# How is this Sudoku not correct?

I used an IOS app from Brainium to do a Sudoku puzzle. I completed the puzzle but it did not indicate it was complete. So I clicked Hint and it highlighted all the incorrect entries in red.

But the puzzle has 1-9 in each box, row, and column. That is by definition a correct solution.

Can there be two solutions to any given Sudoku? Or is this app just buggy?

Your 2nd column has a couple of ‘5s’

Second column has two fives.

The third column has two 3s and no 5s.

nm, found two 3’s and no 5’s in the the third column so it all unravels from there.

slowpokes…

It’s only complete if it’s also correct. If you have repeats of numbers in any row, column or box, or missing numbers, it’s not complete. And yes, there is only one solution for each puzzle.

Your correct answer should look like this (bolds are fixed).

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Yeah, I think we all agree on that.

Nothing to do with this puzzle, but I gave my wife a book of easy Sudoku puzzles I found on Amazon - and most of them did not have unambiguous solutions. So buggy Sudoku generators do not exist.
I wrote them a scathing review. Never heard from them.

You would think they would check them before publishing, but, yes, I have encountered a Sudoku puzzle in a booklet with more than one valid solution.

So have I, but I wonder exactly how you check that a puzzle has a unique solution? It’s easy to check that a proposed solution is correct.

There’s one way you can know for sure: if it has fewer than 17 clues, the puzzle will have more than one solution. It’s been proven that there are no single-solution sudokus with fewer than 17 clues. (However, that does not mean that all 17+ clue sudokus have one solution.)

A very good question with these NP-complete puzzles, but with 9x9 puzzles computers can easily deal with it; for example the solver on http://www.math.ie/checker.html will print out all solutions to a given puzzle (that’s how they checked there are no valid puzzles with fewer than 17 clues, by running a similar program).

You verify that a Sudoku has only one solution by solving it, of course.

That doesn’t follow. Solving it only proves that it has a solution. It says nothing about whether there’s another solution.

The easiest way a sudoku can have two solutions is where there’s four cells with these properties:

1. are on the corners of a rectangle
2. all four are initially blank
3. there’s only two different numbers in those four cells
4. the four cells are contained in two 3x3 blocks

In those cases, there are two solutions with the four cells having these two solutions:

a b
b a

b a
a b

As I understand it, most sudoku generators go through a check stage where they look for this situation and eliminate any puzzles that have it.

However, you can have more elaborate double solutions involving 6, 8, 10, etc. cells in half that many blocks. At the extreme, it involves all 9 blocks, which means that there would be two numbers that are not in the initial puzzle. This last condition is easily checked, and I expect the generators eliminate those as well. I don’t know if they check for the intermediate double-solution conditions as well. I suspect they don’t.

There’s a difference between finding a solution to a Sudoku and solving it. Take the extreme example: Given a Sudoku grid with no numbers filled in, it’s really easy to find a solution, but it’s impossible to solve it. When you solve a Sudoku puzzles, you go through lines of reasoning that say that this square must be a 9, and so this square must be a 7, and so on. If you only fill in numbers (or at least, only use ink, or whatever) when you know that those numbers must be correct, then once you fill in the last number, you’ll know that there was only one solution, because it’s the one you found.

I was going to say this. When you solve a puzzle, for each square you determine the number it contains because it cannot logically contain any other number. Therefore if you solve a puzzle, it must necessarily have only one solution.

If you have a puzzle with more than one solution, you reach a dead end in the chain of reasoning and you cannot solve it. In such a case you would have to go outside puzzle-solving logic to determine what the alternate solutions could be.

That’s how I always reasoned it. I think what these other folks are saying is that you need a certain minimum number of clues in order to solve the puzzle correctly.