Advanced sudoku techniques: naked pairs, candidates and X-Wing

By Miguel Arechaga

There comes a moment in every sudoku player's life when the basic techniques stop working. You scan the board over and over, you go through every cell... and nothing. That wall usually shows up on hard and expert levels, and you don't get past it with more patience, but with better tools. These are the techniques that make the difference between finishing an expert sudoku and giving up on it.

The foundation of everything: noting candidates

Advanced techniques don't work without this habit: writing down, in every empty cell, the numbers it could still hold (its "candidates"). On paper they're written small in a corner; playing online you can keep track of them mentally in the tricky areas. It might look like extra work, but it's the opposite: candidates turn invisible information into visible information, and every technique that follows is built on them.

Naked pairs: the technique that unblocks hard sudokus

Imagine that in the same row there are two cells whose candidates are exactly the same two numbers, say {2, 8} and {2, 8}. You don't know which one holds the 2 and which holds the 8, but you do know something very valuable: between the two of them, the 2 and the 8 are accounted for. So you can erase the 2 and the 8 from the candidates of every other cell in that row. That removal usually leaves some cell with a single candidate, and the sudoku moves forward again. The same works with triples: three cells that only contain combinations of {1, 4, 9} eliminate those three numbers from the rest of the unit.

Hidden pairs: the shy sibling

The hidden pair is the reverse version. It happens when two numbers can only go in two specific cells of a row, column or block, even though those cells have more candidates noted down. If the 3 and the 7 can only fit in two cells of a block, those two cells are for the 3 and the 7 no matter what: you can erase all their other candidates. It's harder to spot than the naked pair, but it shows up constantly in expert-level sudokus.

Intersections: when a block points to a line

Look at a 3x3 block and a number that isn't placed in it yet. If every candidate for that number inside the block falls on the same row, then that number will go in that row for sure (even if you don't know which cell yet). Conclusion: you can remove it as a candidate from the rest of that row outside the block. This technique, called "intersection reduction" or "pointing pairs", is simple to apply and surprisingly powerful.

X-Wing: the crown jewel

The technique with the most spectacular name is also the most satisfying to pull off. Look for a number that has exactly two candidate positions in one row, and another two in a different row, such that the four cells form a perfect rectangle (the same two columns). Since each of those rows must place the number in one of the two columns, between the two rows they'll occupy both columns. Result: you can eliminate that candidate from every other cell in those two columns. Once you pull off your first successful X-Wing, you can call yourself an advanced player.

What order should you apply them in?

1. Exhaust the basic techniques first: scanning and single candidate (covered in our beginner's guide).
2. Note down the candidates in the stuck areas.
3. Look for naked pairs and triples, which are the easiest to spot.
4. Check block-line intersections.
5. Look for hidden pairs.
6. And if nothing works, go for the X-Wing.

With this arsenal you can solve practically any published sudoku, including MundoSudoku's expert level. Even more exotic techniques exist (Swordfish, XY-Wing, chains...), but they're needed so rarely that almost no player uses them. Master these five and no board will resist you.