For easy reference, the name xy-chain is proposed without knowing whether the technique has already got a proper name.
xy-chain is a special case of forcing chains, and can be considered as another simplest application of such technique which involves more than 4 cells.
Similar to the xy-wing, the xy-chain makes use of the special property of cells with 2 candidates.
If three cells A, B, C, with A-B in the same unit and B-C in the same unit, have candidates zx, xy and yz respectively, then z can be removed from the candidates of all the cells (different from A, B, C) contained in the intersection of two units that contain A and C respectively. This is the definition of an xy-wing (refer Nick's thread on xy-wing).
We can now describe the xy-chain:
If n cells A, B, C,.............X, Y, Z, with A-B in the same unit, B-C in the same unit, C-D in the same unit,................. and Y-Z in the same unit, have candidates za, ab, bc,..............wx, xy, yz respectively, then z can be removed from the candidates of all the cells (different from A, B, C,........., Z) contained in the intersection of the two units that contain A and Z respectively.
The reason for that is similar to the xy-wing: Y contains either x or y. If Y=y then Z=z. If Y=x, then X=w,................, then C=b, then B=a, then A=z. Therefore either A or Z contains z. Therefore a cell located inside the intersection of two units containing A and Z cannot contain z.
Consider the following examples:
Example 1 is an xy-wing; z can be removed from r4c5.
Example 2 is an xy-chain;
r2c5=a => r8c5=z.
r2c5=b => r2c7=c => r6c7=d => r4c9=e => r4c3=z.
Therefore r8c3 not=z. Therefore, z can be removed from r8c3.
Example 3 is a very long xy-chain. Although z can be removed from r3c8, the process of searching for such long chain can be quite tedious. Long xy-chain should not be considered unless all other techniques are unsuccessful.
Let's see how a very hard puzzle can be solved by a simple xy-chain:
The xy-chain is r4c2-r5c2-r5c4-r5c8-r6c8, with candidates 76-62-23-35-57.
If r5c2=6, then r4c2=7.
If r5c2=2, then r5c4=3, then r5c8=5, then r6c8=7.
As either r4c2=7 or r6c8=7, r6c2 cannot be 7.
Therefore r6c2=2 and the rest is simple.
