Concerning Crazy's puzzle i think we all agree that my solution was the first but worst

I like both Vidar's and Carcul's.

Maria, for solving sudokus proof by contradiction *is* constructive, you can eliminate a concrete candidate. So let me call a deduction that directly leads to a number or elimination "positive" and one which needs proof by contradiction "negative". This reflects the fact that the vast majority (including my friend) prefers the first one. I accept that, because i can understand it psychologically.

But no one seems to have a problem with using techniques like x-wing or UR. How would i explain, why they can be used? I would say, "if you put x there, you get a contradiction to the rule that the number must be only once in the row" or "if you dont put x there, you get a contradiction to the assumption that the puzzle is unique". Are there "positive" explanations? If not, using the techniques is just a hidden form of using proof by contradiction.

[Edit:]

Ok, you can say "you already do have a number for this row, so you can eliminate the other candidates", this sounds positively, though its the same in other words. Same for pairs, coloring, locked candidates etc. So most techniques are "positive".

Fact is that the harder puzzles are, the less you can apply positive techniques. So we have positive and negative puzzles

And: As you said, normally also you would apply a UR instead of a lot of forcing chains. In this case (and there are many others) the negative method is far more elegant.