Single’s chains is the first and simplest in a family of powerful chaining strategies such as
X-Cycles, XY-Chains, 3D Medusa, Alternating Inference Chains,
etc. They all follow a similar idea
- we “assume” that a given candidate in a cell will either be the value of that cell or will not be.
From this assumption we make deductions to other linked candidates and sometimes we may reach
a specific setting which allows to eliminate from or place candidates in specific cells.

To simplify the notation and make spotting and understanding such strategies easier,
we give the first candidate a colour - either **ON** or **OFF**, i.e. we either think of the
candidate as the true value for the cell (**ON**) or not (**OFF**).
By following simple rules we can then turn other linked candidates **ON** or **OFF**.
Such links will be represented by a line.

Single’s chains is the simplest such method since we only consider a single fixed candidate and we make
“deductions” in a pretty straightforward way.

It is important to know that when we create the chains, initially, we do not know which colour is **ON** and which colour is **OFF**.
We could have used other two random colours. When we reach a specific setting of candidates and colours, the correct colours can
be deduced or they may not matter at all. What matters is that if one candidate is, say, red it will imply that the next linked is, say, green.
The green candidate will imply that the next is red and so on.

Let us first see how chains are constructed by considering the following board and the candidate **9**.

Load in Solver

For other chaining strategies implications do not necessarily work both ways, which will be important at a later stage. In Single’s chains implications always work both ways so let us just ignore the above comment for now and continue with the example. Since there are only two candidates

Now, that we have learned how to create chains, let us consider the rules which will help us eliminate or place candidates. Note, that we number the rules 2 and 4, since they will correspond to the same rules in 3D Medusa.

Once we have created a chain, if we notice that two candidates share a house and have the same colour,
then this colour must be the **OFF** colour. The logic here is relatively simple: if the colour was **ON** then
a house would have two values placed which are the same. Let us look at an example illustrating this rule.

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To see the logic behind the rule suppose, once again, that we flip the colours, given in the picture, i.e. we start with a

If we cannot use rule 2 we can still make eliminations from outside of the chain. Rule 4 states that if a
candidate not in the chain, say **X**, is seen from two candidates which have different colours,
then we can eliminate **X**. As this rule sounds slightly more complicated, let us directly see it in an example.

Load in Solver

In this article we have described the simplest chaining strategy. It introduces the very useful concepts of chains, links, colourings and the corresponding in-chain (rule 2) and off-chain (rule 4) eliminations. Other methods use and extend these concepts, where the rules for creating chains and the rules for eliminations differ slightly. We will explore these in further articles.