The rules, and how to read the board
Every technique in this handbook builds on the few ideas below. Read them once, then come back when you meet an unfamiliar word, or look it up in the glossary.
The rules
Fill the 81 cells with the digits 1 to 9 so that every row, every column and every 3×3 box contains all nine digits, with no repeats.
- Each row (9 cells across) contains 1–9.
- Each column (9 cells down) contains 1–9.
- Each box (the 3×3 areas with thick borders) contains 1–9.
A proper Sudoku has exactly one solution and can always be solved by reasoning, without guessing. The bold digits on the board are the givens; you may not change them.
There is no arithmetic involved: the digits are just nine different symbols. You could replace them with nine colours and the game would be exactly the same.
Names: rows, columns, boxes, cells
Rows are numbered 1–9 from top to bottom, columns 1–9 from left to right. A cell is named by its row and column: r3c5 is the cell in row 3, column 5 (highlighted yellow). The Vietnamese pages write the same cell as H3C5.
Boxes are numbered 1–9 in reading order: box 1 top left, box 3 top right, box 9 bottom right. r3c5 is in box 2.
Rows, columns and boxes are all called units. The board has 27 units, and every cell belongs to exactly three — the blue outlines show the three units of r3c5.
Three boxes side by side form a band across or a stack down. Many techniques scan by band because its three boxes share the same three rows (or columns).
Cells that see each other
Two cells see each other when they share a row, a column or a box. Two cells that see each other can never hold the same digit.
Every cell sees exactly 20 others: 8 in its row, 8 in its column and 4 more in its box. The board shades the 20 cells that r5c5 sees.
Most advanced eliminations end with the same sentence: “a cell that sees both of these cells cannot be that digit”. Training your eye to find the cells two cells both see is well worth the effort.
Writing pencil marks
Candidates are the digits a cell can still take: those not already in its row, column or box. Players write them small inside the cell; these are called pencil marks.
Each digit has a fixed spot in the cell, like a phone keypad: 1 top left, 5 in the middle, 9 bottom right. That way you recognise patterns from positions at a glance, without reading every digit.
The board alongside has every candidate written in. Early in a puzzle, many players only note a digit when it has exactly two places in a box (Snyder notation): fewer marks, but it still reveals hidden singles and locked candidates.
Once you move on to 2-step reasoning, write them all. After placing a digit, immediately erase it from the candidates of the 20 cells it sees.
A routine for solving a puzzle
This order goes from the cheapest techniques to the most expensive. Whenever you make progress, go back to step one.
- 1
Scan for hidden singles digit by digit
For each digit 1 to 9, cast rays from its copies and look for a box, row or column with only one place for it. Start with digits that appear often.
- 2
Pick off nearly full cells and units
A unit with one empty cell, or a cell whose row + column + box already hold 8 digits, can be filled right away. After each placement, look again at its row, column and box.
- 3
When stuck, write all candidates
Write every possible digit in every empty cell. Do it carefully once: missing or extra candidates make later techniques give wrong results.
- 4
Look for locked structures
Try locked candidates (pointing, claiming), then naked and hidden pairs and triples. Each elimination may open up a new hidden single — go back to step 1.
- 5
For hard puzzles, follow the links
List strong links and bivalue cells, and look for X-Wings, XY-Wings, colouring and chains. After every elimination, return to the simplest techniques.
Colour key for the example boards
- Given digit
- Digit placed by reasoning
- Cell being studied, or just solved
- Cells that make up the pattern
- Blocking digit: blocks its row, column and box
- Blocked cell: cannot take the digit in question
- Eliminated candidate
- Two mutually exclusive options (blue or orange)
- Blocking ray: the direction a blocking digit rules out
- Dotted line: the eliminated cell sees this pattern cell
- Strong link: if not this cell, then that one
- Weak link: if this cell, then not that one