# X-Chain

Part of a family: Chaining-Family

Hard

## Description

The X-Chain technique extends the Simple Coloring technique. It uses alternating strong and weak links to find candidates that can be eliminated.

Don't know what weak and strong links are? See Simple Coloring for a quick description.

For Simple Coloring we had a chain of strong links where we could say: "Either all candidates are correct, or all candidates are correct."

Every candidate that sees (weakly links to) both colors can be eliminated.

For X-Chains we allow every second link to be a weak link.

The additional 4 inside the center box makes it so that the 4s in the center box are only connected via weak link.

The same logical conclusion can still be drawn: a candidate that sees both colors ( and ) can be eliminated.

In this case, 4 can be eliminated.

## Why does it work?

The drawback of weak links is that they provide fewer information.

When there is a strong link we know that if one is false, the other one must be true.

With weak links we don't know that. When one candidate is false, there is always more than one candidates that could be true. (4 makes all the difference here.)

And yet a chain where strong and weak links alternate is still as useful as a chain consisting only of strong links.

But why?

It helps to view it from the elimination candidate's perspective.

Lets look at this example with a longer chain.

If 4 were correct...

... then the 4s would be false...

(because for any type of link we know that if one candidate is true, the other one must be false)

... then the next two 4s would be correct again...

(because for a strong link we know that if one candidate is false, the other one must be true)

... and the last two 4s would be false again, which must not be, because they are strongly linked to each other.

Therefore we know that our starting candidate that we assumed to be correct is in fact not correct, so we can eliminate it.

But again: why does our chain work with weak links?

It is due to how we start. We always start from our elimination candidate with a weak link, then strong, then weak, etc. (in both directions)

Since we start with weak links, and since we start by assuming the elimination candidate is correct, we will always arrive at a weak link with a candidate that is assumed to be correct ()."

And if that one node of the link is correct, then we are allowed to deduce that the other one is false (even for weak links).

You now see why it is important that the links alternate. Because if a weak link is at a wrong position, then the whole thing does not work.

So now we know that weak links cannot be anywhere in the chain. They are only allowed at very specific positions (every second position). So strong links cannot be exchanged for weak links.

Weak links, however, can always be replaced by strong links, because weak links tell us nothing that strong links do not also tell us.

So these X-Chains are just as valid as the examples above:

## Types of X-Chain techniques

Other websites divide X-Chain techniques into different types. All those types can, however, be reduced to the one type described on this page, which I will show for each type.

### Continuous Nice Loop

A Continuous Nice Loop is a loop where weak and strong links perfectly alternate. There will be no consecutive weak links, and there will only be consecutive strong links, if those strong links took the place of a weak link.

Every cell on the grid (9) that sees both a 9 and a 9 candidate, can be eliminated.

Why can it be reduced to our general X-Chain description?

Because for every 9 we always have two weak links like this. Now it looks like our previous examples.

When a loop is not continuous, and there are two consecutive strong links, then 9 is the solution.

Again, this is nothing else then our general type, since it can be seen as this (strong links are allowed to replace weak links)...

... leading to the Hidden Single.

## X-Chain Examples

/
the chain
the candidates we can eliminate