Chains

A Chain, as everyone knows, is a series of links. In a Sudoku puzzle the links are the inferences that can be made from the consequences of using a value in a cell. These links or inferences are defined as strong links or strong inferences and weak links or weak inferences.

A chain can be built using inferences that are all based on the same value. An example of a chain created using just one value is an X Chain. Chains can also be built using links on different values. Examples of these would be XY Chains or Nice Loops.

A link can be made between two cells in a unit using the inference gained from a value contained in those two cells or a link can be made within a cell based on the inferences between the values contained in that cell.

Candidate reductions can be made when chains are found that form loops and have certain combinations of link inferences.

Chains can also lead to candidate reduction when two or more non looping chains can be created that all begin and end in the same cell or unit.

There is a way of writing the information required to describe a chain. In the Sudoku world it is known as Standard Notation.

The chains that are useful in the Sudoku would are Alternating Inference Chains or AICs. These are chains where the type of inference used on each link alternate, strong, weak, strong, weak etc. When checking for AICs the inference between candidates within a cell has to be considered, even though it is often not displayed on sites like this or included in the Standard Notation.

There is a strong link between any two candidates in a bi value cell. If the first candidate is not placed in the cell then the second candidate has to be.

There is a weak link between any two candidates in a cell, no matter how many candidates are in that cell. If any candidate is placed in the cell then none of the remaining candidates can be.

If a chain starts and finishes on the same cell it becomes a looping chain. A looping chain can either be continuous or dis-continuous.

A continuous looping chain maintains the alternating inference requirement and by definition must always have an even number of links. When counting links you have to include the links within cells.

A dis-continuous looping chain has one, and only one, instance where the alternating inference requirement is not maintained.

Both continuous and dis-continuous chains can lead to candidate placement or reductions.
The example in Figure 1 shows an X Chain where all links are based on inferences using the value 1. Strong inferences are shown as double lines and weak inferences are shown as single lines.

It is easy to see the alternating inference here because the value on the links never changes so there are no inferences within cells used to build this chain.

This chain is a dis-continuous loop. The two links to the yellow cell are both weak an on the same number.

The standard notation for this loop is:

[R4C3] -1- [R4C6] =1= [R6C5] -1- [R1C5] =1= [R1C3] -1- [R4C3]The example in Figure 2 shows a Nice Loop with links based on inferences using different values.

Look through this chain for the alternating inference. When the links to and from a cell are both strong, there is a weak link within the cell between the value of the first link and the value of the second link. When the links to and from a cell are both weak there is a strong link within the cell between the only two candidates in that cell.

This is a continuous loop. In all cases the inference from one link to the next alternates. Remember to consider the links between candidates within a cell even though they are not shown in the diagram or the standard notation below.

The standard notation for this chain is:

= [R1C3] -2- [R1C5] -1- [R3C6] =1= [R4C6] =3= [R4C1] =9= [R4C3] =4= [R3C3] =2= [R1C3] -