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# Forcing chains

Earlier, we considered chains of equal pairs, and observed
that the values must alternate. Let us now consider chains
of the form *ab-bc-cd-...-xy-yz*, of pairs where the second
element of each pair (except the last) equals the first element
of the next pair. If the field corresponding to the first pair
has the second value *b*, then the next has the second value *c*,
etc., and the last has the second value *z*. If that is a contradiction
- for example because the chain was a cycle, the first and last pair
belong to the same square, and the last pair is *ba* -
then the field corresponding to the first pair must be *a*.
**Terminology** The forcing chains described above
are called 'simple bivalue xy-chains'.

For example, the cycle 13-31-13-38-83-31 starting and ending
in the yellow square, forces the digits in red:

Use of the cycle 83-31-15-58-83-38:

Use of the cycle 26-68-86-65-56-62:

A more general way of getting a contradiction, is with a position
that is both first and last element of the chain and may have more
than two possible values. A chain of the form
*Xa-ab-bc-...-za-aX* proves that the first element is not *a*.

A rather common version is the short chain *Xa-ab-bc-cd-dX*.
For example,

**Terminology** This last argument is known as 'XY-wing'.

## Forcing chains and three possibilities

Consider

The yellow square cannot be 1, hence must be 2.

This can be described as a set of forcing chains
*5-51-12 and *4-41-12 and *1-12 all starting in the green square.
Since the green square has value 1 or 4 or 5, the yellow square
is forced to be 2 in all cases.

Of course it doesn't play a role that the yellow square had only two
possible values. The argument just shows that it cannot be 1.

**Terminology** This argument is known as 'XYZ-wing'.

Here a somewhat similar situation:

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