How to solve sudoku: a beginner’s guide

Sudoku has one rule, and everything else follows from it: each of the nine rows, each of the nine columns, and each of the nine 3×3 boxes must contain the digits 1 to 9 exactly once. That is the whole game. There is no arithmetic — the digits could be nine different colours and nothing would change — and there is never a need to guess. If a puzzle is properly made, every cell can be worked out from what is already on the board.

What stops most beginners is not the rule but the first move. Staring at a grid with fifty empty cells, it is not obvious where to start looking. So here is a method that works from the first puzzle you pick up.

Start where the grid is crowded

Do not start at the top-left corner. Start at whichever row, column or box already has the most digits in it. A box with seven of its nine cells filled has only two digits missing, and there is a decent chance those two can only go one way round. That is a free cell, and free cells are how a grid opens up.

Scan the board for the densest region and work outward from it. Every cell you fill in makes its row, its column and its box that little bit more constrained, which is why a sudoku speeds up as it goes: the first ten cells are the slow ones.

The first technique: scanning for a single spot

Pick a digit — 1 is as good as any. Now find a 3×3 box that does not contain a 1 yet. Look at the three rows that pass through that box. If a 1 already appears somewhere in one of those rows, then no cell of that row inside your box can hold the 1. Rule it out. Do the same for the three columns.

Often you will eliminate every cell in the box but one. That last cell must be the 1, and you can write it in with certainty. This is called cross-hatching, and on an easy puzzle it will carry you a long way — sometimes all the way to the end.

Work through the digits one at a time, 1 through 9, then start again at 1. Each pass fills in 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 is just as useful. Instead of asking “where can this digit go?”, pick an empty cell and ask “what could go here?” Look along its row, down its column, and around its box, and cross off every digit you see. If exactly one digit survives, that is your answer.

Beginners tend to favour one of these two views and forget the other. They complement each other: cross-hatching finds cells that are forced by a digit’s position, while this one finds cells that are forced by everything around them. When one stalls, switch to the other before you conclude you are stuck.

When scanning runs out: pencil marks

On a medium puzzle you will reach a point where neither view yields a cell. That is not a wall, it is a signal to change tools. Go through the empty cells and write the possible digits for each one in small figures in the corner — candidates, in sudoku terms. It is tedious the first time and it feels like giving up. It is not: it is the technique the harder half of the puzzle is built around.

With candidates on the board, patterns appear that you cannot see otherwise. The most common is the naked pair: two cells in the same row, column or box that both hold exactly the same two candidates — say 3 and 7. You do not know which is which, but you know that between them they use up both the 3 and the 7 for that region. Every other cell in that region can have 3 and 7 crossed off. That elimination usually cascades into a solved cell somewhere nearby.

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 can only go in one cell of a box, it goes there — regardless of what else that cell might have held. Hidden singles are easy to miss precisely because the cell looks undecided.

Habits that prevent wasted evenings

• Use a pencil. Not for lack of confidence — for the candidate marks. A pen turns a grid into a mess the moment you eliminate something.
• Never guess. If a puzzle has a single solution, and every puzzle from this site does, there is always a logical next move. A guess that turns out wrong will not announce itself for another twenty cells, and unpicking it is worse than being stuck.
• Update your candidates as you go. The most common cause of a contradiction is a stale pencil mark that should have been crossed off three moves ago.
• Put it down. Coming back to a grid after ten minutes away is startlingly effective. You stop seeing the pattern you were convinced was there and start seeing the board.

How to practise

Solve easy puzzles until you stop needing candidate marks to finish one. Then move to medium and stay there until pencilling in a whole grid feels routine rather than laborious. Only then is hard worth your time — attempted too early it teaches frustration rather than technique.

Paper beats a screen for this. On a printed grid you can see all your candidate marks at once, scribble, cross out, and come back to it tomorrow. Print a set of puzzles at the level you are working on — two per page leaves plenty of room for pencil marks — and turn the answer key on so you can check a finished grid instead of wondering about it.

When easy stops being a challenge, the solving techniques guide picks up from here with pointing pairs, box-line reduction and the X-wing.