Sudoku strategy: from singles to Swordfish

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Every technique below is shown at the moment it was the only move left in a real puzzle. Learn them in order: each tier only asks for what came before it, plus one new idea.

How the difficulty works

Difficulty isn't about how many digits are given. A puzzle with 22 givens can be easier than one with 30. What counts is the hardest step you need to finish it.

Easy needs nothing beyond hidden and naked singles.
Medium needs at least one of: pointing, claiming, a naked pair or a hidden pair.
Hard needs at least one of: a naked or hidden triple, an X-Wing, a Swordfish or a Y-Wing.

Nothing here ever needs guessing, trial and error, or the long chains that expert puzzles elsewhere call for. If you're stuck, there's always a move from this page waiting somewhere on the board.

Easy: singles

Hidden single

The first technique everyone learns, and still the one you'll use most. Pick a box, row or column and a digit it's missing. Look at where that digit already sits elsewhere on the board: each copy blocks its own row, column and box. If only one empty cell in your unit is left unblocked, the digit goes there. The digit is "hidden" because that cell might still have several candidates — it's just the only home for this one.

The circled 2s block every other empty cell in box 1, so the 2 goes in the outlined cell. From Easy puzzle #1 — play it

A good habit is to sweep one digit across all nine boxes before moving on to the next digit. Start with the digits that already appear most often: with six or seven copies on the board, the last few are nearly pinned down.

Naked single

Turn the question around. Instead of asking where a digit can go, ask what a cell can hold. List what its row, column and box already use; if eight of the nine digits are taken, the cell has only one option left.

Everything the outlined cell can see is tinted. Between them, its row, column and box use every digit except 6. From Easy puzzle #178 — play it

Naked singles are harder to spot by eye than hidden ones, which is why they tend to turn up when scanning has dried up. This is where pencil marks start to pay off.

A word on pencil marks

Everything from here on works with candidates: the digits a cell could still hold. On Easy puzzles you can usually get by without writing them down. From Medium on, it's worth switching on Notes and filling in candidates, at least for the box or row you're working on. When you place a digit, the game removes it from the notes it affects, so keeping them tidy costs you nothing.

In the diagrams below, small grey digits are candidates, bold blue ones form the pattern, and red ones with a stroke through them are the candidates the technique removes.

Medium: candidates that lock together

Pointing

Sometimes you can't place a digit in a box yet, but you can tell it'll land on one particular line. If all a box's candidates for a digit sit in the same row (or column), then that digit is in that row inside the box — so it can't be anywhere else along that row.

In box 6, the 6 can only go in the two bold cells, and they all lie in row 6. Whichever one it ends up in, the rest of row 6 can't have a 6, so the struck-out candidate goes. From Medium puzzle #2 — play it

Claiming

The same idea, the other way round. If a row's candidates for a digit all fall inside one box, the row "claims" that digit for the box: it has to be in those cells, so the box's other cells can't hold it.

Every possible 9 in row 6 sits inside box 5. Row 6 needs its 9, so it will use up box 5's 9 too — and the other 4 candidates in the box can go. From Medium puzzle #1 — play it

Naked pair

Two cells in the same unit with exactly the same two candidates — say, both can only be 3 or 8. One of them will be the 3 and the other the 8. You don't know which way round yet, but it doesn't matter: those two digits are spoken for, and every other cell in the unit can drop them.

Two cells in row 7 can only be 7 and 9. That uses up both digits, so 4 candidates elsewhere in row 7 are crossed out. From Medium puzzle #1 — play it

Hidden pair

The mirror image of a naked pair. Two digits that, within one unit, can only go in the same two cells. Those cells may list other candidates too, but they're fully booked: one gets each digit, so everything else in them goes.

In row 8, the digits 2 and 9 fit only in the two bold cells. Those cells must hold them, so their other candidates are struck out. From Medium puzzle #49 — play it

Hidden pairs are easy to miss because the cells look cluttered. A quick way to find them: for each digit in a unit, count how many cells could take it. Two digits that both show up exactly twice, in the same places, are your pair.

Hard: patterns across the grid

Naked and hidden triples

Pairs grow into triples. Three cells in a unit whose candidates, taken together, are only three digits form a naked triple — even if no single cell lists all three. Those three digits belong to those three cells, and the rest of the unit can drop them.

Three cells in box 9 share only 4, 5 and 9 between them. The other cells in box 9 can't use those digits. From Hard puzzle #85 — play it

A hidden triple is three digits that, within a unit, fit only in the same three cells. As with the hidden pair, those cells lose every other candidate.

In column 4, the digits 6, 7 and 8 can only go in the three bold cells, so those cells hold nothing else. From Hard puzzle #234 — play it

X-Wing

The first pattern that reaches across the whole grid. Take one digit. Find two rows where it has exactly two possible cells each, and where those cells line up in the same two columns. The digit must sit at opposite corners of that rectangle — one diagonal or the other. Either way, both columns get their copy from these rows, so no other cell in those columns can hold it. (Swap rows and columns and it works just the same.)

Only the 5s are pencilled in. In row 3 and row 7, the 5 has exactly two spots each, in column 1 and column 9. Those two lines get their 5s from the tinted rows or columns, so the struck-out candidates can go. From Hard puzzle #1 — play it

X-Wings are much easier to see if you look at one digit at a time. Some players highlight all of a digit's candidates and squint for rectangles.

Swordfish

An X-Wing with three lines instead of two. Three rows in which a digit's candidates fall within the same three columns (each row may use two or three of them). Between them, the three rows fill all three columns' slots for that digit, so it can be removed from the rest of those columns.

The 4s in row 3, row 6 and row 7 are confined to column 1, column 2 and column 6. Whatever the final arrangement, those three lines take their 4s from the tinted ones, which knocks out 3 more. From Hard puzzle #32 — play it

Y-Wing

The hardest technique our puzzles ask for, and the most satisfying to spot. You need three cells with two candidates each. The pivot holds, say, A or B. It sees one cell holding A or C, and another holding B or C. If the pivot is A, the first wing must be C. If the pivot is B, the second wing must be C. One of the wings is C either way — so any cell that sees both wings can't be C.

The pivot at row 2, column 4 is 7 and 9. Its wings at row 2, column 8 and row 3, column 5 each share one of those digits, and both can be 8. Whichever digit the pivot takes, one wing ends up as 8, so the cell that sees both wings loses its 8. From Hard puzzle #2 — play it

When you're stuck

Run through the list in order, because the easy moves are the ones you overlook. Sweep each digit for hidden singles. Check cells with only a couple of candidates. Look for pointing and claiming in every box. Then pairs, then the patterns. It's surprising how often a hidden single has been sitting there all along.

If that fails, Hint tells you which technique to use and where, without giving away the digit. And if things stop making sense altogether, Check shows whether an earlier mistake is the real reason you're stuck.