Trial and error is rightly frowned upon by sudoku cognoscenti. But what constitutes unacceptable T & E compared to acceptable checking out of different logic chains is always subject to debate. Some devotees swear by “nishio,” in which you look for possible contradictions in your head, without writing anything down beyond your pencil marked list of candidates, while others swear at it as a Japanese euphemism for T & E. I have my own “maybe T & E maybe not” technique, which allows me to do puzzles in an hour or so that are rated as five or six-hour brain-crackers for experts and that would otherwise take me at least as long as that to do. It has similar though less dramatic effects for me in cracking less extreme puzzles. I call it “converging chains,” and it's easily explained.

You start with the converging chain technique when you've gone as far as you can in paring down your pencil-marks with your other techniques. You then pick a square with two candidates that looks critical to the puzzle, in that either answer will allow you to determine a fair number of other squares. You then assume one answer to the square, and follow the chain forward. After doing that for a while—not until you arrive at a contradiction, just long enough to have a number of answers, preferably in different parts of the puzzle—you stop working on that chain and start following the complementary chain that is based on assuming that the other answer to the first square is right. If a step in that second chain produces the same answer for any square as the first chain does, voila! That answer must be right, and you write it in. Then you go back to solving the puzzle with your other techniques.

To keep track of the two chains, I use dots above the numbers for the first chain and dots below for the second chain. There are plenty of possible refinements--if you want to be a purist, you can do the two chains in your head without dots, and if you're eager for a faster solution you should use chains not just with squares with only two candidates but also with boxes, rows, and columns in which a given number can only be put in two places (or you can go further on the chains and find multiple converging squares rather than stopping when you get the first one).

Is converging chains kosher? I don't have an absolute answer, but I think it's worth using when you've gone as far as you can otherwise, and I pass it on for what's it's worth.