## Unsolvable puzzles?

Everything about Sudoku that doesn't fit in one of the other sections

### Unsolvable puzzles?

Does anyone know if it is possible to have a legal board position (ie, none of the clues violate the rules), but which does not have at least one legal solution? In other words, an unsolvable, yet legal, puzzle?

I suspect the answer is no, but I'm not sure; could someone post some sort of rigorous proof if true, and a counterexample if false?

Thanks
Hikaru79

Posts: 4
Joined: 04 April 2006

Hikaru79 wrote:Does anyone know if it is possible to have a legal board position (ie, none of the clues violate the rules), but which does not have at least one legal solution

Actually, it is faily simple to create these.

Take any valid sudoku, and use pencilmarks to show the valid candidates. Now solve the puzzle (or use a program to do it for you). Any candidate that is not part of the solution can be placed without violating the rules, but preventing a valid solution.

Here is a simple example:

Code: Select all
`6 . .|3 . .|. . .. . 8|. . .|. 1 .7 2 .|8 1 .|. . .-----+-----+-----3 . .|. . .|6 9 48 . 4|. 7 5|2 . 39 5 6|. . .|. . 1-----+-----+-----. . .|. 3 9|. 6 7. 9 .|. . .|4 . .. . .|. . 1|. . 2`

Can you spot the odd one out?

Ruud.
Ruud

Posts: 664
Joined: 28 October 2005

Ruud wrote:
Hikaru79 wrote:Does anyone know if it is possible to have a legal board position (ie, none of the clues violate the rules), but which does not have at least one legal solution

Actually, it is faily simple to create these.

Take any valid sudoku, and use pencilmarks to show the valid candidates. Now solve the puzzle (or use a program to do it for you). Any candidate that is not part of the solution can be placed without violating the rules, but preventing a valid solution.

Here is a simple example:

Code: Select all
`6 . .|3 . .|. . .. . 8|. . .|. 1 .7 2 .|8 1 .|. . .-----+-----+-----3 . .|. . .|6 9 48 . 4|. 7 5|2 . 39 5 6|. . .|. . 1-----+-----+-----. . .|. 3 9|. 6 7. 9 .|. . .|4 . .. . .|. . 1|. . 2`

Can you spot the odd one out?

Ruud.

Ah, yes. Obviously there must be a 5 at the middle-right area, but there are no legal places to put it.

Silly me, I should have seen this. Thanks for the response, Ruud!
Hikaru79

Posts: 4
Joined: 04 April 2006