Can Sudoku Have Multiple Solutions?

The short answer is no. Any proper Sudoku puzzle should only have one unique solution that can be found through logical deduction without the need for guessing.

That said, Sudokus can have multiple solutions or only one unique solution based on the initial starting clues and their positions in the grid. 

While people commonly think that Sudokus can only have one unique solution, any puzzle with seven or fewer starting clues must mathematically have multiple solutions. 

This is because in these Sudoku puzzles there cannot be a clue for every digit from 1 to 9 in the initial clue. In other words, there must be at least two digits that do not appear in any clue. 

That means whatever these two digits are, they can be used interchangeably in the finished puzzle, giving at least two distinct solutions. 

Sudoku minimum number of clues problem

But it’s not just puzzles with 7 or fewer clues that can have multiple solutions. 

Finding the minimum number of clues required in order for a Sudoku to have a unique solution is known as the Sudoku minimum number of clues problem. 

The answer to this problem is 17. 

While a concrete mathematical proof has not been found, mathematicians have analyzed every possible Sudoku puzzle with 16 clues through brute force and found that none have a unique solution. 

In other words, every Sudoku puzzle with 16 or fewer clues has multiple valid solutions. 

Sudokus with 17 clues or more may have multiple solutions

Just because the minimum number of clues required for a Sudoku puzzle to have one unique valid solution is 17, it doesn’t mean that Sudoku with 17 or more clues can only have one unique solution. 

It’s possible for these puzzles to have multiple solutions, again based on how the clues are given. 

For example, if every 1, 2, and 3 digits in a grid was given, which would mean there are 27 clues to start with, the other digits could still be used interchangeably. This would give rise to multiple solutions. 

Sudoku variants are a different story

Everything we’ve talked about so far only relates to regular Sudoku puzzles and not any variants with additional constraints. 

Variants like Thermo Sudoku where there are different types of clues or Chess Sudoku where there are different types of constraints complicate matters significantly. 

But again, whether there is a unique solution or multiple valid solutions will in just about all cases will come down to the initial clues. Both in how they’re structured and how many there are.