The Techniques Desk

A reader's catalogue of Sudoku techniques

The named patterns competent solvers recognise on sight, arranged in roughly the order you will need them.

Every Sudoku has a unique solution arrived at by pure logic. In principle you could solve any puzzle by guessing, then backtracking when the guess fails — but that is not how solvers play, and it is not what the puzzle is for. The point of Sudoku is the deduction. Skilled solvers think in patterns: small, repeating shapes of constraint that they have seen often enough to spot in a glance. The patterns have names, and once you know the names you stop guessing.

What follows is a working catalogue. The order is roughly the order in which a difficulty ladder will demand them. An easy puzzle is solvable using only the first two. A medium puzzle wants the next three. By the time you reach expert puzzles you may need every technique on this page and a few more besides.

Singles — the foundation

1. Naked single

A cell where eight of the nine digits are already present in its row, column, or box — leaving exactly one possibility. The most direct technique in Sudoku: scan a row, find the empty cell whose peers cover the most digits, and the missing one belongs there. Used in almost every puzzle, usually many times. Full explanation →

2. Hidden single

A row, column, or box where a particular digit can only legally appear in one of its cells — even if that cell itself has several candidates. Slightly less obvious than a naked single because you look at the digit's perspective, not the cell's. Together, the two singles finish most easy puzzles without anything heavier. Full explanation →

Pairs and triples — when singles run out

3. Naked pair

Two cells in the same row, column, or box that share the same two candidates and nothing else. Between them, those two cells must contain those two digits — so the digits can be eliminated from every other cell in the unit. A naked pair rarely places a digit by itself, but it shrinks the candidate landscape until a single appears.

4. Hidden pair

Two digits that can only appear in the same two cells of a row, column, or box. Mirrors the naked pair: instead of two cells sharing two candidates, two candidates share two cells. The consequence is the same — those two cells are reserved for those two digits, and any other candidate written in them can be erased.

5. Naked / hidden triples

The three-way extension. Three cells, three candidates, distributed any way among them, and again the three digits can be cleared from the rest of the unit. Useful when pairs are scarce but you can spot a tight three-cell cluster.

Intersections — moving information between units

6. Pointing pairs and triples

When a digit's candidates inside a box are all confined to a single row (or column), the digit must lie somewhere on that line inside the box. Therefore it cannot appear elsewhere on that same line outside the box. The constraint points outward from the box along the line. A small move, but the engine that drives many medium-difficulty solves.

7. Box / line reduction

The mirror of pointing. When a digit's candidates in a row (or column) are all confined to a single box, the digit cannot appear elsewhere in that box. Information flows from the line back into the box. Pointing and box/line together cover the most common box-vs-line interactions; serious solvers learn to spot both reflexively.

Fish — the advanced patterns

8. X-Wing

Two rows that each contain a digit's candidate in only two cells — and the cells line up to share the same two columns. The digit must occupy one cell in each row, and the configuration forces it onto two of the four corners of a rectangle. That lets you erase the candidate from the rest of those two columns. An X-Wing is the simplest of the fish family and the first technique that requires looking at the whole board at once rather than just a single unit.

9. Swordfish

Three rows, three columns, the same logic, scaled up. Rare but unmistakable once seen.

10. Jellyfish and beyond

Four-by-four fish. By this point you are deep into expert territory, and almost every puzzle that requires a jellyfish is solvable by some other route — but the patterns are beautiful when they appear.

Wings, chains, and the rest

Beyond the fish family lies a longer tail: XY-Wing and XYZ-Wing, W-Wing, simple colouring, forcing chains, Nice Loops, ALS-XZ, and ultimately Trial-and-Error (which is to say, depth- first search with backtracking — what a computer would do, and what most solvers consider the boundary at which Sudoku has stopped being Sudoku). Whole books are written on the chain techniques; for the dedicated, the HoDoKu solver's technique catalogue is the standard reference.

How difficulty maps to technique

On Very Sudoku, the four difficulty bands correspond roughly to the deepest technique required to crack the puzzle:

• Easy — naked single, hidden single.
• Medium — adds naked pair, hidden pair, pointing.
• Hard — adds box/line reduction, triples, and occasional X-Wing.
• Expert — anything up to swordfish and XY-Wing. Beyond that we declare the puzzle a draft and move on.