X-Cycles is the first slightly more complex colouring strategy that we will discuss. There are three main differences between X-Cycles and other colouring strategies such as Singles's Chains, 3D Medusa and XY-Chains:
In previous methods, we explicitly used the ⇔ sign to denote that a link works both ways.
For example, suppose we have a row with exactly two candidates, e.g. A2 and A5 have a candidate 9, without any
other such candidates in the same row. If we colour (9, A2), this means that we can colour (9, A5)
. The opposite works as well. If we colour (9, A5) then we can colour (9, A2). We denote this relationship
with a solid line between candidates and we write ⇔. This way of linking is said to be strong. With a strong link
the colours are not all that important, as the implication works both ways, independently of which colour denotes the
ON candidate.
Let us now introduce the new concept of a weak link. For the rest of this article a green colour means ON
and a red colour means OFF. Suppose we have the same example as above, but there is an additional candidate in row A,
say (9, A3). Now, if we colour (9, A2), then we can still colour (9, A5), since we have assumed
that green means ON and so we can turn OFF any other candidates in the row.
So we have (9, A2) ⇒ (9, A5). The opposite, however, is not true. If we assume (9, A5)
is OFF, then it is not necessary for (9, A2) to be ON. The reason is because we have an
additional (9, A3) candidate in the row. Summarizing, a weak link is one, where an ON candidate turns
OFF another candidate, but the opposite is not true, due to an additional candidate in the same house.
Let us apply rule 1 for this board. On this screenshot the starting cell is marked in dark blue and all cells with eliminations,
are marked in light-blue. Let us carefully observe , for example, (7, H1). This candidate sees both
(7, H5) and (7, H7). From our observation above, either H5 or H7 will be a 7.
Hence, we can eliminate (7, H1).
This is true for all light-blue cells. Both (7, H2) and (7, H9) see (7, H5) and (7, H7).
Also, both (7, D6) and (7, D9) see (7, D7) and (7, D3).
To summarize rule 1: If we obtain a perfect cycle, we can eliminate all off-chain
candidates X which see two differently coloured candidates X from the chain.
This is a nice board with 6 different X-Cycles in the solution. The illustrated cycle starts with (3, B1) and continues clockwise. We see that the final cell is (3, C3), which we can weakly link to the starting (3, B1). We notice that the colours are not consistent, so the initial colour for B1 is wrong. In other words, we can eliminate the candidate 3 from B1.