~~I like to play Sudoku at work~~ I’m aware of a sidebar gadget on Vista that lets you play Sudoku. (Gotta be careful, my manager reads this. :)) One day when experimenting with this gadget (I was testing it, really…), I discovered a potential solution that, while technically correct, was counted as incorrect.

You see the incorrect values in red. You can go through the row / column / square checking or you can take my word for it – that solution is correct. However, the sidebar gadget seemed to think that this alternative, similar solution was the right one:

In fact, they’re both correct. Which brings up an interesting question to ponder: Under what circumstances are Sudoku solutions unique? When can multiple solutions exist?

Continuing on the train of deep Sudoku thinking, I noticed that the Sudoku games in this gadget don’t start off symmetrically (if you look at Sudoku puzzles in the books or newspaper, the starting values are symmetric along the diagonal). Is there something important about this symmetry? Is the symmetry at all correlated with unique solutions?

I call on some really smart math whiz (Eric) to solve this problem immediately.

I would guess that the fewer squares you get to begin with, the more likely it is that multiple solutions exist. But, setting up an unambiguous puzzle is a fairly deterministic process. You simply have to start placing numbers one by one (and filling all logically deduced squares along the way) while only making a square given when no more can be deduced. I would say that their generation algorithm is faulty. One of your 8 ambiguous squares should have been given. However, maybe they take a normal puzzle and remove a given or two to make it even harder. But, in these cases there will always be multiple solutions.

I’ve heard that the symmetry thing is an easy way to hand-generate a puzzle, but I’ve never done it. I’m sure you could find an explanation on the web.

If you want to try out something new in sudoku, try shendoku, using the sudoku rules but playing two people, one against the other, like battleshipps. They have a free version to download at http://www.shendoku.com/sample.pdf . Anything else they are bringing out or they are working on you can find at http://www.shendoku.com or at they´r blog http://www.shendoku.blogspot.com . Have fun, I am.