How to solve sudoku

Every sudoku here can be solved by logic alone: there is always a next step that follows from what is on the grid, and you never have to guess. This guide goes through the techniques that find those steps, from the two that solve every easy puzzle to the ones an expert puzzle needs. The level of a puzzle is simply the hardest of them it calls for.

Scan for hidden singles

Pick a digit and a box, and ask: where in this box can this digit go? Every copy of the digit in the rows and columns that cross the box rules out a line of cells in it. Often only one cell is left, and the digit goes there. This is called a hidden single: the cell may have other candidates, but the digit has nowhere else to go.

Where does the 5 go in the top-left box? The 5s in the second and third rows rule those rows out, the 5 in the second column rules out that column, and the 3 fills the last cell. Only the corner is left.

Work through the digits one by one, 1 to 9, across all nine boxes, and then do the same along rows and columns. On an easy puzzle, scanning alone gets you most of the way. It is the fastest technique there is, so it is always worth going back to after anything else has changed the grid.

Look for naked singles

The other way round: pick a cell and ask which digits could go here? Strike out every digit already in its row, its column and its box. If only one is left, that is the digit. This is a naked single.

The shaded row, column and box hold 1, 2, 3, 4, 6, 7, 8 and 9 between them. The only digit the cell can take is 5.

Naked singles show up best in crowded parts of the grid: a row, column or box with seven or eight digits already in it.

Pencil in the candidates

When singles run out, write the candidates into the empty cells with the pencil: every digit each cell could still be. It takes a minute, but from here on every technique works on notes, and with them in front of you they are easy to see.

Keep them honest. When you write a digit in pen, rub out that note from its row, column and box (the game does this for you), and a cell whose notes come down to one is a naked single.

Pointing and claiming

If a digit's candidates inside a box all lie in one row (or one column), the digit must go in that row within the box, so it cannot go anywhere else in that row. This is pointing.

The top-left box still needs 2 and 4, and both can only go in its middle row. So the 4 cannot go anywhere else in that row: rub it out of the other cells.

Claiming is the same the other way round: if a digit's candidates in a row (or column) all lie inside one box, it cannot go anywhere else in that box.

Naked and hidden pairs

If two cells of a row, column or box have the same two candidates and no others, those two digits go in those two cells, one each, even if you do not know which way round yet. So neither digit can go anywhere else in that unit.

The two ringed cells can only be 3 or 7, so between them they take both. Rub 3 and 7 out of the rest of the row.

A hidden pair is the same thing hidden among other notes: two digits that fit only in the same two cells of a unit. Those cells must hold those two digits, so every other note in them can go.

The same works with three cells and three digits, a triple, though the three cells do not need all three candidates each: 2, 3 and 3 is a triple too. Triples are where hard puzzles start.

The X-wing

Take one digit. If in two rows it has exactly two places, and they are in the same two columns, then it takes one corner of that rectangle in each row, and either way it fills both columns. So it cannot go anywhere else in those two columns.

In the second and sixth rows, 6 can only go in the third and eighth columns. However it falls, those columns get their 6 in those rows, so 6 comes out of every other cell in them.

The same works with columns and rows swapped. A swordfish is the X-wing with three rows and three columns: a digit that has two or three places in each of three rows, all within the same three columns. It is rarer and harder to spot, and it is what expert puzzles ask for.

The XY-wing

Look for a cell with exactly two candidates, say X and Y (the pivot), that sees two other two-candidate cells: one with X and Z, one with Y and Z (the pincers). Whatever the pivot turns out to be, one of the pincers must be Z. So any cell that sees both pincers cannot be Z.

The pivot in the corner is 4 or 7. If it is 4, the pincer to its right is 9; if 7, the pincer below it is 9. The two cells that see both pincers cannot be 9.

The XYZ-wing is the same with a pivot of three candidates, X, Y and Z; then only the cells that see the pivot too lose their Z.

Colouring

Pick one digit and find every row, column and box where it has exactly two places. Those two cells are linked: one of them is the digit and the other is not. Chain the links together and colour the cells in two colours as you go, alternating. One colour is all true and the other all false. If two cells of the same colour see each other, that colour is the false one; and any other cell that sees both colours cannot hold the digit.

This is the last technique an expert puzzle here needs.

Racing a friend

In a race, speed comes from order. Scan every digit through every box before you reach for the pencil: singles are quick, and every digit you place makes the next one easier. Pencil in candidates only when scanning has stopped giving you anything. And keep an eye on the red: a clash you miss costs far more time later than the second it takes to fix now.

Ready? Play sudoku, or read the rules again.