We have already seen how chains can be built using Grouped Cells and Almost Locked Sets as more complex links, expanding the number of patterns available. In this article I detail how we can use the Unique Rectangle as another type of link. And it's relatively simple! At least for the basic Type 1 Unique Rectangle most often found.
This has been on my to-do list for a long time, but I have to credit David Hollenberg for giving me the inspiration and push to implement it in the solver, having given me some examples with Y-Wing chains.
Anywhere where chains can be used this type of link is valid. In the first example we have a Discontinuous Alternating Nice Loop that starts and ends on C5. By tracing round it can be shown that if 2 on C5 is removed the chain reaction puts the 2 right back, implying it must be the solution. 9 Can be removed from that cell.
But what is going on on F8 and D8? The chain jumps straight from -9 on F8 to +2 on D8. The reason is simple. The four shaded cells DF28 all contain 6/7 plus these other TWO candidates 2 and 9. We know from understanding unique rectangles (type 1) that we cannot allow all four of these cells to be reduced to 6 and 7 alone - that would give two solutions to the puzzles. So one of the extra candidates must exist (or maybe both!).
The chain coming into F8 takes off the 9 there. That forces D8 to be 2 (for the duration of the chain - we don't know yet). +2 on D8 allows us to continue the chain, in this case to D9.
AIC on 2 ((w.UR) Discontinuous Alternating Nice Loop, length 10): -2[C5]+9[C5]-9[E5]+9[F4]-9(UR[DF28])+2[D8]-2[D9]+2[B9]-2[B4]+2[C5] - Contradiction: When 2 is removed from C5 the chain implies it must be 2 - other candidates 9 can be removed
In this next example we have off-chain eliminations.
Removing the 6 from G6 endangers the solution by exposing the unique rectangle, so we can confidently turn ON the 9 in D6. That allows us to continue and close the loop.
(This is a much harder puzzle and might require some clicking to get to the interesting step)
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