannot tell you which of these its puzzle needs. It is choosing difficulty by proxy and hoping.
The floor nobody mentions
There is a hard mathematical limit here: a 9×9 sudoku with a single solution needs at least 17 given numbers. This was settled by exhaustive computer search — not an estimate, a proof by enumeration.
Below 17, no arrangement produces a unique answer. And a puzzle with two valid answers is not a hard puzzle. It is a broken one, and the solver who found the 'wrong' correct answer will be certain your key is at fault.
Uniqueness is the thing to check
This is the check that separates a real generator from a grid shuffler, and it is invisible until it fails. After removing each number, the puzzle has to be solved again, exhaustively, to confirm exactly one solution still exists. If a second appears, the number goes back.
It is slow, which is why it gets skipped. Ours runs it on every 9×9.
We do not run it on 16×16. The search space is large enough that verifying uniqueness in a browser would take long enough to feel broken, so we are telling you rather than pretending.
If you are producing 16×16 puzzles for a book, solve a sample yourself before publishing. The risk is small but it is not zero, and a book is not patchable.
Choosing a size
| Size | Suits | Notes |
|---|---|---|
| 4×4 | Ages 4–7 | The rules without the search. Finishes in a minute. |
| 6×6 | Ages 7–9 | A genuine step up, still a single sitting. |
| 9×9 | Everyone else | The standard. All difficulty levels live here. |
| 16×16 | Enthusiasts | A long sitting. Uses 1–9 and A–G. |
One warning about 16×16 in a printed book: at that size the cells on an 8.5 × 11 page are small enough that pencil marks become genuinely awkward, and solvers past a certain age will not thank you. If you publish them, give them a full page and nothing else on it.
For a printed book
Group by difficulty in ascending order rather than mixing. A buyer who opens to page 3 and cannot start puts the book down; the same buyer who succeeds on page 3 will keep going into the hard ones and enjoy them.
And label the levels honestly. 'Expert' on a puzzle that yields to plain scanning is the fastest way to a review that says the book is mislabelled — which is a harder complaint to recover from than the puzzles simply being easy.