Sudoku Solving Techniques: From Singles to X-Wing

I used to stall on hard Sudoku and start guessing, but once I learned these techniques in order I could solve almost any puzzle without ever taking a wild shot.

1. The one rule that drives everything

Every digit from one to nine appears exactly once in each row, each column, and each three by three box. That is the whole game. Every technique below is just a smarter way of asking where a number is allowed to go and where it is forbidden.

2. Naked singles

This is the first thing I look for. I find an empty cell and check its row, column, and box. If eight different digits already appear among those neighbors, only one digit is left, so it has to go in that cell.

I scan for these whenever I place a new number, because filling one cell often creates a fresh naked single right next to it.

The fastest way I find them is to work box by box rather than wandering the whole grid at random. I pick the box with the most filled cells, since it has the fewest candidates left, and clear the easy wins there before moving to the next. A tidy, systematic sweep finds far more singles than darting around hoping something jumps out.

3. Hidden singles

A hidden single is the reverse view. Instead of asking what fits a cell, I pick a digit, say a seven, and ask where it can go inside one box. If every other empty cell in that box is blocked by a seven in its row or column, the one remaining cell must be the seven, even if other numbers also look possible there.

Hidden singles are easy to miss precisely because the cell looks like it could hold several numbers. The technique I use is called cross-hatching: I take a digit and mentally draw lines through every row and column where it already appears, then see which cells in a box survive. When only one survives, I place the digit and move on. Doing this for one digit at a time across the whole grid clears a surprising number of cells.

4. Pencil marks and pairs

When singles dry up, I switch on pencil marks, writing every candidate digit small in the corner of each empty cell. This turns the puzzle into a logic map I can read.

Naked pairs

If two cells in the same row, column, or box both show only the same two candidates, those two digits are locked into those two cells. I can safely erase both of those candidates from every other cell in that unit, which often unlocks new singles.

5. Pointing pairs

Sometimes a candidate inside a box is restricted to a single row or column. For example, if the only cells that can hold a four in one box all sit in the same row, then the four for that box lives somewhere in that row. That lets me remove four as a candidate from the rest of that row outside the box.

6. The X-Wing

The X-Wing is my favorite advanced move, and it looks scarier than it is. I look for a candidate, say a five, that appears in exactly two cells in two different rows, and those cells line up in the same two columns. They form the corners of a rectangle.

Because the five must take opposite corners of that rectangle, no matter which way it falls, it cannot appear anywhere else in those two columns. I erase the five from the rest of both columns and the puzzle usually cracks open.

The X-Wing also works the other way around, with the candidate locked into two columns that line up across the same two rows, in which case I clean the digit out of those two rows instead. Once I trained my eye to spot these rectangles, hard puzzles that used to stop me cold started falling into place, because the X-Wing almost always reveals a fresh single somewhere on the board.

The reason all of this works is that Sudoku never actually requires luck. Every elimination I make is a fact, not a hunch, and facts stack on top of each other. Each candidate I remove tightens the grid until a single appears, and the puzzle solves itself one logical step at a time.