Can a Sudoku Have More Than One Solution?

A proper sudoku has exactly one solution. A set of clues can allow two or more, but then it is a flawed puzzle, and good sources check every puzzle by computer before publishing it. You can recognise a two-solution puzzle by its typical symptom: near the end, four cells at the corners of a rectangle, spread over two boxes, can hold the same two digits either way round. No logic can choose between them, because both answers are right.

What a puzzle with two solutions looks like

We took a proper puzzle and deleted a single clue, the 3 in R2C8 (row 2, column 8). The puzzle now has exactly two solutions. Here it is with everything filled in except the four cells that differ between them.

• 5 clue
• 5 already solved
• 5 new digit
• 5 pattern candidate
• 5 candidate to remove

The pattern is always the same shape. The four cells sit in two rows, two columns, and exactly two boxes, and hold the same pair of digits. Swapping the pair keeps every row, column, and box complete, so the swap produces a second valid grid. Solvers call this a deadly pattern. Real two-solution puzzles can have bigger versions (six or eight cells in a loop), but the four-cell rectangle is by far the most common.

Why a single solution matters

With one solution, every digit is forced by the clues, and you can always reach it by logic. With two, the last step of the puzzle is a coin flip, and all the careful reasoning before it cannot help. That is why puzzle makers treat uniqueness as the basic requirement. It is also why a proper puzzle can always be solved without guessing, which we cover in can you solve sudoku without guessing.

There is a limit to how few clues a unique puzzle can have. In 2012, Gary McGuire, Bastian Tugemann, and Gilles Civario showed with an exhaustive computer search that no 16-clue sudoku has a unique solution. So 17 is the minimum. Every puzzle with 16 clues or fewer has at least two solutions. Having 17 or more clues does not guarantee uniqueness, though: the puzzle above has 24.

Turning uniqueness into a technique

If you trust that a puzzle has one solution, you can use the deadly pattern backwards. When three corners of a rectangle across two boxes hold only the same pair, the fourth corner must hold something else, or the puzzle would have two solutions. That is the unique rectangle. BUG+1 uses the same idea on the whole board: if every cell but one has two candidates, the odd cell must take the digit that breaks the deadlock. Both have worked examples in what to do when you are stuck with only pairs.

Some solvers refuse these techniques on principle, since they assume something about the puzzle instead of reading it off the grid. Both views are fine. Just do not use them on a puzzle from a source you do not trust, because on a genuinely two-solution puzzle they can remove a correct candidate.

How to check whether a sudoku has one solution

By hand, you mostly cannot prove it in advance. You find out by solving: if logic carries you to the end, the solution was unique. If you end up with a rectangle of interchangeable pairs and nothing else to place, it was not.

A computer checks it directly. A backtracking solver tries every possibility and counts the solutions it finds, stopping as soon as it reaches two. For a 9x9 puzzle this takes a fraction of a second, which is why every serious puzzle source runs it. Our guide to how sudoku solvers work explains the method, and the mathematics of sudoku covers how many grids exist: 6,670,903,752,021,072,936,960 completed grids in total, as counted by Bertram Felgenhauer and Frazer Jarvis in 2005.

What to do if your puzzle has two solutions

First make sure it really does. Check your clues against the source, since a miscopied clue can create a second solution or remove the only one. Then look for the rectangle: four cells, two rows, two columns, two boxes, one shared pair. If you find it and every other cell is solved and consistent, fill in either arrangement. Both are correct, and the puzzle is at fault, not you.

If instead you have a cell with no possible digit, the puzzle is fine and one of your entries is wrong. How to find a mistake in sudoku shows how to trace it.

One solution, every time, in Sudoku Royale

Every puzzle in Sudoku Royale has exactly one solution. Our grader proves it for each puzzle by solving it to the end with logic alone, using techniques that never assume uniqueness, which is only possible when the solution is unique. That covers all 457,000 puzzles across the six Classic levels and the Daily Challenge.