Inference

In Sudoku, inference is the interaction between linked candidates, in particular when they are used in chains or loops.

There is some discussion about the difference between inferences and implications.

There are 2 types of inference. Strong and weak. There are also 2 types of links. Strong links and weak links. The type of link determines what type of inference you can use.

• Weak links allow you to use weak inference only.
• Strong links allow you to use strong inference or weak inference.

Strong inference

Strong inference is caused by strong links, which can be found in bivalue cells or bilocal units. Strong inference can also stem from patterns with the term almost in their name. A well known example is the Almost Locked Set.

For two candidates, named A and B, the following strong inference deductions can be made:

• If A is false, B is true.
• If B is false, A is true.

Strong inference is represented in most notation systems by an equal sign: =.

Weak inference

Weak inference is caused by any type of link, strong or weak. Two candidates in a single cell or two candidate representing the same digit for two peers all have weak inference that we can use.

For two candidates, named A and B, the following weak inference deductions can be made:

• If A is true, B is false.
• If B is true, A is false.

Weak inference is represented in most notation systems by a dash sign: -.

Alternating inference

To create chains or loops, the inference between subsequent pairs of candidates must alternate between strong and weak. Consider the following links:

• A weak link with B
• B strong link with C
• C strong link with D
• D strong link with E
• E weak link with F

You may be inclined to write the following chain:

A - B = C = D = E - F

However, this chain uses the wrong inference between candidates C and D. This might be illustrated by showing the list of implications:

• If A is true, B is false.
• If B is false, C is true.
• If C is false, D is true.
• If D is false, E is true.
• If E is true, F is false.

There is no proper connection between the 3rd line and those preceding and succeeding it. Although C and D have a strong link, we must use weak inference to write a correct chain:

A - B = C - D = E - F

Alternating inference guarantees that the logic is sound from the first to the last node in the chain.

The topic in this article is a still a subject of debate. Parts of the text may not express everybody's opinion. Use the associated Talk page if you do not agree with the opinion of the writer, rather than continuously editing the main article.