LatinSquares FAQ
Difficulty levels
So what's this "difficulty zero" mean? (And why is it so hard?) The basic principle is whether "if-then" thinking is required. At "difficulty zero", you can always find one cell where you can immediately conclude the letter must be "X". For example, if five letters in a column have been filled in, the remaining empty cell must be the sixth. By contrast, if you say for example that "filling this cell with X means that other puzzle piece [row, column] can't be finished", that's an example of if-then reasoning, difficulty level of (at least) one.
Another way of looking at the distinction is whether the given cell must be filled that way even if you stop right there and don't fill in any more letters at all.
Level zero doesn't mean it's easy, not at all. Instead, you have to range over the entire diagram and try to identify the weak spot, the one place where you can convincingly argue that there's no choice but to fill it in with X. Me, I always solve level zero puzzles by limiting myself to "level zero" reasoning. And so I prefer them!
Specifically: take two rules of puzzle solution analysis:
    * If there's only one cell within a given puzzle piece (row, column) for a given letter X to go, because all other blanks in that puzzle piece (row, column) conflict with X.
    * If there's only one letter that can go in a particular cell (each of the five alternatives to X already occupies a cell somewhere in the same puzzle piece or row or column).
Apply these rules repeatedly until no further cells can be filled in. If you can finish the puzzle this way, we have "level zero".
Otherwise, we must use "trial-and-error with backtracking". In that case, the difficulty level is the maximum recursion depth. Difficulty zero, versus the others, is pretty intuitive: you can solve it using only "level zero" (immediate) reasoning. Pen versus pencil, as it were. (If "difficulty 1" means you'll need to fill one cell in with pencil, "difficulty n" means you could need up to n pencilled cells working at once.)
In practice, there are some "reasoning principles" that are deeper than the two above, are completely correct, and seem pretty straightforward to apply. So you might well find that a "difficulty 4" puzzle falls right over.
Difficulty levels