# 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:

## Contents

## 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.

**2**digits with candidates in**2**cells.

**3**digits with candidates in**3**cells.

**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.

- 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

- 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.

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