Sudoku solving techniques

Every way to solve a sudoku, in the order worth learning them. There are 116 in all. 34 of them come with a worked example from a real board.

New here? Start at the top. Tier 1 is all you need for an easy puzzle. How to read a sudoku grid explains the words used below.

Start with a family

Techniques come in families, and the members of a family share one idea. Reading the family first is faster than reading its members one at a time, because once the shared idea lands the individual techniques are mostly variations on where the digits sit.

• SinglesThe three ways a digit can be forced without comparing anything. 3 techniques.
• Locked candidatesA digit trapped in the overlap between a box and a line. 3 techniques.
• SubsetsN cells that between them hold exactly N digits. 10 techniques.
• Basic fishOne digit, N rows, N columns, and nowhere else to go. 7 techniques.
• Single-digit patternsOne digit, chained through the houses where it has only two homes. 8 techniques.
• WingsA pivot and its wings, forcing one digit out wherever both are seen. 8 techniques.
• UniquenessArguments from the fact that the puzzle has exactly one solution. 21 techniques.
• ChainsAlternating implications, followed until both ends say something. 18 techniques.
• Almost locked setsA set one digit short of being locked, used as a chain node. 12 techniques.
• Complex fishFish that almost work, plus a precise account of the exceptions. 8 techniques.
• Exotic and rank-zeroWhole-grid structures for the puzzles nothing else touches. 13 techniques.
• Brute force and theoryNot techniques. How machines solve, rate and generate puzzles. 5 techniques.

Every technique, in tier order

Tier 1: Singles

• Full House exampleA row, column or box has just one empty square left.
• Hidden Single exampleLook at one digit in one row, column or box.
• Naked Single exampleA square has only one digit left that fits.

Tier 2: Locked candidates and subsets

• Hidden Pair exampleTwo digits in one group can only go in the same two squares.
• Hidden Quad exampleFour digits in a group can only go in the same four squares.
• Hidden Subset (general form) exampleSome numbers in a group have only a few homes left.
• Hidden Triple exampleThree digits in a group can only go in the same three squares.
• Locked Candidates (general form) exampleSometimes all the spots for a digit inside one group also sit inside a second group.
• Locked Candidates Type 1 (Pointing) exampleInside one box, every spot left for a digit sits in the same row or column.
• Locked Candidates Type 2 (Claiming) exampleInside one row or column, every spot left for a digit sits in the same box.
• Locked Pair exampleA pair that shares two groups at once, usually a row and a box.
• Locked Triple exampleA triple that sits in two groups at once, usually a row and a box.
• Naked Pair exampleTwo squares in the same group both hold only the same two digits.
• Naked Quad exampleFour squares in one group share the same four digits between them.
• Naked Subset (general form) exampleFind some squares in one row, column or box.
• Naked Triple exampleThree squares in one group share the same three digits between them.

Tier 3: Basic fish

• Basic Fish (base and cover formulation) examplePick some rows where a digit has only a few spots left.
• Jellyfish exampleThe X-Wing idea stretched to four rows and four columns.
• Leviathan exampleA seven by seven version.
• Squirmbag exampleA five by five version of the same idea.
• Swordfish exampleThe same idea as an X-Wing but with three rows and three columns.
• Whale exampleA six by six version.
• X-Wing exampleA digit is stuck in the same two columns in two different rows, making a rectangle.

Tier 4: Single-digit patterns and colouring

• 2-String Kite examplePick a digit.
• 3D MedusaColouring spread across every digit at once, not just one.
• Empty Rectangle exampleIn one box, all the homes for a digit sit in one row and one column.
• Multi-ColoringColouring marks a digit's spots in two shades.
• Simple Coloring examplePick a digit.
• Skyscraper examplePick a digit.
• Turbot Fish examplePick one digit.
• X-ColoringColouring with an extra step.

Tier 5: Wings and pair chains

• Bent Naked SubsetA group of squares bent across two rows, columns or boxes.
• Letter-named wings (M, S, L, H, Split, Hybrid, Local)Short chains that people gave letters to.
• Remote Pairs exampleA string of squares that all hold the same two numbers, each one linked to the next.
• VWXYZ-WingThe same idea stretched to five squares and five numbers.
• W-Wing exampleTwo squares far apart hold the same two numbers.
• WXYZ-WingFour squares holding four numbers between them.
• XY-Wing exampleThree squares, each with two numbers left.
• XYZ-Wing exampleLike the XY-Wing, but the middle square has three numbers instead of two.

Tier 6: Uniqueness

• Avoidable RectangleFour squares can form a rectangle across two boxes.
• Avoidable Rectangle Type 1Three corners of a rectangle were worked out by you, not printed in the puzzle.
• Avoidable Rectangle Type 2The same idea, but two corners are still open and share an extra number.
• Bivalue Universal GraveSay every open square had just two numbers left.
• BUG-LiteThe pattern applied to part of the grid rather than all of it.
• BUG+1Say every open square had two numbers left, and each number had two homes in every group.
• BUG+2The same idea when two squares break the pattern instead of one.
• BUG+3The same again with three odd squares.
• Deadly PatternA shape that would give the puzzle two answers if it ever appeared.
• Extended Unique RectangleA bigger version using six squares and three numbers instead of four and two.
• Hidden Unique RectangleThe rectangle trick again, but the giveaway is hidden.
• Reverse BUGThis one uses the numbers you have already filled in, not the ones still open.
• Unique LoopLike a rectangle but longer, a ring of squares all sharing two numbers.
• Unique Rectangle exampleFour squares in a rectangle across two boxes, all holding the same two numbers.
• Unique Rectangle Type 1Three corners of the rectangle hold only the two numbers.
• Unique Rectangle Type 2Two corners hold only the pair.
• Unique Rectangle Type 3Two corners hold the pair.
• Unique Rectangle Type 4 exampleTwo corners hold only the pair.
• Unique Rectangle Type 5Like Type 2, but the extra number sits in corners that sit across from each other rather than side by side.
• Unique Rectangle Type 6Like Type 4, but the two numbers form an X across the rectangle instead of sitting along one side.
• Unique Rectangle with Missing CandidatesA rectangle where one corner has already lost one of the two numbers.

Tier 7: Chains, loops and nets

• AIC with AHS NodesThe same as above but using sets of numbers that are hiding rather than sets of squares.
• AIC with ALS NodesA chain that can step through a small set of squares holding one more number than it has room for.
• Alternating Inference ChainA chain that switches between two kinds of step.
• Bowman's BingoPick a digit, assume it, and write out everything that follows until you either finish or hit a contradiction.
• Cell Forcing ChainTake one square and try each number it could hold.
• Continuous Nice LoopA loop where the reasoning closes back on itself with no weak spot.
• Digit Forcing ChainTake one number in one square.
• Discontinuous Nice LoopA loop that almost closes but has one weak spot.
• Dynamic Forcing Chain with ContradictionAssume a digit is right.
• Dynamic Forcing Chain with Double ImplicationTry both options for a square and follow each one out fully.
• Forcing Chain (general form)Pick something and assume it.
• Grouped AICA chain where a step can use a whole group of squares at once instead of a single square.
• KrakenTake a pattern that nearly works.
• NishioAssume a number goes in a square, then check whether the rest of that number can still fit everywhere it needs to.
• Nishio Forcing NetThe same check as Nishio.
• Unit Forcing ChainTake one number in one row, column or box and try each home it could have.
• X-ChainA chain about one number only.
• XY-ChainA chain of squares that each hold exactly two numbers.

Tier 8: Almost locked sets

• Aligned Pair ExclusionLook at two squares together and list every pair of digits they could hold.
• Aligned Triple ExclusionLook at three squares together and list every set of three numbers they could hold.
• Almost Hidden SetThe mirror image of an almost locked set.
• Almost Locked CandidatesTwo areas cross.
• Almost Locked SetA small group of squares holding exactly one number more than there are squares.
• ALS ChainA row of almost full groups linked one to the next.
• ALS-XY-WingThree almost full groups arranged like a wing.
• ALS-XZTwo of those almost full groups share numbers.
• Death BlossomOne square is the stem.
• Subset ExclusionThe same idea for any size of group.
• Sue de CoqLook where a row or column crosses a box.
• Sue de Coq (extended)The same counting trick with more squares involved.

Tier 9: Complex fish

• CannibalismWhen a square belongs to both halves of the fish at once.
• Complex Fish (rank framework)The fish idea, but boxes are allowed to join the rows and columns.
• Endo-FinA spare square that sits inside two parts of the pattern at the same time.
• Finned FishA fish pattern with one extra square hanging off it.
• Franken FishA fish where a box is allowed to join in with the rows and columns.
• Kraken FishA fish that does not quite work.
• Mutant FishA fish where rows, columns and boxes can be mixed freely on both sides.
• Sashimi FishA finned fish that is missing a corner.

Tier 10: Exotic and rank-zero

• Broken WingA ring of squares where one number would have nowhere to go if the ring were perfect.
• Domino LoopA ring of pairs that lean on each other all the way round, like falling dominoes.
• Double ExocetTwo of those launcher patterns sharing the same numbers.
• FireworksA number is trapped so that it can only sit in one row, one column, or the square where they cross.
• Gurth's Symmetrical PlacementSome puzzles are built with a mirror pattern.
• Homogeneous ChainsA chain where every link is the same kind.
• Junior ExocetTwo squares act as a launcher and two more as the target.
• Multi-Sector Locked SetsCount several rows, columns and boxes at once.
• Pattern Overlay MethodEvery digit has to land in one of a fixed set of shapes across the grid.
• Rank TheoryA way to count a pattern.
• Senior ExocetA bigger version of the Exocet pattern, with more squares acting as the target.
• SK LoopA ring of almost complete sets passing through the corners of a band and a stack.
• Table-Driven MethodsWork out ahead of time what follows from each single guess, and store it in a table.

Tier 11: Computational

• Difficulty Rating EnginesPrograms give a puzzle a score.
• Exact Cover and Dancing LinksA sudoku can be rewritten as a tidy tick box puzzle.
• Minimal Puzzles and the 17-Clue ResultA puzzle is minimal when you cannot take away any clue without giving it a second answer.
• SAT and Constraint Programming EncodingsThe puzzle is turned into a long list of yes or no questions.
• Unavoidable Sets and Puzzle GenerationAn unavoidable set is a group of squares you could rearrange to get a second answer.