# Design Two Sudoku Puzzles

I’ve recently gotten into Sudoku, and I’ve been wondering about some things.

1. Is there a theoritical minimum number of spaces that can be filled in on a Sudoku puzzle so that the puzzle is solvable? If so, can you point to one such puzzle (or design one yourself)?

Conversely…

1. Is there a theoretical maximum number of spaces that can be filled in on a Sudoku puzzle, but yet the puzzle remains unsolvable? Again, if so, can you point to one such puzzle (or design one yourself)?

Yes, there is. The theoretical minumum for a 9x9 sudoku square is 17 clues. This site has a brief discussion on it and links to such sudokus.

According to this site (warning: pdf), the maximum number of clues for an unsolvable 9x9 sudoku is 33; there is not an example though.

That’s quite a confusing definition of solvable you’re using there. I’d say a sudoko puzzle is solvable if a solution exists. That is, if all the open spaces can be filled in in such a way that there are no contradictions.

So the answer to question 1 would be zero (the solution of any sudoko would be a solution) and 81 would be the other answer.

What you’re questioning here is the uniqueness of solutions.

Not that this answer is of any help…

The rule of Sudoku is that there is always a unique solution. So by definition, a Sudoku puzzle which doesn’t have a unique solution is unsolvable.

As your link mentions, that result is more empirical than theoretical. As for the maximum, I think I’ve seen a grid with only four empty spaces and two solutions, but I can’t find a reference right now.

Four empty spaces and two possible solutions is relatively common – I have found a few of these in newspapers and such – you can use logic and deduction to complete everything except the final four squares. I will see if I can find one.

``````
3 456 789
654 987 321
789 123 456

5 678 934
436 295 178
897 314 562

541 762 893
972 831 645
368 549 217

``````

The above puzzle does not have a unique solution. You can fill the four blanks with 12/21 or 21/12.

As for the answer to the OP… well, I don’t know. But I think it is sensible to distinguish between legal solutions and unique solutions. A well designed puzzle will have a unique solution, but I think the OP was talking about legal solutions.

Another rule of sudoku is that the clues don’t contradict each other. Combining our two rules, a sudoku puzzle is, by definition, solvable. But using the word (un)solvable in this way doesn’t make much sense in the context of the OP.

Her second question makes clear that she was talking about unique solutions. If she were talking about the other way in which the puzzles can be unsolvable (inconsistent clues: no legal solutions), the question doesn’t make any sense. Once you have such a sudoko you can just keep on adding clues and it remains inconsistent, so it would be a bit strange to ask for a maximum. I think she meant to ask:

What’s the maximum number of clues in a consistent sudoku, such that by adding a clue the sudoku would become either inconsistent or uniquely solvable?

Your example, combined with the fact that a consistent sudoku with three open spaces has a unique solution, makes clear that the answer is 77.

I’m a he.

The first rule of Sudoku is: DON’T TALK ABOUT SUDOKU!

Do you mean solvable as in, once x number of squares have been filled in you can use logic to solve the puzzle without resorting to guesses? If that’s what you meant I guess there would be a minimum, and going by j_sum1’s post it would be 3 blank squares.

Yes, that’s exactly what I meant. And that it would have a unique solution.