All nodes in a grouped xy-chain, except for the node where the elimination(s) take(s) place, are strong nodes.
Grouped xy-chain is equivalent to all current techniques that use strong nodes as placeholders:
- xyz-wing - Discontinuous grouped xy-chain of length 3
Almost locked set xz rule - Discontinuous grouped xy-chain
SueDeCoq - Double continuous grouped xy-chains
In a grouped xy-chain, nice loop propagation always follows pure links with 'weak inference' (drawn as broken lines, each with a -ve label) connected by strong nodes (refer definitions of "strong node", "link", "strong link", "strong inference" and "weak inference" here), ie.
.........[node A]-a-[node B]-b-[node C]-c-[node D]-d-[node E]-e-[node F]-f-[node G].........
where the notation :
[node X] = [cell 1] or [cell 1|cell 2] or [cell 1|cell 2|cell 3] or [cell 1|cell 2|.......|cell N]
'-x-' is a link with weak inference of -ve label x
In a grouped xy-chain nice loop, if the links propagate in a cyclic manner (ie. no adjacent links are of same label), the loop is said to be 'continuous'.
With continuous grouped xy-chain, the labelled candidate of a link can be eliminated outside the loop but within the unit of the link as demonstrated below:
Nice loop notation:
example 1 : -[r2c2]-1-[r2c9]-5-[r9c9]-79-[r7c7|r7c8]-2-[r7c2]-4-[r2c2]-
example 2a, SueDeCoq-part 1 : -[r7c4]-2-[r1c4|r1c6|r3c4]-9-[r7c4]-
example 2b, SueDeCoq-part 2 : -[r1c6]-1-[r1c4|r3c4|r7c4]-6-[r1c6]-
A 'discontinuous' grouped xy-chain nice loop has exactly one pair of adjacent links carrying the same label; the discontinuity is located between these 2 links.
At the discontinuity between 2 adjacent links of same label, the labelled candidate can be eliminated from the node as demonstrated below:
Nice loop notation always starts from the discontinuity:
example 3, xyz-wing: [r9c6]-5-[r7c6|r8c4]-3-[r2c6]-5-[r9c6] => r9c6<>5
example 4, Almost locked set xz rule: [r4c3]-3-[r6c3]-1-[r5c1]-7-[r5c9|r4c7|r4c9]-3-[r4c3] => r4c3<>3
