SadMan Software: Techniques for Solving Sudoku

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Solving Techniques

These are some of the techniques that can be used to solve Sudoku puzzles. They're listed in roughly increasing order of complexity. Many published sudokus won't require any technique beyond hidden subsets, but the more complex techniques are extremely useful against the very hardest puzzles.

See the glossary for an explanation of the terms used.

Naked Single / Singleton / Sole Candidate

It is often the case that a cell can only possibly take a single value, when the contents of the other cells in the same row, column and block are considered. More ť

Hidden Single / Unique Candidate

If a cell is the only one in a row, column or block that can take a particular value, then it must have that value. More ť

With the notable exception of forcing chains, the remaining techniques are all about reducing the number of candidates for cells. The aim being to reduce the candidates to such an extent that the first two techniques can be used.

Block and Column / Row Interactions

Sometimes, when you examine a block, you can determine that a certain number must be in a specific row or column, even though you cannot determine exactly which cell in that row or column. More ť

Block / Block Interactions

If a number appears as candidates for two cells in two different blocks, but both cells are in the same column or row, it is possible to remove that number as a candidate for other cells in that column or row. More ť

Naked Pair, Triplet, Quad / Naked Subset / Disjoint Subset

If two cells in the same row, column or block have only the same two candidates, then those candidates can be removed from other cells in that row, column or block. This technique can also be extended to cover more than two cells. More ť

Hidden Pair, Triplet, Quad / Hidden Subset / Unique Subset

This technique is very similar to naked subsets, but instead of affecting other cells with the same row, column or block, candidates are eliminated from the cells that hold the hidden subset. More ť

X-Wing

This is another method of reducing the candidates when two rows have the same candidate only in the same two columns. More ť

Swordfish

Swordfish is on the same principle as X-wings, but extended to three columns or rows. More ť

XY-Wing

This is similar to a short forcing chain with only two links for each candidate. More ť

XYZ-Wing

This is a variation of an XY-wing. More ť

Colouring

Colouring considers cells where a particular candidate occurs for only two cells in a unit. More ť

Remote Pairs

This technique is a combination of naked pairs and colouring. More ť

XY-Chain

XY chains allow you to make eliminations by following a chain of cells that have only two candidates each. More ť

Forcing Chains

Forcing chains is a technique that allows you to deduce with certainty the content of a cell from considering the implications resulting from the placement of each of another cell's candidates. More ť

Nishio

This is limited form of trial and error. It asks what are the effects of putting a particular number into a particular cell? More ť

Trial and Error

There are some that would argue trial and error is not a logical technique, and is no better than guessing. When further moves seem impossible, trial and error may be the only way forward. More ť

Other Techniques

Turbot fish: This is somewhere between X-wings and swordfish, and so is also related to colouring.
Jellyfish and Squirmbag: tongue-in-cheek names for extensions of the X-wing and swordfish techniques to a greater number of cells.
Tabling: this is an exhaustive search using "structured trial-and-error". Only possible using a computer solver.
Uniqueness: Assumes that the puzzle is well-formed and so has only one solution, then makes deductions along the lines of "this cell cannot be X, or else the puzzle would have multiple solutions."