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Unsolvable Sudoku Puzzles

All too often I try to solve a Sudoku and end up believing it is impossible to solve and give up. I go through every strategy and trick I can think of and still can not find a square to solve.

Mistakes

One reason that my puzzle may be 'unsolvable' is that at some stage an incorrect allocation has been made. This may not be immediately obvious when you made the mistake. Here is an example of a puzzle which can not be solved, consider square Gg, all squares in row G are allocated except for this one, so according to this row Gg must take a 9, as it is the only choice in the row. But Dg is already set to 9 and another 9 can not go in Gg as column g would then have two 9s in it.

Unsolvable Sudoku

In this case the mistake was assigning Gc to 3 it should have been a 9.

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SudokuDragon will spot this incorrect allocation using its hint facility and let you use undo to go all the way back and correct the square allocation.

Here is another example unsolvable puzzle. In this case we have two squares Bi and Ih that are in a bad state, no numbers can be allocated to these squares at all. If you run through the numbers 1 to 9 you will find there is already one of them allocated in a shared row, column or region. What again has happened is that an incorrect choice was made for a square and the repercussions of this has only become apparent at this point.

Unsolvable Sudoku

Going back and 'Undoing' all the allocations, it turns out that allocating Dd to 8 was the mistake, it should have been a 9.

Sudoku Dragon has a hint that will show any incorrect allocations that you have made.

Multiple Solutions

Another example of a puzzle that can not be solved is when it has multiple legitimate solutions. Following all the rules you end up with two or more different solutions from the same starting squares.

Here is the puzzle and two solutions, both of which fulfill all the rules. We have highlighted the squares that differ in the solution.

Multiple Solution Sudoku Puzzle
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A puzzle with multiple solutions can often be created by accidentally missing off a square when copying a puzzle from a book or newspaper. Or it may be the case that it is a badly formed puzzle, and the person/program generating it has made a mistake. It is hard to devise puzzles that are challenging enough and yet have a unique solution.


Multiple Solution Sudoku Solution 1 Multiple Solution Sudoku Solution 2

Hard Puzzles

Sometimes we shout 'unsolvable puzzle' where we fail to spot the way forward. Consider this 'hard' puzzle, after solving all the 'easy' squares it reaches this point.

Hard Sudoku - hidden twins
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There are no more obvious squares to solve. However it is still solvable as there are more complex strategies available that yield more squares, in this case it is hidden twins and shared subgroups that permit further squares to be allocated. So often it is not spotting possible usage of a rule that can make you think you have an unsolvable puzzle.

As too often happens after you have solved a few more squares you get stuck again. This time the twin; shared subgroup rules although excluding some possibilities do not allow any more squares to be solved

In fact the full force of alternative pair analysis needs to be used to be able to solve any more squares. A look at the squares where 1s can go gives a complex and on the face of it not much help. However Ef and Gf form a pair, and Gf and Hd form a pair within the region. To complete the chain Hd and Hi also form a connected pair. This enables the possibility of a 1 in intersecting square Ei to be excluded. The square Ei must therefore take a 2 and after that difficult step the puzzle can be solved with no more problems.

Hard Sudoku - alternate pairs

This is the real issue with hard puzzles, they can seem 'unsolvable' to all but the committed Sudoku expert. We sometimes receive puzzles that our users consider 'unsolvable', but with a bit of study they can usually be solved.

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