How to Play Sudoku: The Rules, and the Techniques That Solve Almost Every Puzzle

Sudoku looks like arithmetic and contains none. The digits 1 to 9 are just nine distinguishable symbols; you could play with nine colours or nine animals and nothing about the puzzle would change. What you are actually doing is elimination, and there are a small number of ways to do it — small enough that four techniques cover almost every puzzle you will ever meet.

This is the whole rule: fill the grid so that each row, each column and each three-by-three box contains the digits 1 to 9 exactly once. That is it. Everything else in this article is a consequence of that sentence.

Every square belongs to three groups at once — its row, its column and its box. Those three groups are called its units, and the twenty other squares they contain are its peers. A digit that appears anywhere among those twenty peers cannot go in that square.

Almost every sudoku technique is a statement about units. That is why the same reasoning keeps reappearing in different costumes: a rule about rows works for columns and boxes too, because all three are just units.

It is also why the small grids work as a teaching tool. A four-by-four sudoku with the digits 1 to 4 has exactly the same units — rows, columns, and two-by-two boxes — and exactly the same logic, in a space a six-year-old can see all of. Nothing is simplified; it is only smaller.

Look at one empty square and list what could go in it. Cross off every digit already in its row, its column and its box. If one digit survives, it goes there. That is a naked single, and it is the entire game in miniature.

The practical skill is not the logic but the looking. Beginners check the row, then the column, then the box, and lose track. Experienced solvers scan a single box and ask "where can the 7 go in here?", which is the next technique and a much faster habit.

On an easy grid — one that starts with around forty numbers filled in — naked singles alone will often take you most of the way, each one you place opening another.

Turn the question around. Instead of asking what can go in this square, ask where this digit can go in this unit. Take a box with three empty squares; if two of them already see a 4 somewhere along their row or column, the 4 must go in the third — even if that square still has several other candidates of its own.

This is why it is called hidden: the square did not look decided, but the digit had nowhere else to be. Hidden singles are what solvers are doing when they scan across a grid with a finger, checking one digit at a time against every box.

Together, naked and hidden singles solve every easy sudoku and most medium ones. If you learn nothing else, learn these two properly, because the rest are refinements.

Sometimes two squares in the same unit have exactly the same two candidates — say both can only be a 3 or a 7. You do not know which is which, and it does not matter: between them they own the 3 and the 7. No other square in that unit can be either, so both digits can be struck off every other square in it.

That single deduction often cracks a grid that has stalled, because the eliminations it produces create naked singles somewhere else. The same argument extends to three squares sharing three candidates, and to four sharing four, though those are rare enough that you can safely learn them later.

This is the first technique that requires writing candidates down. Up to here you can hold it in your head; from here, pencil marks earn their keep.

These are two halves of one idea: when a digit is confined to the overlap between a box and a line, the overlap decides both.

If every possible home for the 5 in one box sits in the same row, then the 5 in that box must be in that row — so the 5 cannot appear in that row anywhere outside the box. That is a pointing pair, and it eliminates candidates across two other boxes.

Run it the other way: if every possible home for the 5 in a row sits inside one box, then the 5 in that box must be in that row, so the 5 can be struck from the rest of the box. That is box-line reduction.

With these four techniques you can solve the overwhelming majority of published sudoku, including most puzzles labelled hard. What remains — X-wings, chains, colouring — belongs to the small class of genuinely difficult grids, and is worth learning only once these are automatic.

A properly constructed sudoku has exactly one solution, and that property is what makes guessing unnecessary. If the answer is unique, then at every point there is some deduction available that follows from the givens alone — you may not have found it, but it is there.

This is also the line between a real puzzle and a broken one. A grid with two valid completions can be filled in legally and still be marked wrong, which feels like the puzzle cheating, because it is. A grid with none is worse. Uniqueness is not a difficulty setting; it is the minimum standard for the thing being a sudoku at all.

Good generators enforce it directly. The puzzles on QTNest are built by starting from a complete valid grid and removing numbers one at a time, keeping each removal only if the puzzle still has exactly one solution — so a hard grid here is sparse, never ambiguous.

The number of givens is the crude measure, and it is what most difficulty labels are based on: around forty for easy, low thirties for medium, mid twenties for hard. It correlates with difficulty but does not define it.

The honest measure is which techniques a grid requires. A puzzle solvable entirely with naked and hidden singles is easy no matter how few numbers it starts with. One that needs a naked pair to get past a wall is medium. One that needs box-line reduction or worse is hard, and that is true even if it looks generously filled in.

It is measurable, too, which is the useful part: run a solver that only knows the basic techniques and see whether it finishes. Every easy grid on our sudoku pages passes that test by construction — a generated grid that fails it is thrown away and another dealt — while about half the hard ones fail it, which is precisely what earns them the label.

Count digits, not squares. Keeping track of how many 7s are already placed tells you when 7 is nearly resolved and worth a scan. Having eight of them down makes the ninth trivial, and it is the cheapest bookkeeping in the game.

Work the fullest units first. A box with seven numbers in it is close to resolving; a box with two is not worth staring at. The same goes for rows and columns, and the habit of going where the information is saves more time than any technique.

Do not pencil in candidates too early. Marking every square at the start produces a grid so dense it hides the singles that were already there. Solve what you can see first, then mark the region that has stalled.

And resist correcting with guesses. If you are stuck, the answer is a technique you have not applied yet, not a digit worth trying. Guessing in sudoku does not fail immediately — it fails twenty minutes later, with no way back to where it went wrong, which is why it is the one habit worth breaking early.

Start where the two basic techniques are guaranteed to be enough: our easy sudoku deals grids with up to forty-seven numbers already placed, each one checked before you see it to be solvable with singles alone, and flags a wrong digit the moment you place it so a mistake cannot quietly ruin the next twenty minutes.

Then move to the main sudoku page, where medium grids will occasionally need a naked pair and hard grids usually need more, with the mistake flagging switched off — at which point the discipline of not guessing starts to matter.

For a child, or for anyone who wants the logic without the size, the four-by-four and six-by-six grids on sudoku for kids use the identical rule set and can be finished in a couple of minutes. It is the same game, and it is a much better introduction than a nine-by-nine grid that takes an hour to fail at.