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Techniques

A guide to solving Sudoku puzzles. Here we show you various techniques to solve Sudoku puzzles of all levels. You will need to upgrade your membership to access this section.

Basic Strategies

Naked Pairs

A Naked Pair (also known as a Conjugate Pair) is a set of two candidate numbers sited in two cells that belong to at least one unit in common. That is they reside in the same row, box or column.

Naked Triples

Naked Pairs can be extended to Naked Triples. Any three cells in the same unit that contain the same three candidate numbers will be a Naked Triple. The rest of the unit can be scrubbed clean of any of those numbers.

Naked Quads

A Naked Quad is rarer, especially in its full form but still useful if they can be spotted. The same logic from Naked Triples applies to combinations of numbers.

Hidden Pairs

Hidden Pairs are real work horses for removing candidates and they are the basis for more advanced strategies as well.

Hidden Triples & Quads

Hidden Pairs can be extended to Hidden Triples or even Hidden Quads. A Triple will consist of three pairs of numbers lying in three cells in the same unit (row, column or box)

Pointing Pairs/Intersection Removal

We’re already familiar with Pairs, Triples and Quads but previously we were only looking for them in a single row, column or box (unit). There are sound logical reasons for looking at the overlap of two units now which is where the name ‘intersection removal’ comes from.

Box/Line Reduction

As an extension of Intersection removal we should also be looking to remove candidates within a box because of a row or column. This is sometimes called Box/Line Reduction but is really just a type of Intersection Removal.

 

Uniqueness Strategies

Unique Rectangles

There is a family of strategies which rely on Sudoku puzzles having one single solution. Unique Rectangles is one of those techniques.

Empty Rectangles

This is a clever but obscure spin on uniqueness in a Sudoku. An Empty Rectangle occurs within a box where four cells form a rectangle and do NOT contain a certain candidate.

 

Chaining Strategies

X-Wing Family

X-Wings are the start of a family of strategies with such evocative names as Sword-Fish, Jelly-Fish, Squirm-Bag and Burma. They are often considered an advanced strategy but they all operate on a single number so I believe they are mid way and relatively easy to spot. Only X-Wings are common but we’ll discuss the whole family to be thorough and logical and you’ll soon see the pattern.

Sword-Fish

X-Wing operates on two rows (or columns) aligned with two columns (or rows). Can it be extended to three rows and columns? Logically yes, and we’re taking about a grid of nine cells here, not four, but interestingly not everyone of the nine cells needs to have the number under consideration.

Jelly Fish

For a real hair-pulling screamer that’s still logical, not a lot beats a Jelly-Fish – except the fabled Squirm-Bag which uses five rows and columns. Jelly-Fish is the fourfold extension of X-Wing and Sword-Fish.

Simple Coloring or Chains

As a preamble to many of the more complicated logical strategies it is worthwhile understanding how cells ‘see’ each other. This allows us to use more familiar language than technical terms like ‘overlaying’ or ‘congruence’. There are 81 cells in a normal Sudoku and they all belong to one row, column and box. Any cell can always ‘see’ all eight other cells in the box they reside in, plus six other cells in the row and column, making a sphere of influence of 20 cells.

Multi-Coloring

Now we’ve seen how chains are formed and can be applied we can take the idea to the next level. This is where more than one chain is present on the board.

Y-Wing Strategy

This is an excellent candidate eliminator (and is also known as XY-Wing). The name derives from the fact that it looks like an X-Wing - but with three corners, not four. The forth corner is where the candidate can be removed from.

XYZ-Wing Strategy

This extends Y-Wings which have three bi-value cells to bi-tri-bi, in other words the hinge contains three candidates, not two.

WXYZ-Wing

This is an extension of XYZ-Wing that uses four cells instead of three. Each possible value of the hinge cell results in a Z value in one of the cells in the WXYZ-Wing pattern, thus leaving no room for a Z on any cell all four can 'see'.

Y-Wing Chains

Y-Wings have a ‘hinge’ and two pincers. The pincers allow us to target certain cells but the hinge is not involved in the elimination. It only holds the pincers together in a vice of logic. There is no reason why the hinge has to be a single cell. An odd number of identical pairs linked together will also work. In fact a Y-Wing is just a Y-Wing Chain of length 1.

XY-Chain Strategy

Y-Wing chains are actually part of a larger family of chains of bi-value cells that’s rather like a platted thread. Numbers intertwine from one end to the other. If the two ends contain a candidate in common then we can show that one end or the other must contain that number and any cells seen in common cannot.