Impossible

Alternating Inference Chains (AIC for short) is a very powerful method. Many of the previously discussed methods can
be represented with such chains: Naked Pairs, Intersection Removal, X-Wing, Y-Wing, Swordfish, Jellyfish,
X-Cycles, among others.
AIC is a direct extension of X-Cycles, where instead of creating chains with a single candidate, we can link different candidates
within the same cell. AIC incorporates the previously mentioned **strong** and **weak** links and the rules are identical to
X-Cycles. If you have any troubles understanding these concepts, please refer back to our discussion on X-Cycles.

Let us first list all possible ways to link candidates. The first two rules are just a repetition of the X-Cycles linking rules.

- We can link two candidates
**X**with a**strong**link if they appear only twice in a given house. - We can link an
**X****ON**candidate and another**X**candidate with a**weak**link, if there are more than these two candidates in a house. - We can link a candidate
**X**and a candidate**Y**with a**strong**link, if they are the only two candidates in a given cell. - We can link an
**X****ON**candidate and another**Y**candidate with a**weak**link, if both candidates are in the same cell and there are other candidates in that cell.

Let us consider an example of a cycle. In this case we have a perfect loop and will be used as our main example for rule 1, but let us focus first on each link of the chain. Please note, that when we weakly link candidates in the same cell, we will use a capsule with a dashed border.

The chain begins with

Let us now look at the first rule. The elimination logic is the same as in X-Cycles. We could have started with
with the opposite initial colour and complete the cycle (anticlockwise). Hence, either the green candidates are **ON**
(the case on the image), or they are **OFF** (starting with opposite colour for the first cell). We do not know which
colour represents **ON**, but what we do know is that any off-chain candidate **X**, which sees two differently coloured
candidates **X** has to be eliminated. Again, if you are not convinced, try to place any of the **3**s marked in orange and check
what happens to the candidates in the chain.

In this specific case, **(3, D2)** sees both **(3, D3)** and **(3, D9)**. Hence, it can be eliminated.
The same holds for **(3, D5)** which sees the same two candidates from the chain.

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Because of the additional in-cell chaining strategy, there is another type of elimination which we can use as rule 1. Looking at **G3**,
there are two differently coloured candidates. Either **9** will be placed or **2** will be placed in that cell. In both cases
**6** cannot be the correct value of **G3** and we can eliminate it.

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Let us summarize rule 1, where we will denote **"Rule 1A"** to be off-chain elimination, and **"Rule 1B"** to be the
in-cell elimination:

**Rule 1A:** In a perfect loop, we can eliminate any candidate **X**, which is not in the chain, and which sees two differently coloured
candidates **X** from the chain.

**Rule 1B:** In a perfect loop, if there is a cell with two weakly linked candidates **X** and **Y**, we can eliminate any
other candidates in that cell, which are not a part of the chain.

Rule 2 is very similar to X-Cycles rule 2. We start with an initial colour for a candidate and the cycle implies the opposite colour.
This board illustrates the second rule. Let us write out the chain:

**(3, C6)** **⇔** **(3, C9)** **⇔** **(6, C9)** **⇔**
**(6, B9)** **⇒** **(6, B4)** **⇔** **(7, B4)** **⇒**
**(7, G4)** **⇔** **(7, G6)** **⇒** **(3, G6)** **⇔**
**(3, C6)**.

We notice that the initial colour for **(3, C6)** is **OFF** and the chain implies that
the candidate should in fact be **ON**. Imagine this as a series of **"if-then"** statements. If **(3, C6)** is **OFF**,
then **(3, C9)** is **OFF** and so on. Eventually, when we complete the cycle, we have implied the opposite colour.
In other words, the initial colour for **(3, C6)** is wrong and so **(3, C6)** is **ON** and **3** is the
correct value for **C6**

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The second example is almost the same, but we start with an **ON** candidate and the chain implies that it should be
**OFF**. In this case we start with **(3, B4)** and we continue anticlockwise to **(4, B4)** with a
**weak** link. Completing the cycle yields a contradiction, or in other words, **(3, B4)** should be **OFF** and we should
remove the candidate from the cell.

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These examples are handpicked to include nicely looking short chains, but it is very common for chains to be quite long and complex. If you are applying this method in practice on a piece of paper it is a good strategy to always aim to complete the cycle as soon as possible. If this cannot be the case, keep a careful track of which candidates you have used, so that if needed you can go back. Here is an example of a board with 19 Alternating Inference Chains. One of them is quite long and is illustrated here. You will notice, that it takes longer for our solver to show you the board, as AIC is quite resource-intensive method.

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