Every Sudoku Technique, Ordered From Easiest to Hardest

Every Sudoku technique worth learning, easiest first: cross-hatching, naked and hidden singles, naked pairs and triples, hidden pairs and triples, pointing pairs and box-line reduction, X-Wing, then swordfish, XY-Wing and the chains. Eight rungs, and the order matters more than the list.

The rule for using it is one sentence. Work the lowest rung that still produces something. A technique four levels up is not a better move, it is a slower one, and hunting for an X-Wing while a hidden single sits unfound is how people spend twenty minutes on a grid that had five easy placements left in it.

Levels 1 and 2 finish an easy puzzle by themselves. Levels 3 to 5 finish nearly every newspaper hard. Level 6 and above belong to the small share of grids graded expert or extreme, and plenty of strong solvers never learn level 8.

Each technique below gets a definition, a worked example on real cells, and the thing it actually buys you: a placement, or an elimination.

Want somewhere to practise a rung? Sudoku Master grades 4,000 classic puzzles easy to extreme, so you can pick a tier that forces the technique you are learning instead of hoping one turns up. Works with no signal, and no ads while you solve.

The ladder on one screen

Notice where the placements stop. Only the first two rungs put digits on the grid. Everything from level 3 up deletes candidates, and the placements come afterwards, from singles that the deletions expose. If you are looking for a technique that fills a cell directly on a hard puzzle, it does not exist.

The grid the examples use

One puzzle carries levels 1 to 5. Dots are empty cells.

5 3 . . 7 . . . .
6 . . 1 9 5 . . .
. 9 8 . . . . 6 .
8 . . . 6 . . . 3
4 . . 8 . 3 . . 1
7 . . . 2 . . . 6
. 6 . . . . 2 8 .
. . . 4 1 9 . . 5
. . . . 8 . . 7 9

Rows run 1 to 9 from the top, columns 1 to 9 from the left, so r4c7 is row 4, column 7. Boxes are numbered 1 to 9 reading left to right and then top to bottom, so box 1 is the top left and box 9 the bottom right. If any of that is unfamiliar, the words this series uses are collected in one place.

Level 1: cross-hatching

Pick a digit, pick a box, and ask where in that box the digit can go. Rows and columns carrying that digit already cross out most of the candidates, and sometimes only one cell survives.

Take the 1 in box 9, the bottom right. Four cells there are empty: r7c9, r8c7, r8c8 and r9c7.

• Row 8 already holds a 1, at r8c5. That kills r8c7 and r8c8.
• Column 9 already holds a 1, at r5c9. That kills r7c9.

One cell left. Place a 1 at r9c7, and you did it without writing a candidate anywhere.

Cross-hatching needs no notation and works on a paper grid at a bus stop. Run all nine digits across all nine boxes and an easy puzzle mostly falls apart. Cross-hatching in detail covers the scanning order that wastes the least time.

Level 2: naked singles and hidden singles

Two ways to prove a cell is forced, and they are opposites.

A naked single is a cell whose peers have used up eight of the nine digits. You find it by looking at the cell. Take r7c9. Row 7 holds 6, 2 and 8. Column 9 holds 3, 1, 6, 5 and 9. Box 9 holds 2, 8, 5, 7 and 9. Between them the peers cover every digit but one. Place a 4.

A hidden single is a digit with one cell left in some unit. You find it by looking at the unit, not the cell. Row 3 is empty at r3c1, r3c4, r3c5, r3c6, r3c7 and r3c9. Where can its 5 go?

• r3c1 is out: column 1 has a 5 at r1c1.
• r3c4, r3c5 and r3c6 are all out: box 2 already holds a 5, at r2c6.
• r3c9 is out: column 9 has a 5 at r8c9.

So r3c7 takes the 5. Its own candidate list is {1,3,4,5,7}, five digits wide, which is exactly why nobody spots it by staring at the cell. Hidden singles are the most commonly missed move in Sudoku, and a grid that looks stuck usually has one in it. Both singles, worked slowly goes through more of them.

Level 3: naked pairs, triples and quads

From here on you need pencil marks, and one region at a time is enough. Writing candidates into all 81 cells before you start is a good way to spend ten minutes producing nothing. The maintenance rules matter more than the writing.

A naked pair is two cells in the same unit whose candidates are the same two digits. Between them they will use both, so those two digits leave every other cell of that unit. A naked triple is three cells holding three digits between them, and no cell needs all three. A quad is four.

Column 2 of the grid has one. Its six empty cells mark up like this:

r2c2 {2,4,7}
r4c2 {1,2,5}
r5c2 {2,5}
r6c2 {1,5}
r8c2 {2,7,8}
r9c2 {1,2,4,5}

Look at r4c2, r5c2 and r6c2. Three cells, and between them only the digits 1, 2 and 5. Those three digits are going into those three cells in some order, so no other cell in column 2 can hold any of them.

Cross 2 out of r2c2, leaving {4,7}. Cross 2 out of r8c2, leaving {7,8}. Then cross 1, 2 and 5 out of r9c2, which was {1,2,4,5} and is now a 4. Place it.

There is a second payment. All three cells of the triple also sit in box 4, so the same argument clears 1, 2 and 5 from the rest of that box. r4c3 was {1,2,5,9}. Now it is a 9. Two placements from one pattern, and neither cell was reachable any other way on this grid. Naked pairs, triples and quads has the counting rule that keeps you from seeing triples that are not there.

Level 4: hidden pairs and triples

Same idea read backwards. A hidden pair is two digits that appear in only two cells of a unit. Those cells belong to those digits, so every other candidate in them gets deleted. The pair hides because the cells are usually carrying four or five marks each.

Here is a row from a harder grid, marks already written:

r5c2 {1,4,5,7,9}
r5c3 {2,4,5,7,9}
r5c5 {1,2,4,5}
r5c6 {1,4,5}
r5c7 {2,4,5}
r5c8 {1,2,5}

Nothing looks locked. Now count the 7s: r5c2 and r5c3, and nowhere else in the row. Count the 9s: the same two cells. Two digits, two cells, so those cells are the 7 and the 9 in some order and everything else in them goes.

r5c2 drops to {7,9}. r5c3 drops to {7,9}. The row itself has gained no placement, but both cells sit in the same box, and a {7,9} pair in a box clears 7 and 9 out of the other six cells there. If r6c1 was {7,8}, it is now an 8.

That two-step is the shape of most progress above level 3. The technique makes a small cell, the small cell makes a placement somewhere else. Hidden pairs and triples covers the triple, which hides even better.

Level 5: pointing pairs and box-line reduction

These two are one idea pointed in opposite directions, and the whole family is also called locked candidates or intersection removal. Every case lives in the three cells where a box overlaps a line.

Pointing goes box to line. If a digit can only go on one row or column inside a box, then it lands on that line, so it leaves the rest of that line outside the box.

Box 6 of our grid, the middle right, has six empty cells. Where can its 7 go? Row 6 already holds a 7 at r6c1, so r6c7 and r6c8 are out. Column 8 holds a 7 at r9c8, so r4c8 and r5c8 are out. That leaves r4c7 and r5c7, and both are in column 7.

Wherever box 6 puts its 7, it is somewhere in column 7. So 7 comes out of the rest of column 7: r2c7 drops from {3,4,7,8} to {3,4,8}, and r3c7 drops from {1,3,4,5,7} to {1,3,4,5}. Now look back at row 3. Its 7 had two homes, r3c7 and r3c9. One is gone. Place a 7 at r3c9.

Box-line reduction goes line to box, and is also called claiming. Column 9 of the grid can only take its 8 at r1c9 or r2c9, both inside box 3. So box 3 puts its 8 in column 9, which means 8 leaves r1c7 and r2c7. Row 2 then has exactly one place for its 8, at r2c9. Place it.

Level 5 is the rung most solvers stop at, and it is enough for almost every puzzle a newspaper prints. Pointing and claiming, side by side sets out both directions with more cases.

Level 6: the X-Wing

Above here the techniques stop being about one unit and start being about a rectangle. The X-Wing works on a single digit at a time, so the easiest way to see it is a map of that one digit.

Below, a 4 marks a cell where the 4 is still possible. Every other cell is a dot. Rows 3, 5 and 9 are blank because their 4 is already placed.

. . 4 . . . . 4 4
. . 4 . . . . 4 .
. . . . . . . . .
4 . 4 . 4 . . . .
. . . . . . . . .
4 4 . . 4 . . . .
. . 4 . . . . 4 .
. 4 . . . . . 4 4
. . . . . . . . .

Read row 2. Its 4 goes at r2c3 or r2c8, nowhere else. Read row 7. Its 4 goes at r7c3 or r7c8, nowhere else. Four cells, one rectangle, two columns.

Now the argument. Row 2 needs a 4 and row 7 needs a 4, and both are trapped in columns 3 and 8. Two rows, two columns, one 4 each. Whichever way round it falls, columns 3 and 8 have used up their 4s on rows 2 and 7. So every other cell in those two columns loses the 4.

Out goes the 4 at r1c3, r4c3, r1c8 and r8c8. Four eliminations, and they pay immediately: row 1 now has one place left for its 4, at r1c9, and once that is written, row 8 has one place left, at r8c2.

Columns work the same way with rows and columns swapped. If a digit sits in exactly two cells in each of two columns, and those cells share two rows, the rows get cleared instead. The X-Wing on real grids is the long version, including how to spot one without checking every digit.

Level 7: swordfish and the XY-Wing

The swordfish is the X-Wing with three lines instead of two. Three rows, each of whose candidates for a digit fall inside the same three columns, mean those three columns are spoken for. A base row may have two positions or three, as long as none of them escapes the three columns.

Here is a map of the 3s:

. . . . . . . . .
3 . . . 3 . . . .
3 3 . . 3 3 . . .
. . . . . . . . .
. . . . 3 . . . 3
. . . . 3 3 . 3 3
. . . . . . . . .
3 . . . . . . . 3
3 3 . . . . . 3 3

Rows 1, 4 and 7 are blank because their 3 is placed. Now read rows 2, 5 and 8. Row 2 keeps its 3 in column 1 or column 5. Row 5 keeps its 3 in column 5 or column 9. Row 8 keeps its 3 in column 1 or column 9. Three rows, three columns, three 3s to share out, so the rest of those columns is clear: out go the 3s at r3c1, r9c1, r3c5, r6c5, r6c9 and r9c9.

Six eliminations and not one placement. That is normal up here, and it is worth saying plainly so you do not think you have done it wrong. At level 7 you buy eliminations and hope a single falls out two moves later. Swordfish, explained covers the four-line version, the jellyfish, which almost nobody needs.

The XY-Wing is a different animal: the first technique that chains cells rather than scanning lines. It needs three cells with exactly two candidates each. One is the pivot; the other two are wings the pivot can see.

Say r5c2 holds {4,7}, r5c8 holds {4,9} and r6c3 holds {7,9}. The pivot r5c2 sees r5c8 along row 5 and sees r6c3 inside box 4.

Now test both halves of the pivot. If r5c2 is a 4, then r5c8 cannot be, so r5c8 is a 9. If r5c2 is a 7, then r6c3 cannot be, so r6c3 is a 9. The pivot has to be one or the other, so a 9 lands on r5c8 or r6c3 either way.

Any cell that can see both wings therefore cannot hold a 9. That is r5c1 and r5c3, which share row 5 with one wing and box 4 with the other, and r6c8, which shares row 6 with one and column 8 with the other. Three eliminations from three small cells. The XY-Wing, slowly works through the same argument with the grid drawn out.

Level 8: colouring, chains and the rest

Past the wings the techniques stop having tidy shapes and start being bookkeeping. The main families:

• Simple colouring. Take one digit, find the units where it has exactly two possible cells, and chain those pairs together, tinting them alternately. The two tints are the two ways the chain can fall. If a tint contradicts itself, every cell in it is false. If a cell outside the chain sees both tints, it cannot hold that digit.
• Remote pairs. A chain of cells that all hold the same two candidates. Anything seeing both ends loses both digits.
• XYZ-Wing and W-Wing. Close relatives of the XY-Wing with an extra candidate or an extra link.
• Forcing chains. Assume a candidate, follow the consequences, and see whether every branch reaches the same conclusion. Slow, general, and the point at which many solvers decide the puzzle has stopped being fun.
• Uniqueness patterns, such as the unique rectangle and the BUG. A pattern that would give the puzzle two solutions cannot sit on a properly made grid, so some candidate in it must go. These are sound on a published puzzle, since a proper puzzle has exactly one solution. They argue about the setter rather than about the grid, though, and they collapse on a badly generated one.

Almost nobody needs this rung. Graders reserve it for the top of the extreme range, and if a newspaper puzzle seems to demand a forcing chain, check your last few placements first. A wrong digit produces exactly this feeling. Advanced techniques for extreme puzzles is the deeper tour, and hard Sudoku strategy is about choosing between them under time pressure.

How to know when to move up a rung

The ladder only works if you climb it in order, and the discipline is simpler than it sounds.

1. Scan all nine digits. Every box, every digit. Most stuck grids end here.
2. Take every single you can see, naked and hidden, and re-scan after each placement. A placement changes twenty other cells.
3. Write marks for one region only, the fullest one, and look for pairs and triples.
4. Check the box-line overlaps. Pointing and claiming are cheap and they are the most common technique on a hard newspaper grid.
5. Only now go hunting for a fish or a wing, and pick one digit to hunt rather than sweeping the grid.

One habit sits underneath all five. When a technique finally pays, go straight back to step 1 rather than carrying on up the ladder. Grids unravel from below.

A note about worked examples, the ones on this page included. An easy grid usually offers several routes to the same cell, so a technique you have just learned will often turn out to be the slow way to a digit a scan would have found. That is not a fault in the technique. On the puzzle where the cheap route is missing, the pattern is all you have.

Which level does your puzzle actually need?

Difficulty is set by the hardest technique the grid forces on you, and by how many times it forces it. It is not the number of clues, which is the most common mistake in Sudoku writing. A 30-clue puzzle hiding one X-Wing is harder than a 24-clue puzzle that never asks for more than singles.

Treat that as a rough guide rather than a promise, because publishers share no difficulty scale. One paper's hard is another's medium and neither of them is lying. What makes a puzzle easy, medium, hard or extreme explains how graders decide.

I built Sudoku Master to have somewhere to practise this ladder without a printer. It carries 4,000 classic 9x9 puzzles across four difficulties, all bundled in the app, so the harder tiers are where levels 5 and up start earning their keep. One limit worth stating: the solver in it returns the finished grid, not a technique-by-technique walkthrough. It will settle an argument about an answer. It will not teach you the X-Wing you missed, which is what the pages linked above are for.

What is not a technique

Guessing. A properly made puzzle has exactly one solution and a chain of logic that reaches it, so a guess is a confession that you have not found the move yet. The case against it is in solving without guessing.

Counting the clues. It tells you nothing about the difficulty. See above.

Trial and error with a pencil. Picking a candidate, following it until something breaks, and reversing out is what a computer does, and a computer is very fast at it. On paper it is slow and it teaches you nothing about why the answer is the answer. The backtracking solver is written out if you want to see the machine version.

Learning them all before you start. Take one rung at a time and stop when your puzzles stop resisting. Nobody is grading you on the swordfish.

Questions people ask

Which technique should I learn next after singles?

Naked pairs, then pointing pairs. Those two clear more hard newspaper grids than everything above them combined. Leave the X-Wing until a puzzle actually refuses to move without one.

Do I need pencil marks to use these techniques?

Not for cross-hatching or singles, which is why they come first. Everything from level 3 up is a statement about candidates, so it needs them written down. Mark one region at a time rather than the whole grid, and rub out marks the moment a placement kills them.

Is there one technique that cracks every Sudoku?

No single named technique does. Every proper puzzle can be finished by logic, but on the hardest grids that logic can run deeper than any named pattern, and the general method is a chain: assume, follow, check. That is why the list ends in chains rather than in a trick.

Why do I keep missing hidden singles?

Because you are reading cells instead of units. A hidden single sits in a cell with four or five candidates, so nothing about the cell is remarkable. Ask where a digit can go in a box, a row or a column, and it appears at once.

How long does it take to learn all of these?

The first two rungs take an afternoon. Subsets take a handful of grids before the shapes stop needing to be spelled out. The fish and the wings take as long as it takes to meet puzzles that need them, which for a lot of people is never, because the puzzles they do never ask. Stopping at level 5 is a normal place to stop and it is not a smaller hobby.

Do these techniques work on Killer and Jigsaw Sudoku?

Mostly, yes. Anything built on rows, columns and boxes carries straight over to Killer, and Jigsaw only changes the shape of the box, not the logic. Killer adds cage arithmetic on top, which is a whole extra layer of elimination that classic play has no use for.

Do computers solve Sudoku this way?

No, and that is the interesting part. A solver usually runs backtracking, which tries a digit, recurses, and reverses out of every dead end. It is unreadable as an explanation and it finishes a 9x9 grid in milliseconds. How a solver works covers when reaching for one is reasonable.

Keep reading

• Cross-hatching: how to read a grid quickly, the rung that does the most work
• Naked singles and hidden singles, with worked grids, for the move people miss most
• The X-Wing, explained with real grids, when level 5 stops being enough
• Stuck on a Sudoku? Work this checklist in order, for the grid that will not move tonight
• Hard Sudoku: the techniques that actually move the grid, on picking between the rungs under time pressure
• Sudoku, from the rules up, if you would rather see the whole ladder at once

Get it: Sudoku Master, free on iPhone and Android. The link sends you to whichever store your phone uses.