Sudoku solving techniques
Every Sudoku can be solved with a small set of logical techniques. This index lists the techniques that the Bare Sudoku solver and its hints use, in the order the solver tries them: singles first, then eliminations within one unit, then patterns that span several rows, columns and boxes, including patterns that rely on the puzzle having exactly one solution. Each technique has its own page with a definition, how to spot it, a real example from a generated puzzle and the common mistakes.
The level of a puzzle is decided by the hardest technique it needs. Easy and Medium puzzles need only singles, Hard puzzles need locked candidates or pairs and triples, Expert puzzles need at least one advanced pattern, such as a fish, a wing, a Skyscraper or a Unique Rectangle, and Master puzzles need chains or other techniques beyond this guide. Whatever the level, singles do most of the work; the advanced technique removes a candidate or two and singles carry on from there.
Cells are written as r3c7: row 3, column 7. Rows are counted from the top, columns from the left, and boxes 1 to 9 from the top left, row by row.
- Naked singleEasy
A naked single is an empty cell with only one possible digit. Learn how to find naked singles quickly, with a real example step by step and the usual mistakes. - Hidden singleEasy
A hidden single is the only cell in a row, column or box where a digit can still go. Learn to spot hidden singles by scanning one digit at a time. - Locked candidatesHard
Locked candidates: when a digit's candidates in a box all lie in one row or column, or the reverse, the digit is removed elsewhere. Two real examples. - Naked pairs and triplesHard
Naked pairs and triples: two or three cells in a unit that share the same two or three candidates. Learn to spot them and what to remove, with real examples. - Hidden pairs and triplesHard
A hidden pair is two digits that can only go in the same two cells of a unit, so their other candidates go. Learn to find hidden pairs and triples step by step. - X-WingExpert
X-Wing: a digit with exactly two candidate cells in two rows, lined up in the same two columns, is removed from the rest of those columns. Real example. - SwordfishExpert
Swordfish: a digit confined to three columns across three rows is removed from the rest of those columns. The size-three fish, explained with a real example. - SkyscraperExpert
Skyscraper: a digit with exactly two candidates in each of two rows, aligned in one column only, is removed from cells seeing both other ends. Real example. - 2-String KiteExpert
2-String Kite: a digit with two candidates in a row and two in a column, joined in one box, is removed from cells that see both free ends. Real example. - Y-WingExpert
Y-Wing (XY-Wing): a pivot with two candidates and two wings that each share one of them force a digit out of every cell that sees both wings. Example. - XYZ-WingExpert
XYZ-Wing: a pivot with three candidates and two wings that share one of them force that digit out of every cell seeing all three. Real example, step by step. - W-WingExpert
W-Wing: two cells with the same two candidates, linked by a digit that has only two places in a unit, remove the other digit from cells that see both. - Unique RectangleExpert
Unique Rectangle: when three corners of a rectangle in two boxes hold only the same two candidates, the fourth loses them, or there would be two solutions. - Finned X-WingExpert
Finned X-Wing: an X-Wing with an extra candidate, the fin, in a base row. The digit is removed from the cover cells that lie in the fin's box. Real example. - Empty RectangleExpert
Empty Rectangle: a box where a digit's candidates lie in one row and one column, joined to a strong link, rules the digit out of one cell at the far end. - Unique Rectangle Type 4Expert
Unique Rectangle Type 4: two corners hold only the pair A and B, and A is confined to the other two corners in a shared unit, so B leaves both of them. - Hidden RectangleExpert
Hidden Rectangle: four corners hold the pair A and B, one of them nothing else, and A is confined to the rectangle in two lines, so the far corner loses B. - Finned SwordfishExpert
Finned Swordfish: a Swordfish with an extra candidate, the fin, in a base row. The digit is removed from cover cells that lie in the fin's box. Real example. - JellyfishExpert
Jellyfish: one digit confined to the same four columns in four rows, or the same four rows in four columns. It is removed from the rest of those lines. - WXYZ-WingExpert
WXYZ-Wing: four cells with exactly four candidates between them, bent across two units. The one digit not confined to cells that see each other is removed. - BUG+1Expert
BUG+1: every unsolved cell has two candidates except one with three. Since the puzzle has one solution, that cell takes the digit seen three times. - X-ChainMaster
X-Chain: follow one digit through alternating strong and weak links. One end of the chain must hold the digit, so cells that see both ends lose it. - XY-ChainMaster
XY-Chain: a path of cells with two candidates each, linked by shared digits. Both ends can hold the same digit Z, so cells that see both ends lose Z. - AICMaster
Alternating Inference Chain: candidates joined by strong and weak links across digits and cells. One end is true, so cells that see both ends lose that digit. - ALS-XZMaster
ALS-XZ: two almost locked sets share a restricted digit X, so one of them is locked. Digit Z must then be in one set and leaves every cell that sees all its Z.
Frequently asked questions
In which order should I try Sudoku techniques?
Cheapest first, in the order of this index: singles, then locked candidates, then naked and hidden pairs and triples, then the advanced patterns, starting with X-Wing. Bare Sudoku tries its techniques in exactly this order and always applies the first one that makes progress; a hint names the hardest technique it needed to reach the next digit.
Do I need all these techniques to solve Sudoku?
For Easy and Medium puzzles, singles are enough. Hard puzzles add locked candidates and subsets. Expert puzzles need at least one advanced pattern, such as a fish, a wing, a Skyscraper or a Unique Rectangle, but usually only once or twice per puzzle. Master puzzles go further and need chains or other techniques that this guide does not cover.
Are there harder techniques than the ones in this guide?
Yes. Players have described many more techniques for the hardest puzzles, and new ones keep appearing. Bare Sudoku uses several of them beyond the pages here: more fish, wings and rectangle patterns at Expert level, and chains, almost locked sets and forcing nets at Master level. The hint always names the technique it used, so you can look it up even when this guide has no page for it yet.
What is a candidate?
A digit that could still go in an empty cell because nothing in its row, column or box rules it out yet. Notes in the game are candidates. Every technique beyond singles reasons about candidates, so keep them complete and accurate.