Glossary of Sudoku

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A Sudoku puzzle (image hyperlinked to solution)

This is a glossary of Sudoku terms and jargon.


List organization and conventions[edit]

This list provides a brief glossary of Sudoku terminology. Items are listed thematically, and usually only once, with a brief description and possibly a link to a detailed description. Links to example usage are provided as in-line numbered references (like [1]). Here the default usage of Sudoku refers to the prominent 9×9 format, as illustrated.

Grid layout and puzzle terms[edit]

A Sudoku grid has 9 rows, columns and boxes each having 9 cells. The full grid has 81 cells. Cells are commonly called squares, but in technical descriptions the term square is avoided since the boxes and grid are also squares. Boxes are also known as blocks or zones.[1] Three vertically stacked blocks make a stack. Three horizontally connected blocks make a band. A chute is either a band or a stack. A grid has three bands, three stacks and six chutes.

The use of the boxes to partition the grid can be generalized to other equal-sized partition shapes, in which case the sub-areas are known as regions, zones, subgrids, or nonets. See Variants below. In some cases the regions are only equal sized, not equal shaped.

Rows, columns and regions are collectively referred to as units or scopes, of which the grid has 27. The One Rule can then be compactly stated as: "Each digit appears once in each unit".

Size refers to the size of a puzzle or grid. Often a composite row × column designation is used, e.g. size 9×9. In technical discussions size may mean the number of cells, e.g. 81. Since the number of cells in a region must be the side dimension of the square grid, e.g. nine cells per block for a 9×9 grid, it is convenient to just use the region size, e.g. 9.

Puzzle terms[edit]

A puzzle is a partially completed grid. The initially defined values are known as givens or clues. A proper puzzle has a single (unique) solution. A proper puzzle that can be solved without trial and error (guessing) is known as a satisfactory puzzle. An irreducible puzzle (a.k.a. minimum puzzle) is a proper puzzle from which no givens can be removed leaving it a proper puzzle (with a single solution). It is possible to construct minimum puzzles with different numbers of givens. The minimum number of givens refers to the minimum over all proper puzzles and identifies a subset of minimum puzzles. See Mathematics of Sudoku – Minimum number of givens for values and details.

Sudoku variants[edit]

The classic 9×9 Sudoku format can be generalized to an

N×N row-column grid partitioned into N regions, where each of the N rows, columns and regions have N cells and each of the N digits occur once in each row, column or region.

This accommodates variants by region size and shape, e.g. 6-cell rectangular regions (The N×N Sudoku grid is always square). For prime N, polyomino-shaped regions can be used. The requirement to use equal sized regions, or have the regions cover the grid entirely can also be relaxed.

Other variation types include additional value placement constraints, alternate cell symbols (e.g. letters), alternate mechanism for expressing the clues, and composition with overlapping grids. This page provides a simple list of variants. See Sudoku – Variants for details and additional variants.

For rectangular regions the row-column dimensions of the region may be used to describe the grid as whole, e.g. 3×2, since each of the grid side dimensions must be the product of row×column, e.g. for a 3×2 rectangular region, the grid must be 6×6. For rectangles of size N×1 or 1×N, the region is a row or column, and Sudoku becomes a Latin square.

Sudoku types and classes[edit]

Sub Doku
[2] Grids smaller than 9×9. Sometimes referred to as "Children's Sudoku" (especially the 4×4 variant) as the reduced number of possibilities makes them easier to solve.
Super Doku
[2] Grids larger than 9×9.
Prime Doku
[2] N×N grid where N is prime. Generally constructed with polyomino regions, e.g. Go Doku and pentominos.
Maximum Su Doku
[2] The class of puzzles which have the maximum number of independent clues needed to allow a complete and unique solution.
Minimum Su Doku
[2] The class of puzzles which have the minimum number of clues needed to allow a complete and unique solution.
Proper puzzle
[2] A puzzle that has a unique solution.
Satisfactory puzzle
[2] A puzzle that does not require trial and error. Note: the level of trial and error is usually not explicitly defined, see trial and error below.
Jigsaw Sudoku
regular 9×9 Sudoku that row and column rules apply, but instead of a 3×3 grid they are nine Jigsaw shapes.

Variants by size[edit]

Polyomino
A shape composed of equal sized, side-adjacent squares. Often used for Sudoku region variants. Polyominos are named by size: (5) pentomino, (6) hexomino, (7) heptomino, (8) octomino, and (9) nonomino.
Du-sum-oh
[3] 5×5, 6×6, 7×7, 8×8 or 9×9 grid with irregular, polyomino, shaped regions and minimal number of clues.

Du-Sum-Oh puzzles are also known as Latin Squares Puzzles (invented by Mark Thompson), Squiggly Sudoku, Jigsaw Sudoku, Irregular Sudoku, or Geometric Sudoku. These puzzles typically have anywhere from 5 to 9 rows. The number of rows is always equal to the number of columns. The regions are polyominos made of the same number of squares that are in any one row of the puzzle. The irregularity of the regions compensates for the relatively small number of givens.

4×4[edit]

Shi Doku
[2] Four 2×2 regions. Shi is Japanese for 4.

5×5[edit]

Go Doku
[2] 5×5 grid with pentomino regions. Go is Japanese for 5.
Logi-5
5×5 grid with pentomino regions

6×6[edit]

These use six 2×3 rectangular regions:

Roku Doku[2]
(unnamed)
featured at the World Puzzle Championship
Sudoku X – with unique main diagonals

7×7[edit]

(unnamed)
7×7 grid with six heptomino regions and a disjoint region, featured at the World Puzzle Championship.

8×8[edit]

Super Sudoku X – Four 4×2 + four 2×4 rectangular blocks.

9×9[edit]

Sudoku
Classic 9×9 grid with nine 3×3 regions.
Jigsaw Sudoku
9×9 grid with nonomino regions.
Du-sum-oh
[3] 5×5, 6×6, 7×7, 8×8 or 9×9 grid with irregular, polyomino, shaped regions and minimal number of clues.

Only "One Rule" variant puzzles with simple givens are listed in this section. For variants with other clue mechanisms, see Constraint and clue variants.

12×12[edit]

Maxi
Twelve 3×4 rectangular blocks.

16×16[edit]

Number Place Challenger
Sixteen 4×4 regions.

25×25[edit]

A 25×25 Giant Sudoku puzzle (link to solution)
Sudoku the Giant
Twenty-five 5×5 regions.

100×100[edit]

Sudoku-zilla
[4] 100 10×10 regions.

Constraint and clue variants[edit]

Puzzles with additional constraints on the placement of values including various forms of expressing the constraints (e.g. < > relations, sums, linked cells, etc.).

Main diagonals unique
the cell values along both main diagonals must be unique, see Sudoku X.
Relative digit location
digits use the same relative location within selected regions. The matching cells or regions are often color-coded.

Mathematics of Sudoku has identified numerous additional constraints as analytic possibilities.

Samunamupure (clue sums)
Regions of various shapes and sizes. The usual constraints of no repeated value in any row, column or region apply. The clues are given as sums of values within regions (e.g. a 4-cell region with sum 10 must consist of values 1,2,3,4 in some order).

Terms related to solving[edit]

The meanings of most of these terms can be extended to region shapes other than blocks. To simplify reading, definitions are given only in terms of blocks or boxes.

Scanning
The process of working through a puzzle to look for or eliminate values.
Cross hatching
Process of elimination that checks rows and columns intersecting a block for a given value to limit the possible locations in the block.
Counting
Process of stepping through the values for a row, column or block to see where they can or cannot be used.
Box line reduction strategy
A form of intersection removal in which candidates which must belong to a line can be ruled out as candidates in a block (or box) that intersects the line in question.
Candidate
Potential value for a cell.
Contingency
A condition limiting the location of a value.
Chain
A sequence of contingencies connected by alternative values.
Higher circuits
Related locations outside the immediate row, column and grid. The locations are related by value contingencies.
Independent clues
A set of clues that cannot be deduced from each other. Often depends on the order of choosing the clues for a given grid.
Intersection removal
When any one number occurs twice or three times in just one unit (or scope) then we can remove that number from the intersection of another unit. For example, if a certain number must occur on a certain line, then occurrences of that number found in a block that intersects this line can be ruled out as candidates. Sometimes called Pointing (or matched) Pairs (or twins)/Triples (triplets) as they point out a candidate that can be removed.
Trial and error
The process of guessing successive candidate values in conjunction with deductive elimination. A.k.a.: what-if, bifurcation, garden of forking paths, depth first search, exhaustive search, back-tracking search, Ariadne's thread. Note: there is no clear boundary between trial-and-error and the use of pattern recognition strategies to eliminate values (higher circuits), the latter being a condensed form of analysis based on elimination by contradiction, i.e. the same as what-if.[citation needed]
Nishio
What-if method of elimination, where the use of a candidate that would make its other (necessary) placements impossible is eliminated.
The One Rule
Fill in all (blank) cells so that each row, column and box contains the values 1-9. Same as: fill in the grid so that each row, column and box contains the values 1-9 exactly once, without changing the clues.
Single[5] or singleton [1] or lone number [2]
The only candidate in a cell.
Hidden single
[5] A candidate that appears with others, but only once in a given row, column or box.
Locked candidate
[5] A candidate limited to a row or column within a block.
Naked pair
[5] Two cells in a row, column or block, which together contain only the same two candidates. These candidates can be excluded from other cells in the same row, column or block.
Hidden pair
[5] Two candidates that appear only in two cells in a row, column or block. Other candidates in those two cells can be eliminated.
Trio [3]
Three cells in a unit sharing three numbers exclusively. See "Triples and quads".
Triples and quads
The concepts applied to pairs can also be applied to triples and quads.
X-wing
[5] See N-fish (with N=2).
Swordfish
[5] See N-fish (with N=3).
N-fish [4]
Analogues of hidden pairs/triples/quads for multiple rows and columns. A pattern formed by all candidate cells for some digit in N rows (or columns), that spans only N columns (rows). All other candidates for that digit in those columns (rows) can then be excluded. Names for various N-fish:
  • 2-fish: X-wing
  • 3-fish: Swordfish
  • 4-fish: Jellyfish
  • 5-fish: Squirmbag – For 9×9 Sudoku, there's no in point naming higher-order (>4) fish, since every N-fish comes paired with a 9−N fish whose effect is the same (thus any 5-fish is paired with a jellyfish; any 6-fish with a swordfish; any 7-fish with an x-wing; any 8-fish with a hidden or naked single). Nevertheless, a 5-fish is occasionally called a squirmbag.
  • 6+ fish: 6-gronk, 7-gronk ... [5] – these patterns are only useful for Sudoku larger than 9×9.
Remote Pairs [6]
When a long string of naked pairs that leads around the grid exists, any cells that are in the intersection of the cells at the beginning and the end of the string may not be either of the numbers in the naked pairs, for example, 4 and 7.

Cell reference schemes[edit]

  • 1...81 or 0...80
  • Row & column
  • Box & cell

Math related terms[edit]

  • Latin square – Related puzzle with only row and column constraints.
  • Constraints – Rules or conditions. In Sudoku, the rule(s) requiring each digit appear once in each row, column and region.
  • Triplet – The set of 3 values in a row or column within a block.

See also[edit]

References[edit]

  1. '^ The Toronto Metros daily Sudoku
  2. ^ a b c d e f g h i j Mathematics of Sudoku, Names section
  3. ^ a b Dusumoh
  4. ^ Eisenhauer, William (2010). Sudoku-zilla. CreateSpace. p. 220. ISBN 978-1-4515-1049-2. 
  5. ^ a b c d e f g Sudoku hints

Notes[edit]