Sudoku with multiple solutions
A Sudoku has multiple solutions when at least two different completed grids satisfy the row, column and box rules while preserving every given. Finding one valid completion proves solvability; it does not prove uniqueness. A uniqueness check must continue looking for a second completion.
The worked example
Coordinates start at the top left: r = row, c = column. Small digits are candidates; outlined cells are discussed below.
Check this exampleView puzzle text
Read left to right across each row, then top to bottom. A 0 means a blank.
003456789456789003789003456034567890567890034890034567345678900678900345900345678Keep every given and change only the blanks
Take the completed grid shown below as completion A. Remove every 1 and every 2 to get the example puzzle. There are nine of each, so 18 cells become blank and 63 givens remain.
Now swap every 1 and 2 throughout completion A: each 1 becomes 2, and each 2 becomes 1. This produces completion B. All digits 3–9 remain untouched, so both completions preserve every given in the puzzle.
Why the second completion is valid
Each row, column and box in completion A contains one 1 and one 2. Swapping their names everywhere still leaves one of each in every unit. None of the other seven digits moves. That proves completion B is a valid completed Sudoku, not just a conflict-free partial board.
The answers differ at r1c1. Two valid answers prove that the puzzle has multiple solutions; there may be more.
63 clues still do not guarantee uniqueness
This deliberately constructed example has many more givens than a typical newspaper puzzle, yet it is ambiguous. The count tells you how many cells are specified; it does not tell you whether those givens distinguish all valid completions.
Giving the two versions different difficulty labels would not fix the issue. Difficulty is about the work needed to reach a solution under an agreed method. Uniqueness is about how many valid completions exist. Establish uniqueness before attaching a normal puzzle difficulty rating.
Interpret the uniqueness check
A solver may stop as soon as it finds one completed grid. That is enough for ‘a solution exists’. To report ‘unique’, it must also rule out every alternative. Sudoku Slate’s checker searches until it has either exhausted the possibilities or found a second completion.
When the result is multiple solutions, do not treat the first returned answer as the intended answer. If you copied a puzzle that was meant to be unique, look for omitted givens. If you are designing the puzzle yourself, revise the givens and check uniqueness again.
Adding a given taken from one valid completion eliminates completions that disagree with it, but may leave several others. A fresh uniqueness check is still required. Never infer a unique result just because the extra digit removed the two examples you happened to inspect.
Check your understanding
Would writing r1c1 = 1 eliminate completion B? Would that fact alone prove the revised puzzle is unique?
Show the reasoning
It would eliminate completion B, because B has 2 at r1c1. But ruling out one known alternative is not a proof that no other alternatives exist. The revised puzzle needs its own complete uniqueness check.
About this example
This teaching board keeps many givens to make the deduction clear and has no difficulty rating. Diagnosis examples deliberately include the stated conflicts or ambiguity.
Rules and terminology
- Nikoli: Sudoku rules
The standard row, column and 3 × 3 box rules used in these examples. The board and worked explanation on this page were created separately.