Solving Technique

These are the methods used to solve a Sudoku. There are many different solving techniques discovered in the last few years, mainly because there is a large community of puzzlers who share their knowledge on the various Sudoku Forums.

Because there are so many, a rough subclassification has been made:

Singles

Full House

A house with a single empty cell.

Last Digit

The last instance of a digit.

Hidden Single | Pinned Digit

A single candidate remaining for a digit in a house.

Naked Single | Forced Digit | Sole Candidate

A single candidate remaining in a cell.

Intersections

Locked Candidates | Intersection Removal | Line-Box Interaction | Pointing (Pair, Triple) | Claiming

Candidates are locked in the intersection of a line and a box.

Locked Pair

A Naked Pair located in a single intersection.

Locked Triple

A Naked Triple located in a single intersection.

Almost Locked Candidates

A box-line intersection where the line or the box contains an Almost Locked Set, and the remaining cells in the line or box outside the intersection does not contain digits from the Almost Locked Set.

Subsets

Naked Subset

N cells with candidates for N digits.

Naked Pair

2 cells with candidates for 2 digits.

Naked Triple

3 cells with candidates for 3 digits.

Naked Quad

4 cells with candidates for 4 digits.

Hidden Subset

N digits with candidates in N cells.

Hidden Pair

2 digits with candidates in 2 cells.

Hidden Triple

3 digits with candidates in 3 cells.

Hidden Quad

4 digits with candidates in 4 cells.

Fish

Basic Fish

Rows and columns only.

X-Wing

2 rows vs. 2 columns

Swordfish

3 rows vs. 3 columns

Jellyfish

4 rows vs. 4 columns

Squirmbag

5 rows vs. 5 columns

Finned Fish

Fish patterns with additional candidates in a single box.

Sashimi Fish

Incomplete basic fish patterns with a fin.

Franken Fish

Fish patterns that include box constraints.

Mutant Fish

Fish patterns with mixed sets of constraints.

Kraken Fish

A fish pattern with indirect connections to a candidate which can be eliminated.

Single Digit Patterns

Skyscraper

Two parallel strong links, weakly connected at the base.

2-String Kite

Two crossing strong links, weakly connected in a box.

Empty Rectangle

A single-digit technique that makes use of a box whose candidates for that digit are contained within the union of a boxrow and a boxcol.

Coloring

Simple Colors

Uses only 2 colors to form a single color cluster.

Multi-Colors | Supercoloring

Uses multiple colors (4, 6 or a higher multiple of 2) to form multiple color clusters.

Weak Colors

Extends Simple-Colors by the use of hinge linkages.

X-Colors

Uses only 2 colors, but also takes the implications for each color into account.

Color Trap

A technique that uses a single cluster to eliminate candidates outside the cluster.

Color Wrap

A technique that uses a single cluster to detect a contradiction in one of the colors.

Color Wing

A technique that uses multiple clusters to eliminate candidates outside these clusters. Same as Multi-Colors.

3D Medusa Coloring | Advanced Coloring | Ultracoloring

A set of techniques that uses colors on multiple digits.

Uniqueness

Uniqueness Test

A set of techniques that avoids the Unique Rectangle deadly pattern.

Avoidable Rectangle

A set of techniques that avoids the Unique Rectangle deadly pattern. Perhaps unique in solving techniques in that it employs solved cells as well as those unsolved.

Bivalue Universal Grave

A set of techniques that avoids the Bivalue Universal Grave deadly pattern.

BUG Lite

A set of techniques that avoids the BUG Lite deadly pattern.

Uniqueness Controversy

These solving techniques assume that the Sudoku puzzle has a unique solution. It is not universally accepted that this assumption is a valid one.

Chains and Loops

Forcing Chain

Generic term for any type of chain.

X-Chain

Single digit chain of cells.

XY-Chain

Chain of bivalue cells.

Remote Pairs

Simplified form of XY-Chain, involving only two digits.

Fishy Cycle | X-Cycle

Single digit continuous loop.

Broken Wing

Eliminations caused by loops of odd length.

Nice Loops

Several types of loops formed by cells following strict rules and a notation system.

Double Implication Chain | DIC

There are 2 interpretations circulating. The first originates from a reliable source [1].

• A Forcing Chain that has implications in both directions.
• 2 Forcing Chains starting from a bivalue cell or a bilocal unit showing a verity.

Alternating Inference Chain | AIC

A chain where each node is a candidate, with alternating strong and weak inference.

Oriented Chains

Oriented chains are generalisations of the basic xy-chains. The central idea is that the information collected from previous candidates in a chain (and/or from the target) can be used to select the next candidates. Oriented chains are either 2D or 3D:

- oriented 2D chains: xyt-chains, xyz-chains and xyzt-chains, together with their "hidden" counterparts: hxyt-chains, hxyz-chains and hxyzt-chains;

- oriented 3D chains: nrct-chains, nrcz-chains, nrczt-chains.

(The original reference to all these chains is the book "The Hidden Logic of Sudoku". Further web references forthcoming)

Wings

XY-Wing

Three cells with pivot cell XY and two pincer cells XZ and YZ.

XYZ-Wing

Three cells with pivot cell XYZ and two pincer cells XZ and YZ.

WXYZ-Wing

Four cells with pivot cell WXYZ and three pincer cells WZ, XZ and YZ.

W-Wing

Four cells in a chain: a cell WX, a cell with X as a candidate, another cell with X as a candidate, another cell WX, such that the two cells containing X as a candidate have a strong link.

Almost Locked Sets

ALS-XZ rule

2 Almost Locked Sets with restricted common digit X perform eliminations for common digit Z.

ALS-XY-Wing rule

3 Almost Locked Sets with 2 restricted common digits Y and Z perform eliminations for common digit X.

ALS-XY-Chain

Very similar to XY-Chain, except that the nodes in the chain are Almost Locked Sets.

Death Blossom

A stem cell of N candidates pointing to N petals, each an Almost Locked Set.

Enumeration on Selected Cells

Aligned Pair Exclusion

Checks combinations of digits in a pair of cells located in an intersection.

Aligned Triple Exclusion

Checks combinations of digits in a triple of cells located in an intersection.

Subset Exclusion

Extension of Aligned Pair Exclusion to arbitrary sets that need not be aligned.

Miscellaneous

Sue de Coq

Uses two intersecting sets A and B, where A is a set of N cells with N candidates in a line, B is a set of N cells with N candidates in a box, and the sets A - B and B - A have no common candidates.

Subset Counting | Extended Subset Principle

Considers the number of times a digit can be placed in a selected subset of cells.

Braid Analysis | Braiding | Traveling Pairs

Analyzes the distribution of digits in a chute, especially how pairs repeat in a chute.

Constraint Subsets

Generalized view of subset and fish techniques.

Equivalence Marks

Similar to coloring, but uses marks instead of colors to represent parity.

Gurth's Symmetrical Placement

Technique for puzzles with rotational symmetry.

Techniques of Last Resort

Most of these techniques are controversial. They are either too complex to be used by human solvers, or they require a lot of extra work, or they are not based on logic.

Forcing Net

A general term for techniques that investigate branching chains.

Tabling | Trebor's Tables

All implications for each move are collected in tables.

Graded Equivalence Marks | GEM

A system of marking candidates starting from 2 complementary starting positions.

Bowman Bingo

A systematic approach to investigate implications.

Trial & Error | Ariadne's Thread | Bifurcation

The implications for a single move are investigated.

Nishio

Trial & Error limited to single digit.

Templating | Pattern Overlay Method | POM

All possible ways to place a digit in the remaining candidate space are investigated.

Guessing

Random moves are made, without any logic behind them.

Brute Force

These are primarily designed for computer solver programs.

• Backtracking Algorithms.

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