DanO wrote:In my opinion, this technique should at least be defined and given a name. It took me less time to discover it from the uniqueness principle than it did to locate somewhere it was being discussed. Wether or not you use it is a personal decision.

It does have a name: the uniqueness test. Or rather, tests. It's now a family of tests at this point, with variants 1-4 and 2b-4b. In theory the logic of these can also extend to higher-order forms 3x3, 4x4, and 6x6.

If you are on the side that believes that uniqueness isn't a given then a modification of this technique should be able to show when the solution is not unique and will provide required information for logically finding the sets of solutions.

That doesn't really make sense. This technique relies on uniqueness as one of its base premises; you can't simply modify it to prove the premise is wrong. And if it is wrong, what's the point in looking for all the solution sets anyway? A multi-solution grid is broken; it is not a valid sudoku. The only point in exploring multiple solutions is to count them or just confirm multiples exist.

For a better view of the basic patterns involved, it is easier to start from a solved board and work backwards. Take any solved board and highlight all the 1's and 2's. Now look at the patterns of the buddy chains between the 1's and 2's. Most often these will form a single chain. Occasionally there will be 2 or more separate chains. Every chain must have at least one exposed number to prevent the chain from flipping into an alternate solution.

There are some other constraints known for the givens as well:

1) At most only one digit can be missing from the givens.

2) At most only one column can be missing from the givens.

3) At most only one row can be missing from the givens.

Multiple boxes may be missing, which is clear if you take a solution grid and remove boxes 1, 5, and 9. You do however need at least 6 of the boxes to contain givens. This provides some support for the theory that no fewer than 17 givens are possible in a valid sudoku. If you arrange 8 givens diagonally you will satisfy the three constraints above, and if you add a few more you can be sure to have enough givens to fill in three missing boxes. I suspect that if 17 really is the minimum, it's to provide just enough more positional information for the 16 other givens to be of any use at all.