DanO wrote:dukuso wrote:When you have a multi-solution puzzle, then "unavoidable" has no meaning at all, it's not defined.
I thought "unavoidable" was a set of cells in a valid grid that can be permuted to form a different valid grid. Having a multi-solution puzzle means that from any solution there is set of cells that can be permuted to form another solution (ie, an unavoidable set that doesn't have a clue).
Yes, you are right about the definition. Note that an unavoidable set is a set of cells *and* the digits in those cells. It's hard to word this properly.
What dukuso is referring to (I think) is that an unavoidable set is a property of a completed grid, not a property of a puzzle.
If a puzzle has more than one solution, it doesn't determine any particular grid.
You might then argue that although a puzzle with multiple solutions will determine several completed grids, those grids will all have the same unavoidable set in common. Namely, the one that "caused" multiple solutions.
In that case, yes the *cells* are determined by the puzzle but not the individual *digits* in the cells.
So we don't actually have an unavoidable set unless we specify the digits as well.
DanO wrote:A question I have is, how does such a permutation affect the other intersecting unavoidable sets?
Usually they are destroyed. After the permutation, other intersecting unavoidable sets are no longer unavoidable sets.
You might create new unavoidable sets as well.
For example, in this grid (which I call MG)
937856241
562194387
481273569
823647915
615932478
749581623
378469152
196725834
254318796
there is an unavoidable set in {11,15,21,25} i.e. rows 1 and 2, columns 1 and 5.
9...5....
5...9....
If you switch the 5 and the 9 in this unavoidable set, you destroy the unavoidable set {11,13,61,63}.
But you create a new one at {15,16,75,76}.
