A properly made sudoku has exactly one solution, and that is what lets you solve it by reasoning rather than guessing: every digit you place is forced by the ones already there. A puzzle needs at least 17 givens to have a single solution (a fact proved by computer search in 2012), and the number of givens has surprisingly little to do with how hard it is.
Why one solution matters
If a puzzle had two solutions, at some point you would reach a square where two digits both work and nothing on the board can decide between them. The only way on would be to guess. Puzzle makers treat a unique solution as part of the definition of a sudoku for exactly that reason: it guarantees that logic alone always reaches the end.
Every puzzle in Sudoku is checked before you see it. The generator starts from a complete grid and removes givens a pair at a time. After each removal it checks that the puzzle still has only one solution (for easy and medium, by solving it with singles alone, which can only succeed when the answer is unique) and puts back any removal that would allow a second one.
Seventeen clues
For years nobody knew the smallest number of givens a sudoku with one solution could have. Many 17-clue puzzles were found, but no 16-clue one. In 2012 Gary McGuire, Bastian Tugemann and Gilles Civario settled it with an exhaustive computer search: there is no sudoku with 16 givens and a single solution. Seventeen is the minimum.
How many sudoku grids are there?
In 2005 Bertram Felgenhauer and Frazer Jarvis counted every possible completed grid: 6,670,903,752,021,072,936,960 of them. Many are the same grid with its digits relabelled or its rows and columns shuffled within the rules; counting those as one leaves 5,472,730,538 essentially different grids, a figure worked out by Ed Russell and Frazer Jarvis. Each one can be turned into a great many different puzzles by choosing which squares to leave empty.
Fewer clues is not the same as harder
It is tempting to think the number of givens sets the difficulty. It does not: what matters is which techniques the puzzle forces you to use. Our own puzzles show it:
| Level | Typical givens | Techniques needed |
|---|---|---|
| Easy | about 38 | singles |
| Medium | about 30 | singles |
| Hard | about 27 | pairs and pointing digits |
| Expert | about 27 | beyond pairs |
Hard and expert puzzles start with about the same number of givens, yet an expert one cannot be finished with the techniques a hard one needs. A 17-clue puzzle can be easy, and a 30-clue one can be very hard. That is why our levels are set by grading each puzzle with a solver that uses human techniques, not by counting squares.
Is guessing ever needed?
For a puzzle with one solution, never in principle: the answer is always forced. In practice the very hardest puzzles need long chains of reasoning that are hard to hold in your head, and some solvers prefer to try a digit and see whether it leads to a contradiction. That is still logic (a proof that the digit is wrong), and undo makes it free in our game.
For the techniques themselves, see how to solve sudoku; to start from the beginning, sudoku for beginners. Or play Sudoku.