denis_berthier wrote: .... Actually, uniqueness of the global puzzle doesn't imply uniqueness for a sub-puzzle (even for one defined by a 1-cut). ....

If the 1-cut is a singularity (in my narrower sense), uniqueness of the overall puzzle does, of course, imply uniqueness for the sub-puzzle. Otherwise, once you find an overall solution, you could substitute the other solution for the sub-puzzle, and you'd have a second solution for the overall.

Of course, this doesn't work for doubularities. In that case, all that can be established is either the sum or the difference of the two cells in the doubularity. And that's not enough to guarantee uniqueness in the sub-puzzle.

.... when I say it can be solved independently, I mean that, after assigning proper sums to the parts of the segments it includes (and that have been cut into two parts), it can be solved. ....

I wish I could understand sentences like that.

.... For the NW puzzle, for instance, this means assigning sum 5 to the trivial 1-cell horizontal sector starting at the right of r5c4 (the remaining two cells have been cut out). ....

The only reason you can assign 5 to r5c5 (and the reason it is trivial) is that r5c5 is a singularity (in my narrower sense).

.... The reason why uniqueness is not guaranteed .... in the general case .... is that the constraints that should be inherited by the global segments are (momentarily) forgotten ....

Again, that's too vague to allow me to figure out what you're saying.

.... (the simplest example is the two small 2x2 NE and SW sub-puzzles). ....

The reason uniqueness is not guaranteed in these two cases is extremely simple: There is no singularity (in my narrower sense) associated with either of these two. There is only a doubularity -- r3c12-r4c12 in the NE case, and its mirror image in the SW case.

.... From my experience of trying to solve independent sub-puzzles, it is sometimes easier to solve the largest one than the smallest ....

If the smaller puzzle is cut off from the larger puzzle as the result of a singularity, then of course that's not true. Once the division has occurred, solving either puzzle is truly independent of solving the other. The solution of the larger provides no information that helps solve the smaller, nor vice versa. It makes absolutely no difference which one is solved first.

I must admit, though, that once I have isolated a singularity, I prefer to solve the smaller puzzle first. That way, in case I made a computational error in evaluating the singularity, I will discover it sooner, and waste less time.

Bill Smythe