# Sudoku Variants

This wiki page provides an overview of common (and some less common) sudoku variants, organized into categories based on their characteristics.

# Introduction

Sudoku variants augment or modify classic sudoku rules, possibly changing the geometry/topology or adding global or local constraints.

## Geometry and Topology

Sudoku variants can feature different geometries and topologies, such as varying grid sizes and shapes, diverse cell shapes (for example triangles or hexagons), or alternative grid layouts like multiple interacting or overlapping grids. Grids can also be visually altered or governed by additional rules to exhibit topologies other than the common planar one. Examples include cylindrical, toroidal, Möbius band or Klein bottle topologies, as well as embeddings on surfaces of or within three-dimensional objects like cubes and other shapes.

## Global Constraints

Global constraints are rules that apply to the entire puzzle. These can include extra regions, such as diagonals or disjoint groups, or additional constraints that span the entire grid, such as chess-based rules.

## Local Constraints

Local constraints are rules that apply to individual cells or specific groups of cells. These can include given digits, line-based constraints (renban, thermometers, etc.), indexing rules, shading rules inspired by pencil puzzle elements, outside clues (sandwich, X-sums, etc.), and more.

## Constraints Defined by Additional Weak Links

A large class of global and local constraints can be defined by supplementing the weak links already inherent in standard sudoku rules (in other words: adding edges to the weak link graph). As a result, more eliminations can potentially be made if a candidate with extra weak links is true.

For instance, a global nonconsecutive constraint introduces weak links from a candidate n in a cell to the candidates n-1 and n+1 (if they exist) in all orthogonally connected cells.

Similarly, a German Whispers line establishes weak links between the candidates of each cell on the line and all candidates in neighboring cells on the line that have a numerical distance of less than 5 to the first candidate.

## Constraints Defined by Arithmetic

Arithmetic constraints require carrying out mathematical operations, which frequently involve but are not limited to addition, on designated groups of cells. Variants featuring such constraints include killer sudoku, little killer sudoku, and arrow sudoku.

Weak links between candidates are generally insufficient to describe these constraints. One way to define them is by extending the weak link graph to a hypergraph, with (hyper)edges connecting two different *subsets* of all candidates. However, in practice, both human and computer solvers typically perform the arithmetic operations for the cases that arise to limit combinations and may employ shortcuts, such as utilizing triangular numbers.

Conditional summation constraints like Sandwich- or X-Sums cannot be captured by a weak link graph or hypergraph alone.

## Other Constraints

Some sudoku variants incorporate unique or specialized constraints that don't easily fit into the aforementioned categories. These can range from shading puzzles that merge rules from Japanese pencil puzzles and sudoku, to "chaos constructions" where the shape of the regions must be determined using clues, as well as various other innovative constraints.

In recent times, entirely new game mechanics inspired by board and video games have been applied to sudoku. Examples include Fog of War or shifting puzzles like Sokoban, Rush Hour, and more.

# List of Sudoku Variants

This list contains descriptions of some of the more popular sudoku variants. It may be biased towards variations that have gained popularity in recent times or possess more interesting mathematical properties and warrant theoretical examination.

It is by no means exhaustive, as new sudoku variants continue to emerge every day. Rules can also be combined or modified to create hybrid variants.

When solving puzzles, please refer to the variant rules provided by the puzzle author, as they might not exactly coincide with the rules described for the variant here.

## Other Geometries

- Samurai Sudoku

## Differently Shaped Regions

- Jigsaw Sudoku (Irregular Sudoku, Nonomino Sudoku)

## Chess Constraints

- Anti-Knight Sudoku
- Anti-King (no touch) Sudoku
- Anti-Queen / Bishop Sudoku
- Fairy-chess / general chess constraints
- Pro-chess constraints
- Manhattan / Taxicab distance

## Extra Regions

- Windoku (Window Sudoku, Hyper Sudoku, NRC Sudoku)
- Diagonal Sudoku (Sudoku X, X-Sudoku)
- variant: Anti-Diagonal Sudoku

- Disjoint Groups Sudoku
- Hashtag

## Arithmetic Constraints

- Killer Sudoku
- Sum Frame Sudoku
- Little Killer Sudoku
- Arrow Sudoku
- Variants with other operations (*, /, etc)
- Double arrows
- Inequalities
- Doublers
- Lunchbox
- Young Tableaux

## Forbidden Orthogonal Pairs

- Non-Consecutive Sudoku
- Kropki Sudoku (kropki pairs)
- XV Sudoku

## Line Constraints

- Renban (Consecutive Sudoku)
- German Whispers
- Chinese Whispers
- Dutch Whispers
- Modular Lines
- Region Sum Lines
- Palindrome Sudoku
- Thermometer Sudoku
- Slow Thermometers
- Between Lines
- Entropic Lines
- Parity Lines
- Product-Sum Lines

## Outside Clues

- Sandwich Sudoku
- X-Sums Sudoku
- Numbered Rooms Sudoku
- Skyscraper Sudoku
- Serbian Sums
- Japanese Sums
- Full Rank

## Indexing

- Row indexing
- Column indexing
- Box indexing
- Cyclic indexing
- Other indexing
- Friendly cells
- Activated Cells

## Shading / Pencil Puzzle based

- Tapa
- Yin-Yang
- Masyu
- Akari
- Shimaguni
- Fillomino
- Slitherlink
- Norinori
- Nurikabe
- Nurimisaki
- Aquarium
- Nansuke
- Spiral Galaxy
- Star Battle
- Country Road
- Cave

## Other Constraints

- Clone Cells
- Odd/Even Sudoku
- Magic Square Sudoku
- Quadruples Sudoku
- Fortress Sudoku (Minimum/Maximum Sudoku)
- Snakes
- Loops
- Chaos Construction
- Schrödinger Cells
- Entropy
- Fog of War
- Knightmare
- Look-and-Say
- Liar / Wrogn
- Pencilmark / Sukaku
- Knapp daneben