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This is the entire puzzle. Numbers in red are the clues posted in the West Australian paper. It is not the first time that this paper has posted a grid with multi-solutions, except it was not obvious to me this time.

Assuming this grid had a unique solution, I eliminated 79 in r9c7 and 6 in r2c7, but was stuck at that point. I completely overlooked a second close loop of the 46 pairs. Otherwise, I could have reached a solution by assuming the grid has one unique solution and eliminated 7 in r9c9 or 9 in r1c8. But then again, I would have thought that the assumed solution was the only solution, and not bothered to go back to check the validity of the assumptions. In fact, this grid has 3 solutions.

From this exercise,

I learned how to identify potential multi-solution grids of more complex patterns.

I learned that I should always go back and check whether the assumption of unique solution was valid.

I learned more about forcing chains.

Thanks for your help.

- Jeff
**Posts:**708**Joined:**01 August 2005

Hmm, 3 solutions (not 5).

I looked into whether there was some plausible logical argument for preferring one of the solutions. Something like "any of group 1 of comparable choces give an ambiguous solution, but any of group 2 of comparable choices give a unique solution. Therefore make a group 2 choice and the group 2 choice solution is the unique one".

But no luck. Assigning any tripleton value selects one of the 3 solutions, so no basis there to prefer one choice over another. You can exclude 9 from the quad node, as 9 gives a contradiction.

For the doubleton nodes, one choice led to a unique solution and the other choice left an unresolved pair of solutions. If the choices giving a unique solution consistently gave the same solution, that would be a good argument for claiming that as the unique solution to this puzzle. But not this time. Making a double node choice leading to a unique solution can get two of the three solutions.

For really perverse "logic"(?), you might argue that the third solution, not reachable as a unique solution from any doubleton choice, is really the special and therefore "unique" solution. But's that's too weird a stretch for me.

As another meta-argument, you might say if there is a unique solution, it must be one of the two reachable by a choice giving a unique solution form a doubleton node. By this meta-argument, you could eliminate the tripleton choices giving the third solution. But this just leaves you all doubleton nodes and two solutions, with no basis for any further distinction (and sort of contradicting the rationale for getting to this point).

I hope there is some sudoku with multiple solutions, and with an argument along these lines for one of the solutions to be preferred as the unique one. But this puzzle isn't it.

I looked into whether there was some plausible logical argument for preferring one of the solutions. Something like "any of group 1 of comparable choces give an ambiguous solution, but any of group 2 of comparable choices give a unique solution. Therefore make a group 2 choice and the group 2 choice solution is the unique one".

But no luck. Assigning any tripleton value selects one of the 3 solutions, so no basis there to prefer one choice over another. You can exclude 9 from the quad node, as 9 gives a contradiction.

For the doubleton nodes, one choice led to a unique solution and the other choice left an unresolved pair of solutions. If the choices giving a unique solution consistently gave the same solution, that would be a good argument for claiming that as the unique solution to this puzzle. But not this time. Making a double node choice leading to a unique solution can get two of the three solutions.

For really perverse "logic"(?), you might argue that the third solution, not reachable as a unique solution from any doubleton choice, is really the special and therefore "unique" solution. But's that's too weird a stretch for me.

As another meta-argument, you might say if there is a unique solution, it must be one of the two reachable by a choice giving a unique solution form a doubleton node. By this meta-argument, you could eliminate the tripleton choices giving the third solution. But this just leaves you all doubleton nodes and two solutions, with no basis for any further distinction (and sort of contradicting the rationale for getting to this point).

I hope there is some sudoku with multiple solutions, and with an argument along these lines for one of the solutions to be preferred as the unique one. But this puzzle isn't it.

- Scott H
**Posts:**73**Joined:**28 July 2005

Scott H wrote:Hmm, 3 solutions (not 5).

- Code: Select all
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857291364312465987964378215195734628728956143643182579589617432476523891231849756

- angusj
**Posts:**306**Joined:**12 June 2005

Nick70 wrote:The 1s in box 7,8 and 9 should not have been placed there.

The starting point I used was the black numbers in Jeff's grid at the top of this page. Do I understand that this is not the same as somewhere else?

Edit: Ahh ... just twigged. Looks like I should have used the RED numbers as the givens. Sorry for the confusion. Using the reds as givens I get 3 solutions too.

- angusj
**Posts:**306**Joined:**12 June 2005

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