Sudoku has a wonderfully short rulebook. Fill the nine by nine grid so that every row, every column and every three by three box contains the digits 1 to 9 exactly once. No arithmetic is needed; the digits could just as well be letters or colours. And yet there is a world of difference between an easy puzzle that fills itself in and a hard one that leaves you staring at a half-empty grid.

The good news is that getting better at Sudoku is not about being clever. It is about learning a small number of techniques, in order, and using them patiently. Here they are, from the first one every solver uses to the ones that crack most hard puzzles.

A note on the puzzle itself

The modern form of the puzzle was first published in an American puzzle magazine in 1979 under the name Number Place. It took the name Sudoku in Japan in the 1980s and became a worldwide craze in the mid-2000s. A properly made Sudoku has exactly one solution, and it can always be reached by logic alone. Every puzzle in our Sudoku is generated with a unique solution, which means you never need to guess.

1. Scanning, or cross-hatching

Pick a digit, say 5, and look at one box. For every 5 already placed elsewhere in the same rows and columns, imagine a line running through that row or column. Often only one empty square in the box is left uncovered by those lines, and the 5 must go there.

Work through each digit from 1 to 9 in turn, and through each box. On easy puzzles, scanning alone will finish the grid. It is fast, needs no notes, and is always the first thing to try when you sit down with a new puzzle.

2. The single candidate

Now flip the question around. Instead of asking where a digit can go, pick an empty square and ask which digits it could hold. Look along its row, its column and its box, and rule out every digit you see. If only one digit is left, that square is solved. Solvers call this a naked single.

This works best in crowded parts of the grid, where a row, column and box between them already contain eight different digits.

3. Pencil marks

Medium and hard puzzles will stall if you work only in your head. The answer is to write candidates, the small digits that are still possible in each empty square. Our Sudoku has a notes mode for exactly this.

Some solvers note candidates everywhere at once. Others prefer to mark only squares where there are two or three possibilities. Either way, the important habit is to update your notes the moment you place a digit: remove that digit from the notes of every square in the same row, column and box. Out-of-date notes are the most common cause of mistakes.

4. The hidden single

Sometimes a square has several candidates, yet one of them appears nowhere else in its row, column or box. If the only place a 7 can go in a row is this square, then the 7 goes there, even though the square could in theory hold a 2 or 4 as well. It is hidden because the other candidates disguise it.

Hidden singles are the technique most people discover last and use most. When you are stuck, go through each row, column and box and ask, for each missing digit, how many squares could take it. If the answer is one, you have a move.

5. Naked pairs and triples

If two squares in the same row hold exactly the same two candidates, say 3 and 8, then those two digits must go in those two squares, one each. You do not yet know which is which, but you do know that no other square in that row can be a 3 or an 8, so you can cross them out everywhere else in the row.

The same idea works with three squares sharing three candidates between them, a naked triple, and in columns and boxes as well as rows. These are some of the most powerful cleaning-up tools you have.

6. Hidden pairs

The mirror image of a naked pair. Suppose in one box the digits 1 and 6 can each only go in the same two squares, even though those squares show other candidates too. Then those two squares must be 1 and 6, and you can erase every other candidate from them. Hidden pairs are harder to spot, so look for them when the obvious techniques run out.

7. Pointing pairs and box line reduction

Within one box, if all the possible places for a digit lie along a single row, then that digit must sit in that row inside this box. So it cannot appear in that row anywhere outside the box, and you can remove it from those squares. This is called a pointing pair, or a pointing triple when three squares are involved.

It also works the other way. If, within one row, all the places for a digit fall inside a single box, then that digit cannot go anywhere else in that box. Together these two techniques, often grouped under the name locked candidates, unlock a surprising number of hard puzzles.

8. A glimpse of the X-wing

For the hardest puzzles there are patterns with names like X-wing and swordfish. In an X-wing, a digit can go in only two places in each of two rows, and those places line up in the same two columns. The digit must occupy one diagonal or the other of that rectangle, so it can be removed from the rest of both columns. You will not need this often, but when an expert puzzle stalls, it is worth knowing.

Habits of good solvers

  • Do not guess. If you find yourself trying a digit to see what happens, step back and search again for a hidden single or a pair.
  • Work the busiest areas first. Rows, columns and boxes with many digits give the most information.
  • Rotate your attention. After each placement, look at the row, column and box it affects; new moves often appear right there.
  • Take a break when stuck. A puzzle that looks impossible after twenty minutes often gives way in two after a cup of tea.
  • Move up gradually. Our Sudoku has four difficulty levels; when an easy puzzle is finished by scanning alone, it is time for the next one.

When you want a change: killer sudoku and friends

Killer Sudoku keeps all the usual rules but adds cages: dashed outlines with a target sum, and the digits inside each cage must add up to it with no repeats. Two facts make it far less intimidating. First, every row, column and box adds up to 45, since 1 to 9 sums to 45. Second, some cages have only one possible set of digits: two squares totalling 3 must be 1 and 2, two squares totalling 17 must be 8 and 9, and three squares totalling 6 must be 1, 2 and 3.

Futoshiki is a smaller Latin square puzzle where greater-than and less-than signs between squares do the work of the givens. Kakuro is a crossword built from sums, and Takuzu uses only 0s and 1s. Each uses the same kind of thinking you have just practised: narrow down the possibilities, look for the one place something must go, and let every certain answer lead to the next.

Most of all, enjoy the slow click of a grid coming together. Sudoku rewards patience more than speed, and every technique you learn makes the next puzzle more satisfying.