Sudoku Strategies, From Singles to X-Wing and Beyond

The main sudoku strategies, in the order solvers meet them, are: singles (full house, hidden single, naked single), pointing and claiming, naked and hidden pairs and triples, X-Wing and Swordfish, XY-Wing and XYZ-Wing, and finally simple colouring and chains. Each one either places a digit or removes candidates. Always try the easier rungs first. This guide explains every technique with a worked example, so it suits you once you can finish an easy grid.

The sudoku technique ladder at a glance

The sudoku technique ladder runs from patterns you can spot in seconds to patterns that need a full set of notes. The table is the whole guide in one view; the sections below take each rung in turn.

A word on notation: R4C7 means row 4, column 7, counting from the top left. A unit is any row, column or box. Candidates are the digits still possible in a square; the notes you write are candidates. Two squares see each other if they share a unit.

Puzzle Picnic's Easy and Medium sudokus can be solved with singles alone. Its Hard and Expert grids are classic 9x9 and need harder techniques. The solver that checks every puzzle knows the techniques in this table, from naked and hidden singles to simple colouring, plus forcing chains, and a puzzle's difficulty depends on which of them it needs.

Singles: the techniques that place digits

Singles are the only techniques that put a digit in the grid; everything higher up the ladder just removes candidates until a single appears. HoDoKu's technique pages define the three (HoDoKu: singles):

• Full house: the last empty square in a unit. It takes whichever digit the unit is missing.
• Hidden single: a digit that has only one possible square left in a unit. The square may still have other candidates, which is why it is "hidden".
• Naked single: a square that has only one candidate left.

Hidden singles in boxes are found by cross-hatching: pick a digit and a box that lacks it, then rule out squares along every row and column that already holds the digit.

×	5			6	2
×	×	×			8	4		7
×	×	×	1				8
				7				8
8	2						1	4

The middle-left box (rows 4 to 6, columns 1 to 3) has eight empty squares and needs an 8. The 8 in row 5 removes three of them, the 8 in row 6 three more, and the 8 at the bottom of column 1 removes R4C1. Only R4C3 is left.

Naked singles are found the other way round: you look at one square and count what its row, column and box already contain. If eight different digits are there, the ninth goes in the square. Naked singles are slow to find without notes and fast to find with them, which is one reason to take notes on harder grids. The full beginner method is in how to play sudoku.

Pointing and claiming: where a box meets a line

Pointing and claiming use the overlap between a box and a row or column. Both rest on the same fact: a digit must appear once in every box and once in every line, so if one of those has all its options inside the overlap, the other loses that digit everywhere else. HoDoKu calls them locked candidates type 1 and type 2 (HoDoKu: intersections).

Pointing

Pointing: if, inside a box, every candidate for a digit lies in the same row (or column), that digit can be removed from the rest of that row (or column) outside the box.

	×	×	2		2		×	×
	2	2					2	2
		2					2	2

The middle box (rows 4 to 6, columns 4 to 6) must hold a 2, and both its options are in row 4. Whichever one it is, row 4's 2 is inside that box. So R4C2, R4C3, R4C8 and R4C9 cannot be 2.

Claiming

Claiming (also called box/line reduction) is the reverse: if, inside a row (or column), every candidate for a digit lies in the same box, the digit can be removed from the rest of that box.

8			8		8		×	×
						8	8
8				8		×	×	×

Row 8 must contain an 8, and both options sit in the bottom-right box. That box's 8 is therefore in row 8, and the five other candidates in the box go.

Naked and hidden pairs and triples

Pairs and triples are about squares and digits in one unit matching up in equal numbers. A naked pair is two squares in a unit whose only candidates are the same two digits; those digits must go in those two squares, so they can be removed from every other square in the unit. A hidden pair is two digits that, within a unit, can only go in the same two squares; any other candidates in those two squares can be removed.

Here is a naked pair from row 1 of a puzzle in progress. Bold digits are already placed; the other entries are candidates.

R1C1 and R1C6 both hold only 2 and 6. One is 2 and the other is 6, so no other square in row 1 can be either: R1C2 loses its 6 and R1C5 loses its 2.

And a hidden pair from row 5 of another puzzle:

Row 5 needs a 1 and a 4, and both digits appear only in R5C4 and R5C8. Those two squares must hold 1 and 4 between them, so their other candidates (a 6 in R5C4, a 6 and an 8 in R5C8) are removed.

Triples work the same way with three squares and three digits. A naked triple does not need every square to hold all three digits: three squares with candidates 12, 23 and 13 are a naked triple, because between them they hold only 1, 2 and 3. Quads, with four, exist too but are rare. Worked triples, the link between naked and hidden sets, and a scanning routine are in naked and hidden pairs, explained.

X-Wing and Swordfish: patterns across lines

Fish patterns use one digit across several rows and columns at once. An X-Wing is two rows in which a digit has exactly two possible squares each, and those squares sit in the same two columns. The digit must then take one corner on each diagonal of the rectangle, so it fills both columns, and it can be removed from every other square in those two columns. The same works with rows and columns swapped (SudokuWiki: X-Wing).

				6	6
6			6
		6	×	6	6
×		6	×		6
6			6

Row 4's 6 is in column 1 or column 4. So is row 9's, and they cannot both be in the same column. Either way, columns 1 and 4 get their 6s from rows 4 and 9, so R5C4, R7C1 and R7C4 cannot be 6.

A Swordfish is the three-line version: three rows in which a digit's candidates all fall within the same three columns (two or three per row). Then those three columns get their copies of the digit from those three rows, and the digit leaves the rest of the three columns. SudokuWiki notes that a four-line version, the Jellyfish, also exists, and that larger fish are never needed (SudokuWiki: Swordfish). Both patterns, with a scanning method, are in X-Wing and Swordfish made simple.

XY-Wing and XYZ-Wing: three-square patterns

Wings link three squares with few candidates. An XY-Wing (also called a Y-Wing) has a pivot square with two candidates, say A and B, and two pincer squares that the pivot sees: one with A and C, the other with B and C. The pincers do not see each other. Whatever the pivot turns out to be, one pincer becomes C, so any square that sees both pincers cannot be C (SudokuWiki: Y-Wing).

1	34	7	9	38	6	5	2	48
8	2	9	34	5	7	1	34	6
6	35	345	1	2	48	9	347	78

Follow both cases. If the pivot R4C5 is 3, then R4C2 in the same row cannot be 3, so it is 4. If the pivot is 8, then R6C6 in the same box cannot be 8, so it is 4. One of the two pincers is a 4 either way. R6C3 shares a box with R4C2 and a row with R6C6, so it sees both, and its 4 goes. R6C3 drops to 3 or 5.

An XYZ-Wing gives the pivot a third candidate: the pivot holds X, Y and Z, and the pincers hold X and Z, and Y and Z. Now the pivot itself might be Z, so a square can only lose Z if it sees all three squares, pivot included (SudokuWiki: XYZ-Wing).

4	5	13
389	389	7
139	6	2
128	7	4
39	39	5

If the pivot R6C1 is 1, its box-mate R4C3 must be 3. If the pivot is 9, R8C1 in its column must be 3. If the pivot is 3, it is the 3 itself. In every case one of the three squares is a 3, and R5C1 sees all of them (it shares column 1 with the pivot and R8C1, and the box with R4C3). So R5C1 drops to 8 or 9. R5C2 also holds a 3 but does not see R8C1, so it keeps its 3.

Simple colouring and chains

Simple colouring follows one digit through a chain of conjugate pairs: units where the digit has exactly two possible squares, so exactly one of them is true. Give the two squares of a pair different colours, say A and B, and keep going from each square to the other pair it belongs to, alternating colours. In the finished chain, either every A square holds the digit or every B square does (SudokuWiki: simple colouring).

Two rules then give removals:

• Colour trap: a square outside the chain that sees one A and one B cannot hold the digit, because one of those two is true.
• Colour wrap: if two squares of the same colour see each other, that colour must be false everywhere, so the other colour is true.

				8	8	B		8
			8	8				8
	8				8
	8		8		8
				A				B
			8	8		A

The chain runs through three conjugate pairs: row 7 (R7C5 and R7C9 are its only 8s), the bottom-right box (R7C9 and R8C7) and column 7 (R8C7 and R1C7). So either R7C5 and R8C7 are both 8, or R7C9 and R1C7 are. R1C5 shares row 1 with R1C7 and column 5 with R7C5, and one of those is an 8, so R1C5 is not.

Chains in general extend this idea. HoDoKu describes a chain as a stream of implications built from strong links (if one is false, the other is true) and weak links (if one is true, the other is false) (HoDoKu: chains). A forcing chain tries a candidate, follows what it forces, and keeps the result only if every branch agrees or one branch leads to a contradiction. Chains can solve almost anything, but they are slow by hand, so treat them as the top rung and look for everything else first.

A solving routine that uses the ladder

A sudoku solving routine is simply the ladder in a loop: try the cheapest technique, and after any progress go back to the bottom. Here is one that works on every difficulty.

1. Sweep for singles. For each digit from 1 to 9, cross-hatch every box. Then check each nearly full row and column.
2. Take notes. On a 9x9 grid, once singles dry up, fill in candidates, box by box. Write only what you have checked.
3. Check intersections. For each box and digit, ask whether the candidates line up. For each line and digit, ask whether they sit in one box.
4. Look for pairs. Scan for squares with exactly two candidates, then for a twin in the same unit. Then scan each unit for digits with exactly two spots.
5. Look for triples. Units with five or more empty squares are where they hide.
6. Look for fish. For each digit, list the rows where it has two spots and look for two rows that use the same columns. Repeat for columns.
7. Look for wings. Start from two-candidate squares and look for pincers.
8. Colour a digit. Choose the digit with the most conjugate pairs and colour it.
9. After every removal, go back to step 1. One removal often unlocks a single straight away.

The habit that matters most is step 9. Solvers who skip it spend minutes hunting for an XY-Wing while a hidden single sits in plain view.

Mistakes that look like techniques

Many broken sudoku solves come from a technique applied when one of its conditions is missing. Each pattern above has a condition that is easy to skip when you are in a hurry.

• A pair in no shared unit. Two squares holding only 2 and 6 form a naked pair only if they share a row, column or box. Two 26 squares in different rows, columns and boxes tell you nothing.
• A "triple" with four digits. Squares with 12, 23 and 34 hold four digits between them, so they are not a triple. Count the digits, not the squares.
• Pointing with a stray candidate. Pointing needs every candidate for the digit in the box to lie on the line. One forgotten candidate elsewhere in the box, and the removal is wrong.
• An X-Wing that does not line up. The two rows must have their candidates in the same two columns. Rows with 6s in columns 1 and 4 and in columns 1 and 5 make no X-Wing.
• An XYZ-Wing removal that only sees the pincers. In an XYZ-Wing, the square losing a digit must also see the pivot. It is an easy slip to make if you are used to XY-Wings.
• Stale notes. A removal is only as good as the notes it rests on. If you placed a digit and forgot to erase it from a neighbour's notes, a pattern can appear that is not really there.

When a square runs out of candidates, or a unit has nowhere left for a digit, one of these is usually the cause. Go back to the last step you are sure of rather than patching the grid forward.

Frequently asked questions

What is the most important sudoku strategy?

For most solvers it is the hidden single. It needs no notes, it works at every level, and on easy and medium grids it does most of the work: you cross-hatch a box for one digit at a time. If you only learn one technique beyond the rules, make it this one.

Do you need X-Wing to solve hard sudokus?

Often not. What counts as hard varies between publishers, and many hard puzzles need nothing beyond pairs, triples, pointing and claiming. Fish and wings appear in expert grids, and the higher you go the more often you need them.

Is guessing a valid sudoku strategy?

It is not needed in a puzzle with one solution, and it is easy to get wrong. Every step in this guide is a proof that a candidate is impossible. A forcing chain looks a little like a guess, but it is reasoning: you follow both possibilities and keep only what they agree on.

What is the difference between an XY-Wing and an XYZ-Wing?

In an XY-Wing the pivot has two candidates, so the shared digit must be in one of the two pincers, and any square seeing both pincers loses it. In an XYZ-Wing the pivot has three candidates and might be the shared digit itself, so only a square that sees the pivot and both pincers loses it.

In what order should I learn sudoku techniques?

In the order of the ladder: singles, pointing and claiming, naked and hidden pairs, triples, X-Wing, Swordfish, XY-Wing, XYZ-Wing, then colouring. Learn one, play a few puzzles until you spot it without effort, then move up.

Where to go next

Pick the rung just above the one you are comfortable with and practise it for a week. If you are still learning the rules, start with how to play sudoku. If you solve 9x9 grids with singles but stall, read naked and hidden pairs. If pairs are second nature, X-Wing and Swordfish is next. To see how puzzles are graded by the techniques they need, read easy to expert: what changes at each level.

You can practise every rung in Puzzle Picnic's sudoku: every puzzle has exactly one solution, and when you are stuck a hint explains the next logical step. More guides are on the sudoku topic page.

Sources

• HoDoKu: Singles
• HoDoKu: Intersections (locked candidates)
• HoDoKu: Chains and links
• SudokuWiki: X-Wing strategy
• SudokuWiki: Swordfish strategy
• SudokuWiki: Y-Wing strategy
• SudokuWiki: XYZ-Wing
• SudokuWiki: Simple colouring (singles chains)