This question is about what stategy I can employ for hard sudoku puzzles.
I solve the vast majority of Sudokus that I see (published in various newspapers) in my head by using one of three principles.
The one that is far and away most common is what I think of as striking out (within one of the 3x3 subsquares). I consider a subsquare and pick a number that does not appear in that subsquare. From each instance of that number either to the side or directly above/below my selected box, I draw an imaginary line from that number through my box "striking out" unfilled squares. If only one square is left, then I can fill it in with the selected number. I don't do this mechanically. Rather, I look for common numbers that impinge upon a subsquare.
This strategy can be extended. As a simple for example, suppose that the right column of a subsquare is blocked (has been filled in) with, say, non 3s. If you have the 3 impinging on the left column from outside the subsquare, that forces the 3 of the subsquare to be in the middle column, which in turn forces the remaining subsquare's 3 to be in the right column.
There are two other situations that are far less common, and I look for these when I am otherwise stuck with my main strategy above. 2) Pick an arbitrary square. If the numbers impinging on that square from both the column, row, and subsquare that square is in amount to all but one of 1-9, then the square may be filled in with the missing number.
3) Consider a row or column. If a number not in that row or column impinges on all but one of the non filled in squares of the row/column, then that last square will have the given number. This is a variation on the first strategy applied to rows/columns instead of subsquares and is relatively rare.
With these 3 strategies I write nothing down, nor do I have to remember the boatloads of special, fancy named strategies that they talk about on sudoku websites.
However, there is a harder level of sudoku. The only place I've regularly encountered these harder sudokus is at http://apps.facebook.com/challengesudoku
on the 'Harder' level (levels are Easy, Medium, Hard, Harder). You can start one of these by Creating a game, and someone will usually join within one minute. Some of them (over 50%) cannot be solved by the above 3 principles and require a more involved logic. Specifically, http://www.sudokusolver.co.uk/
cannot show a next step because a more advanced strategy is needed.
Therefore (finally) my question is what is the next level of strategy to follow? In other words, how do you expand on the strategy that I've delineated above?
To be clear, I'm after a way of looking at the harder puzzles: what should I be scanning for?
There are some advanced strategies such as at http://www.scanraid.com/Death_Blossom
(though most of this site appears to be undergoing revision), but that doesn't answer my question of what I should be looking for because it doesn't tell me what the next most common type of scenario is.