Sudoku Techniques Explained: From Hidden Singles to X-Wing
Every Sudoku technique is a way of proving one of two things: that a digit must go in a cell, or that a digit cannot go in a cell. Placement techniques finish easy puzzles on their own. Elimination techniques, applied to your pencil marks, are what crack hard, expert and extreme boards. This page explains the techniques in the order you will need them, each with a diagram and the reasoning behind it, so you can recognise the pattern on any grid. Try them on the board above.
Level 1: placement techniques
Last free cell
A row, column or box with eight digits has one empty cell, and the missing digit goes there. Trivial, but worth a dedicated scan at the start of every puzzle and after every burst of placements.
Hidden single (last remaining cell)
Take one digit and one box. Every copy of that digit already on the grid rules out its whole row and column. If only one cell in the box remains unblocked, the digit belongs there even though the cell might look like it has several options. Sweeping all nine digits through all nine boxes is called cross-hatching, and it solves most Easy and Medium boards without any notes.
Naked single (last possible number)
Look at one empty cell. Gather every digit in its row, column and box. If eight distinct digits are present, the cell can only be the ninth. With notes switched on, a naked single is simply a cell that has one candidate left.
Level 2: pair and locked-candidate eliminations
Naked pairs
Two cells in the same unit whose notes are exactly the same two digits. Between them they must use both digits, so neither digit can appear anywhere else in that unit. Erase it from the other cells' notes. Three cells sharing three candidates form a naked triple with the same effect.
Hidden pairs
The mirror image. If two digits appear in the notes of only two cells within a unit, those cells must hold those two digits. Erase every other candidate from those two cells. Hidden pairs are easy to miss because the cells often look busy.
Pointing pairs and triples
Inside a box, if all the candidates for a digit lie in a single row (or column), the digit must be placed in that row within the box. It can therefore be removed from the rest of that row outside the box.
Box/line reduction (claiming)
The reverse direction. If a digit's candidates in a row all fall inside one box, the digit must be in that box on that row, so it can be removed from the box's other cells. Pointing pairs and box/line reduction together are known as locked candidates.
Level 3: advanced techniques
X-Wing explained
Find a digit that has exactly two candidate cells in each of two different rows, and check whether those four cells line up in the same two columns, forming a rectangle. The digit must occupy one diagonal of the rectangle: if it were in the top-left, it cannot be in the top-right (same row) or bottom-left (same column), forcing the bottom-right. Either way, both columns receive the digit inside the rectangle, so it can be eliminated from every other cell in those two columns. The pattern also works with columns as the base and rows as the cover.
XY-Wing
Three cells with two candidates each. A pivot holding XY sees one wing holding XZ and another wing holding YZ. If the pivot is X, the first wing becomes Z; if the pivot is Y, the second wing becomes Z. One of the wings is Z in every case, so any cell that sees both wings cannot be Z.
Swordfish
An X-Wing on three lines. If a digit's candidates in three rows are confined to the same three columns, the digit occupies exactly one cell in each of those columns within the three rows, and can be eliminated from the rest of those columns. Each row may have two or three candidate cells.
Skyscraper and other chains
Two rows with exactly two candidates for a digit that share one column but not the other form a skyscraper: whichever way it resolves, one of the two "roof" cells is true, so cells that see both roofs can drop the digit. Longer versions of this reasoning are called X-chains and, generalised across digits, alternating inference chains. They appear on Extreme boards and rarely elsewhere.
Which techniques does each level need?
| Level | Techniques required |
|---|---|
| Easy | Naked singles only |
| Medium | Naked and hidden singles |
| Hard | Mostly naked singles, some hidden singles |
| Expert | Hidden singles throughout, occasional locked candidates |
| Master | Naked pairs and locked candidates |
| Extreme | Pairs and locked candidates plus X-Wing, XY-Wing or Swordfish |
The generator behind this site checks which of these a puzzle needs before assigning it a level, so the labels mean the same thing on every board. When a technique is not enough, read the tips for when you are stuck.
Frequently asked questions
What is the difference between a naked single and a hidden single?
A naked single is a cell with only one candidate left. A hidden single is a digit with only one possible cell left within a row, column or box, even if that cell still shows other candidates.
How do I spot an X-Wing in Sudoku?
Work one digit at a time. Look for two rows in which that digit has exactly two candidate cells, and check whether those cells sit in the same two columns. If they form a rectangle, the digit can be removed from the other cells in those two columns.
Do I need notes to use these techniques?
Singles can be found by scanning. Pairs, locked candidates, X-Wing and everything beyond depend on complete pencil marks, because they work by eliminating candidates rather than placing digits.
What technique should I learn after hidden singles?
Pointing pairs and box/line reduction. They are simple to see once notes are in place and they unlock most Expert and Master boards.