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Sudoku solving techniques

All 80 solving techniques in sudokUI's catalogue, grouped by the idea behind them. Each one says what the pattern is, why it works and how to spot it.

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Worth learning first

How often a technique is needed, weighted by its rating cost, over the 5,533,732 puzzles sudokUI has generated and rated. Difficulty and frequency are different things: a hard technique that turns up often repays the effort of learning it.

1. Locked Candidates (Pointing)Medium, needed in 34% of puzzles
2. WXYZ-WingHard, needed in 1 in 12 puzzles
3. 3D MedusaUnfair, needed in 1 in 16 puzzles
4. Hidden SingleBeginner, needed in 94% of puzzles
5. SkyscraperHard, needed in 1 in 11 puzzles
6. 2-String KiteHard, needed in 1 in 13 puzzles
7. Nice LoopUnfair, needed in 1 in 25 puzzles
8. XY-WingHard, needed in 1 in 16 puzzles
9. W-WingHard, needed in 1 in 16 puzzles
10. Chute Remote PairHard, needed in 1 in 12 puzzles
11. Naked PairMedium, needed in 14% of puzzles
12. XY-ChainUnfair, needed in 1 in 30 puzzles

Singles

The foundation of every solve: a cell with only one possible digit, or a digit with only one possible cell.

• Full Housenaked and hidden single at onceA row, column or box has eight of its nine cells filled, leaving exactly one empty cell.
• Naked Singlenaked subset of oneAn empty cell has exactly one candidate left, every other digit having been ruled out for that cell.
• Hidden Single1-fish · hidden subset of oneIn a row, column or box, a digit X is a candidate in exactly one cell, whatever other candidates that cell may hold.

Intersections

Where a box crosses a row or column: candidates confined to the overlap clear the rest of the other unit.

• Locked PairNaked Pair clearing two unitsTwo cells in the same box and the same row or column are both bivalue cells with the same two candidates, X and Y.
• Locked TripleNaked Triple clearing two unitsThree empty cells in one box that also share a row or column contain between them only the candidates X, Y and Z, though no cell needs all three.
• Locked Candidates (Pointing)1-fish: box onto lineWithin one box, every candidate for a digit X lies in the same row or the same column, in two or three cells.
• Locked Candidates (Claiming)Box/Line Reduction · 1-fish: line onto boxWithin one row or column, every candidate for a digit X lies inside the same box, in two or three cells.

Subsets

N cells that between them hold only N digits lock those digits in, whether in plain sight or hidden.

• Naked Pairflip side of a hidden subset · in a line: X-Wing sidewaysTwo cells in the same unit that each hold exactly the same two candidates, X and Y, and nothing else.
• Naked Tripleflip side of a hidden subset · in a line: Swordfish sidewaysThree empty cells in the same unit whose candidates, taken together, are exactly three digits X, Y and Z, though no single cell needs to hold all three.
• Hidden Pairflip side of a naked subset · in a line: X-Wing sidewaysTwo digits X and Y, both still missing from a unit, whose only places in that unit are the same two cells.
• Hidden Tripleflip side of a naked subset · in a line: Swordfish sidewaysThree digits, all still missing from a unit, whose places in that unit fall within the same three cells, though a digit need not appear in all three.
• Naked Quadrupleflip side of a hidden subset · in a line: Jellyfish sidewaysFour empty cells in the same unit whose candidates, taken together, are exactly four digits, though no single cell needs to hold all four.
• Hidden Quadrupleflip side of a naked subset · in a line: Jellyfish sidewaysFour digits, all still missing from a unit, whose places in that unit fall within the same four cells, though a digit need not appear in all four.

Basic Fish

One digit whose places in N rows fall in the same N columns (or the reverse), which clears it from the rest of those lines.

• X-Wing2-fish · Naked Pair seen sideways · 4-candidate X-CycleDigit X has exactly two candidates in each of two rows, all four lying in the same two columns, or likewise with rows and columns swapped.
• Swordfish3-fish · Naked Triple seen sidewaysDigit X has two or three candidates in each of three rows, all lying in the same three columns, or likewise with rows and columns swapped.
• Jellyfish4-fish · Naked Quadruple seen sidewaysDigit X has two to four candidates in each of four rows, all lying in the same four columns, or likewise with rows and columns swapped.
• Squirmbag5-fish · flip side of a smaller fishDigit X has two to five candidates in each of five rows, all lying in the same five columns, or likewise with rows and columns swapped.
• Whale6-fish · flip side of a smaller fishDigit X has two to six candidates in each of six rows, all lying in the same six columns, or likewise with rows and columns swapped.
• Leviathan7-fish · flip side of a smaller fishDigit X has two to seven candidates in each of seven rows, all lying in the same seven columns, or likewise with rows and columns swapped.

Finned Fish

A fish with extra candidates, the fin. It still works, but only on cells that also see the fin.

• Finned X-WingTwo rows (or columns) have candidate X at all four corners of a rectangle, and their only other candidates for X, the fins, all lie in one corner's box.
• Sashimi X-WingTwo rows (or columns) have candidate X at three corners of a rectangle, and their only other candidates for X, the fins, all lie in the missing corner's box.
• Finned SwordfishThree rows (or columns) have candidate X only in three shared columns (or rows) plus fins, extra cells all in one box, with at least two non-fin candidates per row.
• Sashimi SwordfishThree rows (or columns) have candidate X only in three shared columns (or rows) plus fins, extra cells in one box, and a finned row has only one non-fin candidate.
• Finned JellyfishFour rows (or columns) have candidate X only in four shared columns (or rows) plus fins, extra cells all in one box, with at least two non-fin candidates per row.
• Sashimi JellyfishFour rows (or columns) have candidate X only in four shared columns (or rows) plus fins, extra cells in one box, and a finned row has only one non-fin candidate.

Complex Fish

Fish that use boxes as well as rows and columns.

• Franken X-WingTwo base units (rows or boxes) share no candidate for X, all their X candidates lying in two cover units (columns or boxes) sharing none, or rows and columns swapped.
• Franken SwordfishThree base units (rows or boxes) share no candidate for X, all their X candidates lying in three cover units (columns or boxes) sharing none, or rows and columns swapped.

Single Digit Patterns

Short chains on a single digit, built from two strong links joined by a weak one.

• SkyscraperTurbot Fish shape · 4-candidate X-Chain · two Sashimi X-WingsTwo rows (or columns) each hold a conjugate pair for digit X; one end of each shares a column (or row), and the other ends do not see each other.
• 2-String KiteTurbot Fish shape · 4-candidate X-ChainA row and a column each hold a conjugate pair for X, four different cells; one end of each shares a box, and the others do not see each other.
• Turbot Fish4-candidate X-ChainDigit X forms two conjugate pairs on four different cells; an end of one pair sees an end of the other, and the remaining ends do not see each other.
• Empty RectangleGrouped Nice LoopA box's candidates for X lie only on one row and one column; a column (or row) conjugate pair outside the box has one end on that row (or column).

Colouring

Candidates joined by strong links take two colours: one colour is entirely true, the other entirely false.

• Simple Colorsall X-Chains in one clusterFor one digit X, cells joined by conjugate pairs form a cluster, coloured in two colours so that the two cells of every conjugate pair get opposite colours.
• Multi ColorsX-Chains and loops across two clustersFor one digit X, two separate clusters of conjugate pairs are each coloured alternately in their own two colours, and a cell of one sees a cell of the other.
• 3D Medusacolouring across all digitsCandidates of several digits, joined by conjugate pairs or by sharing a bivalue cell, form a cluster coloured alternately in two colours so linked candidates get opposite colours.

Wings

A few cells whose candidates guarantee that one of them holds a particular digit, so every cell seeing all of them loses it.

• W-Winga short AIC, not bentTwo bivalue cells, both with candidates X and Z, do not see each other; a conjugate pair for X has one end seeing the first, the other seeing the second.
• XY-WingY-Wing · bent triple · 3-cell XY-ChainA bivalue pivot cell with candidates X and Y sees two bivalue pincer cells, one with candidates X and Z, the other with candidates Y and Z.
• XYZ-Wingbent triple · ALS-XZA pivot cell with exactly three candidates X, Y and Z sees two bivalue pincer cells, one with candidates X and Z, the other with candidates Y and Z.
• WXYZ-Wingbent quad · ALS-XZThree cells in one unit hold exactly the candidates W, X, Y and Z between them; a fourth cell, bivalue with X and Z, sees each of those holding X.

Almost Locked Sets

Groups of cells one candidate away from being locked, played against each other.

• ALS-XZALS chain of two sets · VWXYZ-Wing is a special caseTwo almost locked sets, each within one unit and sharing no cell, both contain two digits, X and Z, and every X in one sees every X in the other.
• ALS-XY-WingXY-Wing made of setsThree non-overlapping almost locked sets: every hinge X sees every X in one wing, every hinge Y sees every Y in the other, and both wings hold another digit Z.
• ALS-XY-ChainXY-Chain made of setsFour almost locked sets form a chain: neighbours share a joining digit whose cells all see each other, each middle set's joins differ, and both ends hold another digit Z.
• Death BlossomCell Forcing Chain through setsA stem cell's candidates each have a separate petal, an almost locked set holding that candidate only in cells seeing the stem; every petal holds Z, which the stem lacks.

Chains and Loops

Inferences passed along strong and weak links until the two ends settle something.

• Remote PairXY-Chain of one pairA chain of at least four bivalue cells that all hold the same two candidates X and Y, each cell seeing the next.
• Chute Remote PairTwo bivalue cells, both X and Y, in one chute (three aligned boxes), not seeing each other, where the three chute cells seeing neither hold no X, solved or candidate.
• X-Chainsingle-digit AICFour or more cells with candidate X in a chain whose links alternate between conjugate pairs (strong) and cells that see each other (weak), starting and ending strong.
• X-Cycles (loops)single-digit Nice LoopA closed loop of cells holding candidate X whose links alternate strong and weak throughout, or except at one cell where two strong or two weak links meet.
• Grouped X-Cyclessingle-digit Grouped Nice LoopAn X-Cycle loop where a group of two or three X candidates in one box and one row or column acts as one cell, true if any member is X.
• XY-ChainXY-Wing is the 3-cell caseA chain of bivalue cells, each seeing the next and joined to it by a shared candidate that changes at every cell, leaving an unused candidate Z at both ends.
• Twinned XY-ChainsSix cells where two rows cross three columns, or the reverse, holding exactly six different digits as candidates between them, with each digit's pattern cells all seeing each other.
• Nice Loopan AIC written as a loopA loop of candidates whose links alternate strong and weak, either perfectly all the way round or everywhere except at one candidate where two strong or two weak links meet.
• Grouped Nice LoopA Nice Loop with at least one group node: two or three cells holding candidate X in one box and one row or column, true if any becomes X.
• Alternating Inference Chainumbrella over X- and XY-ChainsA chain of candidates, which may mix digits, whose links alternate strong, weak, strong and so on, beginning and ending with a strong link.
• AIC with GroupsAn Alternating Inference Chain with at least one group node: two or three cells holding candidate X in one box and one row or column, true if any becomes X.
• AIC with ALSsAn Alternating Inference Chain that uses an ALS as a strong link between two of its digits: if X is nowhere in the ALS, Y must be somewhere in it.

Uniqueness

Patterns that would give the puzzle two solutions, and so cannot occur in a puzzle that has exactly one.

• BUG+1All unsolved cells are bivalue cells except one with three candidates, and every candidate in a unit appears exactly twice, except X, three times in each of that cell's units.
• Unique Rectangle Type 1Four cells forming a rectangle in exactly two boxes, where three hold only X and Y and the fourth holds X, Y and at least one other candidate.
• Unique Rectangle Type 2A rectangle in exactly two boxes where two corners hold only X and Y and the other two each hold exactly X, Y and Z.
• Unique Rectangle Type 3A rectangle in exactly two boxes: two corners hold only X and Y, the others share a unit, and their extras and K other cells' candidates there total K+1 digits.
• Unique Rectangle Type 4A rectangle in exactly two boxes: two corners hold only X and Y, the other two hold extras too and are the only places for X in a shared unit.
• Unique Rectangle Type 5A rectangle in exactly two boxes where one corner holds only X and Y and the other three each hold exactly X, Y and Z.
• Unique Rectangle Type 6A rectangle in exactly two boxes: two opposite corners hold only X and Y, the others hold extras too, and its two rows, or two columns, have no other X.
• Hidden RectangleA rectangle in exactly two boxes with X and Y in every corner, one holding nothing else, and X confined to rectangle corners in the opposite corner's row and column.
• Avoidable Rectangle Type 1Unique Rectangle using cells you solvedA rectangle in exactly two boxes with three solved corners, none of them givens: the two seeing the unsolved corner are both Y, and the one opposite it is X.
• Avoidable Rectangle Type 2Unique Rectangle using cells you solvedA rectangle in exactly two boxes: two corners in one row or column are solved, not givens, and each unsolved corner holds only Z and the digit diagonally opposite it.
• Extended RectangleSix unsolved cells where two rows cross three columns, or two columns cross three rows, in exactly three boxes, holding three digits plus one cell's extras or one extra digit.

Miscellaneous

Rare patterns that fit no other family.

• Sue de CoqTwo-Sector Disjoint SubsetsTwo or three box cells in one row or column hold two more candidates than cells; two bivalue cells, one elsewhere in each unit, split four of those candidates.
• FireworksThree digits missing from a row and a column whose candidates in those two units, outside the box where they cross, all sit in one wing cell of each.
• TridagonFour boxes forming a rectangle each hold three cells on a diagonal, one box slanting against the other three; only one cell, the guardian, has candidates besides X, Y, Z.
• SK LoopFour solved cells forming a rectangle in four boxes; their rows and columns inside those boxes give eight cell pairs, each holding four candidates, two shared with each neighbouring pair.
• Aligned Pair ExclusionXYZ-Wing is a special caseTwo cells that see each other, together with one or more almost locked sets, bivalue cells included, in which every cell sees both of the two cells.

Last Resort

Trial-based methods for positions where no pattern is left to spot.

• ExocetAlong one row or column, two base cells in one box together hold three or four candidates; its other two boxes each have a target seeing neither base nor target.
• Double ExocetTwo valid Exocets whose base pairs sit in different boxes of the same row or column, each pair holding the same four candidates between its two cells.
• Pattern Overlayevery one-digit pattern at onceEvery complete pattern for one digit X is listed: nine cells, one in each row, column and box, containing every placed X and otherwise only cells with candidate X.
• Forcing ChainOne assumption, or each of a set of assumptions that covers every case, such as all candidates of one cell, is followed through the steps it forces.
• Digit Forcing ChainOne candidate is followed both ways, once assumed true and once assumed false, each time placing the naked and hidden singles that follow, and the two results are compared.
• Nishio Forcing ChainOne candidate is assumed true, and the singles it forces end in a contradiction: an empty cell without candidates, or a unit with no place for a digit it lacks.
• Cell Forcing ChainEvery candidate of one cell is assumed true in turn, each time placing the naked and hidden singles that follow, and the results are compared.
• Unit Forcing ChainEvery place for a digit X in one unit is assumed true in turn, each time placing the naked and hidden singles that follow, and the results are compared.
• Forcing NetOne candidate is assumed true and followed through locked candidates as well as naked and hidden singles, any step drawing on several earlier ones, until a contradiction appears.
• Brute ForceNo pattern at all: digits are tried in the empty cells one after another, backing up at every dead end, until the whole grid is filled.