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Grouped X-Cycles

From, the puzzle solver's site
Chaining strategies such as X-Cycles use links that connect cells on the board. It is possible to expand on the idea of a "cell" to something a bit larger, namely a "group" of cells. I prefer the word "node" - which in 90% of cases will be a single cell - but can be two or three cells in the same unit. You might wonder how we can use more than one cell and think of it as a node between two links, but there is some cool logic here.

We must go all the way back to Pointing Pairs and Pointing Triples. They attack cells along the row or column on which they are aligned. They also must be in the same box to be a coherent unit. Our “grouped” cells are just Pointing Pairs/Triples and we’re going to use them as part of a chain or Nice Loop.

Clearing the clutter on an example board, in Figure 1, we have a spread of candidate 4s. All the lettered cells are also candidate 4. There is a continuous Nice Loop starting with A. B-C is a weak link, and so is D-E.

The interesting part is the set of cells [X|Y|Z]. It does not matter which of X, Y, or Z (if any) is the solution; any of them will eliminate A and E. Likewise, if E is true, then all of XYZ are gone – and A is true. We can think of [X|Y|Z] as a single node for the purposes of our logic. This promotes the links from A and E to strong links, and the notation for this part of a loop is:

 E     X or Y or Z     A
Figure 1: Grouped X-Cycle
Figure 1: Grouped X-Cycle
The important characteristic is that the cells are all in the same box. One end of the chain (in this case, A) is pointed to by the node cells; the other (in this case, E) is usually within the same box as the node.

At the point it's also worth re-capping something about Continuous Alternating Inference Chains. Firstly, its doesn't matter which way you walk round the loop - clockwise or anti-clockwise, secondly, it doesn't matter which cell you start with and thirdly, each cell could be ON or OFF - as long as you alternate. Even with the convention of starting with the top left-most cell, there are four ways we could write down the chain: