How to solve sudoku: a beginner’s guide

Sudoku has one rule: each of the nine rows, nine columns and nine 3×3 boxes must contain the digits 1 to 9 exactly once. The puzzle involves no arithmetic, since the digits could be nine colours and nothing would change, and you never need to guess. On a well-made puzzle you can work out each cell from what is already on the board.

Most beginners get stuck on the first move. Faced with fifty empty cells, you have no obvious place to start looking. The method below works from the first puzzle you pick up.

Start where the grid is crowded

Skip the top-left corner and start at the row, column or box with the most digits already in it. A box with seven of its nine cells filled is missing only two digits, and those two often fit only one way round. That gives you a free cell, and free cells open up the grid.

Scan the board for the densest region and work outward from it. Each cell you fill in constrains its row, column and box a little more, so a sudoku speeds up as you go. The first ten cells take the longest.

The first technique: scanning for a single spot

Pick a digit, say 1. Find a 3×3 box that does not contain a 1 yet, and look at the three rows that pass through it. If a 1 already appears in one of those rows, no cell of that row inside your box can hold the 1, so rule those cells out. Do the same for the three columns.

Often you will eliminate all but one cell in the box. That last cell must hold the 1, so write it in. Solvers call this cross-hatching, and on an easy puzzle it can carry you to the end.

Work through the digits one at a time, 1 through 9, then start again at 1. Each pass fills cells that make the next pass more productive. Two or three complete passes will finish most easy printable sudoku grids.

The second technique: the last cell standing

The reverse view works as well. Pick an empty cell and ask which digits could go in it. Look along its row, down its column and around its box, and cross off each digit you see. If one digit survives, write it in.

Beginners tend to favour one of these views and forget the other. Cross-hatching finds cells forced by one digit’s position; the last-cell check finds cells forced by everything around them. When one stalls, switch to the other before you decide you are stuck.

Pencil marks, for when scanning stalls

On a medium puzzle you will reach a point where neither view yields a cell. At that point, change tools. Go through the empty cells and write the possible digits for each in small figures in the corner; solvers call these candidates. The first time it feels tedious, but the harder half of sudoku depends on this technique.

With candidates on the board, you can see patterns that stay hidden otherwise. The most common is the naked pair: two cells in the same row, column or box that hold the same two candidates, say 3 and 7. You do not know which cell gets which digit, but between them they use up the 3 and the 7 for that region, so you can cross 3 and 7 off the region’s other cells. That elimination often solves a nearby cell.

Its mirror image is the hidden single: a candidate that appears in only one cell of a region, even though that cell has several other candidates. If the 4 fits only one cell of a box, it goes there, whatever else that cell might have held. You can miss hidden singles because the cell looks undecided.

Habits that save you time

• Use a pencil. You will erase a lot of candidate marks, and a pen turns the grid into a mess.
• Do not guess. On a puzzle with a single solution, which covers each puzzle from this site, a logical next move exists. A wrong guess can hide for twenty cells, and unpicking it costs more than being stuck.
• Update your candidates as you go. Contradictions usually come from a stale pencil mark you should have crossed off three moves earlier.
• Take a break. After ten minutes away, you come back to the grid without the pattern you had fixed on, and you see the board as it is.

How to practise

Solve easy puzzles until you can finish one without candidate marks. Then move to medium and stay there until pencilling in a whole grid feels routine. After that, hard puzzles are worth your time; if you try them too early, you learn frustration instead of technique.

Practise on paper. On a printed grid you can see all your candidate marks at once, scribble, cross out and come back tomorrow. Print a set of puzzles at your level, two per page for room to write, and turn on the answer key so you can check a finished grid.

Once easy puzzles feel too simple, the solving techniques guide continues with pointing pairs, box-line reduction and the X-wing.