X-Wing
The X-Wing is the usual gateway to expert Sudoku, and the first technique that spans four squares in four different units. It looks intimidating written down and is surprisingly mechanical once you have found one.
The idea
Pick a digit. Find a row where it has exactly two possible squares, then another row where it also has exactly two — and where those two sit in the same two columns. The four squares form a rectangle.
A worked example
Now reason it through. Row 2's 4 is at either r2c3 or r2c7. Say it is at r2c3. Then row 6's 4 cannot also be in column 3, so it must be at r6c7. The other choice gives the mirror image: r2c7 and r6c3.
Both possibilities put one 4 in column 3 and one 4 in column 7. It does not matter which diagonal is the real one — either way, both columns take their 4 from inside the rectangle. So 4 can be erased from every other square in columns 3 and 7.
That is the whole trick, and it is why the technique is sound without ever resolving the ambiguity. You are not finding where the digit is; you are finding where it is not.
How to hunt for one
Take one digit at a time and write down, for each row, how many squares could hold it. Ignore every row that does not answer exactly two. Then look at the rows that do, and check whether any pair of them shares its column numbers.
Then run the whole search again with rows and columns swapped — an X-Wing found on columns eliminates along rows. Solvers routinely check one orientation and forget the other.
When you will need it
Genuinely only on expert puzzles, and even then not every one. If you are reaching for an X-Wing on a hard grid, there is almost certainly a hidden single or a naked pair you have not spotted. Exhaust the cheaper techniques first — they are cheaper for a reason.
Common mistakes
"Exactly two" is not "two that I noticed". If the digit has a third possible square in one of the rows, the rectangle proves nothing, and eliminating on it will corrupt the grid. This is the single most common way people break an expert puzzle.
The second is eliminating in the wrong direction. An X-Wing built from rows eliminates along columns. Getting this backwards feels productive and quietly destroys the solution.
Where to go next
The next technique in the order is colouring and chains — Follow true/false links across the grid until a contradiction settles the digit.
All nine techniques: the techniques index. New to the rules? Start here. Ready to use it? Open a board.