Sudoku Definitions, Terminology and Glossary

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General definitions

Almost Locked Set
A set in a common large container with 1 degree of freedom
Boolean variable
Any statement limited to two values. Usually, {TRUE, FALSE}.
A potential number in a cell
A defined partition of the Sudoku puzzle. A cell, box, row, or column.
degrees of freedom
Consider a set of cells in a common large container:
  • Almost Naked set with (n-m) degrees of freedom
    • A set of n cells limited to m candidates
  • Almost Hidden set with (n-m) degrees of freedom
    • A set of n candidates limited to m cells
Number of native strong sets used to justify an elimination(s) (Sudoku only)
The exclusion of a candidate from a location
forbidding matrix
An n x n table that transparently illustrates an elimination(s)
large container
Box, Column, or Row.
adjective - Appearing within the current possibility matrix.
native strong set
Given the current possibility matrix, one of the following strong sets:
  1. One candidate in one large container
  2. One cell
native weak set
(given only a blank puzzle grid) One of the following weak sets:
  1. One candidate in one large container
  2. One cell
continuous nice loop
A wrap around forbidding chain
A set with zero degrees of freedom. (Thus a hidden or naked pair, triple, etc.)
possibility matrix
The puzzle grid decorated with the remaining possible candidates for each cell
Forbids by sharing a container with
strength in cell - also called naked strength
A reflection of the unwritten sudoku rule:
  • A solved sudoku cannot contain a cell empty of all candidates
strength in location - also called hidden strength
A reflection of the written sudoku rule:
  • A solved sudoku cannot contain a large house empty of any candidate
strong set
noun - A set of Boolean variables such that at least one is true
strong link
The logical operator OR (union). The endpoints of a strong link form a strong set
weak set
noun - A set of Boolean variables such that at most one is true
weak link
The endpoints of a weak link form a weak set
wrap around forbidding chain
A forbidding chain is called wrap around when the endpoints of the chain form a set that is both strong and weak.
Unique Possibility
A candidate that is a singleton in a container
short for Unique Possibility or Unique Possibilities


Binary strong link. (A == B) means (A OR B). Used in forbidding chains.
Binary weak link. (A -- B) means ((notA) OR (notB)). Used in forbidding chains.
because it is a singleton in (container x). Used to describe a Unique Possibility.
does not equal
shorthand for a1,b1
Used in forbidding chains to contain a Boolean variable that partitions more than one native strong set
Used within forbidding chains to help clarify the partioning of a native strong set (strong inference set) that has more than one possible distinct partition. Possible examples:
  • A cell with 3 or more possible canidates
  • A candidate with 3 or more possible locations within a house.

Technique definitions

Unique Possibilities
Finds singular strengths in any container
Locked Candidates
Looks at strength in location of one type of candidate in one large container to deduce strength of that candidate in location
Naked Pairs, Triples, Quads,...
Looks at strength of N candidates in N cells to deduce strength of those N candidates in N locations within a large container(s)
Hidden Pairs, Triples, Quads,...
Looks at strength of N candidates in N locations within a large container(s) to deduce strength of those N candidates in N cells
Xwings, Swordfish, Jellyfish, Squirmbags
Looks at strength of one type of candidate in N rows(location) to deduce strength of that candidate in N columns(location) OR vice versa
Unique Rectangles
Looks at potential symmetrical overlapping strength of candidates in location to deduce strength of any type
Looks at strength of one type of candidate in location to deduce strength of that candidate in location
Y wings
Looks at strength of 3 cells containing exactly 2 candidates each. Considers 3 types of candidates. Deduces strength of one of those candidates in location
Y wing style
Looks at 3 native strengths of any type to deduce strength(s) of any type
Forbidding Chains
Looks at any number of strengths of any type to deduce strength(s) of any type
Advanced Forbidding Chains
A forbidding chain that uses a technique as a Boolean variable
Almost Locked Sets
In most locations, considers two sets with 1 degree of freedom each. Uses a weak link between them (relative location) to deduce strength. More generally, a forbidding chain that uses one or more ntuples as a Boolean variable.
Alternating Inference Chain
A Forbidding Chain


  • Native weak links are independent of the possibility matrix.
  • Native strong links are dependent upon the possibility matrix.
  • The operators, == and --, are not dependent on order. Thus, (A==B) is equivalent to (B==A)
  • (A forbids B) is equivalent to (A -- B), and therefor also (B forbids A)
  • (A -- B == C) => (A => C)
  • Strong and weak are not mutually exclusive properties
  • A set that is both strong and weak contains exactly one truth
  • In most forbidding matrices, each row is a strong set
  • Other sudoku sites may needlessly restrict the definition of
    • strong
    • weak
    • coloring
    • forbidding chains
    • depth
    Please endure the definitions herein
  • Puzzle coordinate system used here can be found at the blog introduction page.

Links to recommended sites

Indicate which comments you would like to be able to see

Please feel free to suggest the addition of anything that you think I should include on this page. If you are here and have a question you thought this page would answer, but it does not - please let me know.

Steve, when I first saw the term 'Boolean variables', my eyes did indeed glaze over in memory of all the symbolic logic that I never learned. I'll print off these definitions and then go back to your piece on FCs. My gut says that when you have two or more possibilities, some of your techniques - More...
Hi Kate!
Mastering these technique certainly does allow one to make better educated guesses.

My goal, however, is to move beyond guessing into the realm of certainty. Nevertheless, if one wants to solve a puzzle most quickly, then guessing may at times be the most time efficient More...

I'm struggling with the definition of a strong link: 'A OR B', or expressed in words as 'at least one is true'.

At least one implies: at least one or possibly both. From what I understand about strong links both can never be true. It's either one or the other, so the More...
HI Sharkie!
Absolutely positively do not, ever, think of a strong link as exactly one is true. This is a common misinterpretation, spread all over the web.

It may be the case that Native Strong Links are always also weak, but.... we are not interested in only native strong links. In More...
Hi Sharkie again!

In the expression:

A == B -- C == D,

If A is true, even if B is false, C could still be false, and D be true. So, if we restrict A == B to meaning exactly one is true, we cannot conclude A == D.

On the other hand, if we do not restrict the More...
About the formula you use to calculate the difficulty of a sudoku. Are the details secret?
Is it difficult to describe?
01/May/07 6:34 AM
Hi Jyrki!


I assign a difficulty rating to each step. The proof rating is then the sum of each step difficulty.

Any step that requires a deduction to make an elimination is counted.

Each step is evaluated on exactly one parameter: The number of strong More...
01/May/07 8:39 AM
Steve, Thanks for the explanation. I will scan your blog for examples to make this clearer to me,
but a quick question checking my understanding.
Given that in a forbidding chain every other linkage is strong, does this mean that in such a step D = one half of the number of links in the More...
01/May/07 4:39 PM
Hi Jyrki!

Sorry it took so long to reply!
First: my rating system is not employed by any other being that I know of.
Secondly: You are quite correct that it utterly fails to differentiate between puzzles that are solved merely by UPS. This is a fault, no doubt.
Finally: The rating More...
13/Jun/07 3:58 PM
I am learning. Your blog does a very job in explaining difficult/confusing ideas.
31/Jul/07 2:07 AM
Hi Steve,
I am familiar with Kraken fish patterns, but when I see terms like Kraken Loop, Kraken Column, and Kraken Cell I am totally lost. What I need is a precise definition of these terms. My attempts to google these terms simply lead me to examples. I can analyze examples of Kraken loops More...
14/Jul/10 11:08 PM
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