Sudoku techniques, in the order you need them

Strategy · about 9 minutes to read

Most Sudoku advice arrives as an unsorted pile of technique names. Swordfish, XY-wing, unique rectangle — impressive vocabulary, useless to someone who is still filling in the easy grid slowly. What follows is the same material arranged as a ladder, in the order the puzzles will actually demand it, with a plain note on each rung about when it applies and what it costs you in time.

Throughout, cells are named by row and column: r4c7 means row four, column seven. Rows are numbered from the top, columns from the left, and the nine 3×3 blocks are called boxes.

Everything comes from three constraints

A Sudoku grid carries exactly three rules. Each row holds the digits 1 through 9 once each. Each column does too. Each box does too. That is the whole game, and every technique in this article — up to and well beyond the hardest ones — is a consequence of those three statements and nothing else.

It helps to say the same rules a second way, because the second phrasing is the one that solves puzzles. There are twenty-seven units on the grid: nine rows, nine columns, nine boxes. Each unit contains each digit exactly once. So every cell belongs to three units at the same time, and every digit you write eliminates that digit from twenty other cells. When you learn to feel that overlap, you stop looking for tricks and start reading the grid.

Rung one: scanning and cross-hatching

Use this when: the grid still has plenty of given digits and empty cells with only one or two possibilities. This is the entire toolkit for an easy puzzle and it never stops being useful.

Cross-hatching means working one digit at a time instead of one cell at a time. Pick a digit that appears often in the givens — say 7 — and look at a horizontal band of three boxes. Suppose box 2 already has its 7 in row 1, and box 3 has its 7 in row 2. Rows 1 and 2 are now spoken for as far as 7 is concerned, so box 1's 7 must be in row 3. Three cells become one candidate row, and then you check the three columns of that row: if any of them already contains a 7 further down the grid, that cell is out too. Often two of the three fall away and the digit is placed.

Then repeat for the vertical band of the same boxes, then move to the next digit. This is slow to describe and very fast to do, and it has one big advantage over hunting for cells: you are only holding one digit in your head at a time, so you make fewer errors.

Rung two: the two kinds of single

Use this when: always. Every puzzle is finished by singles; the harder techniques exist only to create more of them.

A naked single — also called the single candidate — is a cell where eight of the nine digits are already blocked by its row, column and box, so only one digit can go there. You find it by standing on a cell and looking outward.

A hidden single is the same idea rotated. Inside one unit, a digit has only one cell left where it could possibly go. The cell itself may still accept four or five different digits, which is exactly why it does not look special: nothing about that cell announces itself. You find it by standing on a digit and asking where in this row, column or box it can still live.

Beginners find naked singles reliably and miss hidden singles constantly, and the reason is habit rather than difficulty. Scanning cell by cell is the natural way to read a grid, and it only ever surfaces the naked kind. The fix is a deliberate second pass: after you have exhausted the cells, walk the units. Take box 5 and ask, for each digit 1 through 9 that box is missing, how many empty cells could hold it. If the answer is one, you have a placement. Doing this on all nine boxes takes a couple of minutes and usually breaks open a grid that felt finished.

Rung three: pencil marks, and when to start them

Use this when: a full pass of scanning and both kinds of single produces no new placement. Not before.

Pencil marks are the small candidate digits you write into empty cells. They are essential for everything above this rung, and writing them too early is the most common way to waste time on a Sudoku. On a grid that still yields to plain scanning, marking every cell means writing several dozen candidates you are about to erase anyway, and it clutters the grid so badly that the obvious placements get harder to see, not easier.

The habit worth building is to mark selectively first. Fill candidates into one box or one row at a time — whichever region looks most constrained — solve what that reveals, and only expand the marks when you need them. Many medium puzzles never require a fully marked grid.

Once you do have marks, they carry an obligation: every digit you place must be erased from every candidate list it touches, immediately. A stale pencil mark is worse than no pencil mark, because you will trust it. Nearly every "this puzzle is broken" moment traces back to one mark that should have been deleted twenty moves ago. If you are working on paper, write marks small and in a consistent position inside the cell so that scanning for a particular candidate is quick. Our own grid accepts one digit per cell and has no notes layer, which is fine up to Medium; for Hard and Expert, keep a scrap of paper beside the screen rather than trying to hold candidate lists in your head.

Rung four: naked pairs and triples

Use this when: you have candidate marks for at least one full unit and singles have dried up.

Here is a constructed example. In row 4, after marking, two cells have exactly the same two candidates: r4c2 can only be 3 or 8, and r4c7 can only be 3 or 8. You do not know which is which, and it does not matter. Between them those two cells consume both the 3 and the 8 for the whole row, so no other cell in row 4 can be either digit. If r4c5 was carrying candidates 3, 6 and 8, it is now simply a 6. That is the naked pair: two cells, two candidates, one elimination that often cascades.

A naked triple is the same logic with three cells sharing three candidates, and there is one detail that trips people up: each cell does not need all three digits. Suppose three cells in box 5 hold {2, 5}, {5, 9} and {2, 9}. Together they use only the digits 2, 5 and 9, and there are three of them, so those three digits are locked inside those three cells. Every other cell in box 5 loses 2, 5 and 9. The test is not "identical lists" but "three cells whose candidates, pooled together, come to exactly three digits."

The mirrored version: hidden pairs

Hidden pairs are to naked pairs what hidden singles are to naked singles — the same statement read from the digit's side. In box 3, suppose the digits 1 and 6 each appear as candidates in only two cells, r1c8 and r2c9, though both of those cells also carry a 4 and a 7. Since 1 and 6 must both appear in box 3, and only those two cells can take them, those two cells are the 1 and the 6 in some order. The 4 and the 7 can be struck from both. Nothing gets placed, but two candidate lists shrink to two digits each, and you have manufactured a naked pair out of a hidden one.

Rung five: the box and the line

Use this when: pairs and triples stop producing and you have marks across a whole band of boxes.

This rung is about the fact that a box and a line overlap in three cells, so information can travel between them in both directions.

A pointing pair travels from the box outward. In box 7, suppose the digit 4 can only go in r8c1 or r8c3. You do not know which, but both are in row 8, so row 8's 4 is definitely inside box 7. Therefore no cell in row 8 outside box 7 can be a 4, and you can erase that candidate from r8c4 through r8c9. Three pointing cells in a line work identically and are sometimes called a pointing triple.

Box-line reduction travels the other way. In column 6, suppose the only cells that can still hold a 4 are r1c6 and r2c6, both of which sit inside box 2. Column 6 must contain a 4, so box 2's 4 is somewhere in those two cells, and every other cell in box 2 loses its 4 candidate.

These two are worth learning as a pair, because they look similar on the grid and confusing them leads to eliminating the wrong cells. The question to ask is: which unit have I proved the digit lives in? Then clear it from the other unit's remaining cells.

Rung six: a first look at the X-wing

Use this when: everything above has been applied exhaustively and the grid is genuinely stuck. On a well-marked grid this is rarer than technique lists imply.

The X-wing is the honest boundary of this article, and it deserves a warning: it is the point where Sudoku stops being about scanning a neighborhood and becomes about spotting geometry across the whole grid. Everything up to now happens inside one unit or one box-line intersection. This does not.

Take the digit 6. Suppose in row 2 it can only go in c3 or c8, and in row 7 it can also only go in c3 or c8. Those four cells form the corners of a rectangle. Whichever way row 2 resolves, row 7 is forced into the opposite corner: if row 2's 6 sits in c3, then row 7's must sit in c8, and vice versa. Either way, one of the two 6s lands in column 3 and the other in column 8. Both columns therefore already have their 6 accounted for, so every other cell in column 3 and column 8 can lose the candidate.

The pattern works with columns and rows swapped, and the whole family of techniques above it — the swordfish and its relatives — is the same argument with more rows involved. If the X-wing feels like a step up in difficulty, that is because it is one. There is no shame in stopping here for a long while; the great majority of puzzles labeled easy or medium never need it.

What to do when you are stuck

Being stuck almost always means one of three things, and they are worth checking in this order.

1. Recount a unit. Pick the row, column or box you have touched most and read its nine cells aloud against the digits 1 through 9. Duplicates and impossible units are common and easy to miss, and finding one now saves you from solving a grid that cannot be solved.
2. Audit your pencil marks. Look specifically for a candidate you should have erased after a placement. Take the last three or four digits you wrote and re-check every cell in their rows, columns and boxes. This finds the problem more often than anything else.
3. Switch from hunting cells to hunting digits. If you have been asking "what goes in this cell," stop and ask "where can the 4 go in this box" for all nine boxes. This single change of direction is what surfaces hidden singles, pointing pairs and box-line reductions, and it is the cheapest unstick available.

If all three come up clean, the grid needs a technique you have not used yet — and on a medium puzzle, the missing technique is nearly always the hidden single or the box-line reduction rather than anything exotic.

You never have to guess

This is the firmest thing in the article. A well-formed Sudoku has exactly one solution, and it can be reached by deduction alone. If you find yourself about to try a digit and see what happens, you are not out of logic — you have missed something, and the checklist above will usually find it faster than a guess will.

Guessing is also expensive in a way that is easy to underestimate. Once you write a digit you are not certain of, every subsequent deduction inherits that uncertainty, and when a contradiction finally appears twenty cells later you cannot tell whether the guess was wrong or your bookkeeping was. The unwinding costs more than the patience would have.

The uniqueness guarantee is not a nice assumption; on this site it is checked. Our generator builds a complete solved grid, removes digits, and then counts the solutions of what remains. If a candidate puzzle admits more than one solution, the removal is rejected and the digit goes back. You only ever receive a grid that has been verified to have a single answer, which means any dead end you hit is recoverable by reasoning and never requires a coin flip.