Sudoku techniques from beginner to expert, with simple examples

How to read a Sudoku grid

A Sudoku has nine rows, nine columns and nine 3 by 3 boxes, and each must end up holding the numbers 1 to 9 once. In this guide, rows are numbered from the top, columns from the left, and boxes reading like a book, so box 1 is top left and box 5 is the centre. The small numbers you write in a square to show what could still go there are called candidates, pencil marks or notes.

Beginner: the only place and the only number

The easiest step is the last empty square in a row, column or box: the missing number goes there. Two more steps solve most gentle puzzles.

Only place. Pick a box and a number, and ask where the number can go. Say you want the 7 in box 5. If rows 4 and 6 each have a 7 outside the box, the 7 in box 5 must sit in row 5. If columns 4 and 5 also hold a 7 elsewhere, only row 5, column 6 is left. Working through every number in every box like this is called scanning or cross hatching.

Only number. Now look at one empty square instead. If its row has 1, 2, 3 and 5, its column has 4, 6 and 8, and its box has a 9, that is eight different numbers, so only the 7 fits. You have to hold three groups in your head at once, which is why Thinking Cap puts this a level above the only place.

Intermediate: pencil marks, pointing and pairs

When the beginner steps find nothing, write in the candidates. It is slow by hand, but it reveals patterns. These steps cross candidates out until an only place or an only number appears.

• Pointing. If every square in box 1 that could hold a 4 lies in row 2, box 1 will put its 4 in row 2, so you can cross 4 out of the rest of row 2.
• Box line. The reverse. If the only places for a 3 in row 7 lie inside box 8, row 7 gets its 3 from box 8, so you can cross 3 out of the other squares in box 8.
• Naked pair. Two squares in the same row, column or box that both hold only 2 and 6. One will be the 2 and the other the 6, so no other square in that group can be either.
• Hidden pair. Two numbers that fit only the same two squares of a group. Those squares must hold those numbers, so their other candidates can go.

Triples work the same way with three squares and three numbers, even if no square holds all three: 2 and 6, 6 and 9, and 2 and 9 make a naked triple.

Advanced: the X-Wing and the Y-Wing

X-Wing. Look at one number across two rows. Say that in row 2 the 5 can only go in columns 3 and 8, and the same is true in row 7. The four squares make a rectangle, and the two 5s must sit on opposite corners. Either way, columns 3 and 8 each get their 5 from rows 2 and 7, so you can cross 5 out of every other square in those two columns. It works with rows and columns swapped, too.

Y-Wing. This uses three squares with two candidates each. The middle square, the pivot, holds 4 and 9. In its row is a square holding 4 and 6, and in its column a square holding 9 and 6. If the pivot is 4, the row square must be 6. If the pivot is 9, the column square must be 6. Either way one of them is a 6, so any square that sees both of them cannot be a 6.

Expert: Swordfish, kites and chains

Swordfish. An X-Wing stretched over three rows. If, in each of three rows, a number can only go in some of the same three columns, those rows supply that number to all three columns, so you can cross it out of the rest of those columns. A Jellyfish does the same over four.

Strong links and chains. When a number has exactly two places in a row, column or box, it must be in one or the other. A skyscraper or a two-string kite joins two of these either/or links; a chain joins several. Followed through, a chain shows that one end or the other must hold the number, so any square that sees both ends can lose it. These patterns are rare, and you can enjoy years of Sudoku without them.

The techniques at a glance

Why grading by technique beats counting clues

It is tempting to judge a Sudoku by how many numbers it starts with, but a puzzle with few givens can fall to scanning alone, while one with plenty can need an X-Wing. There is a floor, though: in 2012 Gary McGuire, Bastian Tugemann and Gilles Civario showed, by an exhaustive computer search, that a Sudoku with a single answer needs at least 17 givens.

What sets the difficulty is the hardest step a puzzle forces on you, and that is how Thinking Cap grades: a Tricky puzzle needs pointing or pairs somewhere, never anything harder.

Questions people ask

Where Thinking Cap helps, and where it does not

Thinking Cap is our Sudoku and logic puzzle game for Android. Every technique here is in its tutor: tap Hint and it nudges you towards the right part of the grid, then highlights the pattern and explains why the step works. There is a daily set of five short logic puzzles too.

Free: the daily set, today’s Sudoku, unlimited Gentle, Moderate and Tricky Sudoku and every hint. Tough and above, where X-Wings and chains live, need Thinking Cap Pro, a yearly Puzzle Pass or a one-time Lifetime unlock.

What it is good for: learning the steps a level at a time, with no timers unless you want one.

What it will not do: check or explain a puzzle from your newspaper, only its own. Quads are too rare to have a worked example in the guide. It is Android only, and coming soon to Google Play.

Questions or bugs: Email us.

Sources

1. Intersection RemovalSudokuWiki.org, Andrew Stuart · sudokuwiki.org
2. Naked CandidatesSudokuWiki.org, Andrew Stuart · sudokuwiki.org
3. X-Wing StrategySudokuWiki.org, Andrew Stuart · sudokuwiki.org
4. Y-Wing StrategySudokuWiki.org, Andrew Stuart · sudokuwiki.org
5. There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues ProblemMcGuire, Tugemann and Civario, arXiv · arxiv.org