Overview | Single Position | Printable page |

# Techniques For Solving Sudoku

Sudoku is a puzzle involving logic - no arithmatic or guessing is required! The basic idea of completing puzzles is to find cells (the small squares) where you are sure that only one value is a valid placement.

## The Basics

The rules of Sudoku are that you should fill a number in to every cell in the grid, using the numbers 1 to 9. The restriction is that you can only use each number once in each row, each column, and in each of the 3x3 boxes.

Take a look at this incomplete puzzle... | |

The three red questionmarks are in places where there is a missing value - but it is easy to see which number to place in each. If the rest of the line or box is complete, then by a process of elimination, you know which value must be left!
In this case, the only number missing from the horizontal row is a |

## Easy Techniques

The majority of Sudokus of the type seen in newspapers can be solved with use of just two easy techniques, Computer programs that assist players can make these kinds of puzzles very quick to solve, because they help to make it obvious where the next placement can be.

- Technique 1 : Single Position
- Technique 2 : Single Candidate

## Pencilmarks

Most paper-and-pencil Sudoku players tend to come up with their own systems to help them complete the grids. Very few players complete puzzles only by writing in the final numbers!

The most common way to annotate grids is by writing in tiny numbers - usually referred to as "pencilmarks" - which really mean "this number is still possible for this cell." You may be able to narrow down a cell to only contain "5 and 8" but not know which one. If you write a tiny 58 in the cell, then later on you may make another placement which allows you to cross off one of your pencilmarks. So, if you crossed out the 5, you know that the cell can now only contain an 8, so you can erase both pencilmarks and write in the large 8. | |

This example is computer generated, and shows all of the possible pencilmarks that could be added. You don't need to add in all of the pencilmarks - good players find it slows them down too much for the easiest puzzles, but need to use them for trickier puzzles. |

Many of the techniques that follow don't actually provide you with a direct placement, but help you by allowing you to cross out one or more pencilmarks.

Writing in all of the pencilmarks for every cell is quite a laborious task, but a computer program can do this automatically, updating the pencilmarks for cells whenever you add in a new value. Using pencilmarks makes it very easy to spot Single Candidates! (See the lone 3 in the top right 3x3 block)

## Medium Techniques

Going a little further, there are some extra techniques which help you to find either valid placements, or to help you to remove some of the pencilmarks. These are obviously quite tricky to manage without using pencilmarks!

- Technique 3 : Candidate Line
- Technique 4 : Double Pair
- Technique 5 : Multi-Line

### Advanced Techniques

Going a little further, there are some extra techniques which help you to find either valid placements, or to help you to remove some of the pencilmarks. These are obviously quite tricky to manage without using pencilmarks!

- Technique 6 : Naked Pairs/Triples
- Technique 7 : Hidden Pairs/Triples

### Master Techniques

Going a little further, there are some extra techniques which help you to find either valid placements, or to help you to remove some of the pencilmarks. These are obviously quite tricky to manage without using pencilmarks!

- Technique 8 : X-Wing
- Technique 9 : Swordfish
- Technique 10 : Forcing Chains

### Harder Still

So, armed with all of these potential techniques, you'll be able to solve all possible Sudoku? Well, maybe, maybe not. The vast majority of puzzles don't require the trickier techniques, but there are some which just aren't solvable by simple logic alone, and require various forms of guessing to solve. Some argue that guessing is a form of logic, but it does often mean a lot of erasing!

Nishio is a form of guessing, where you look for a guess causing a contradiction, meaning that you can rule it out. Going on from this, it is possible to solve entire Sudoku puzzles from guesses alone, but it can take a long time!