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X-Cycles (Part 2)
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|Here is a rule that applies in the presence of two adjacent strong links:
If the adjacent links are links with strong inference (solid line), a candidate can be fixed in the cell at the discontinuity.
This rule allows us to know the solution of a certain cell absolutely, no matter how many other candidates there may be on that cell. Unlike the case of the first Nice Loop rule, we are not looking at a mass of eliminations outside the loop; instead, this rule tells us something about the loop itself. Let’s look at an example before examining the logical proof.
For discontinuous X-Cycles, the notation always starts with the discontinuity. In Figure 1, our Nice Loop on number 1 is:
Figure 1: Nice Loop on 1: Load Example or : From the Start
|The brown cell is the discontinuity based on two weak links that are next to each other in the loop. We can safely eliminate the 1 from this node. It might not seem much of an elimination considering how powerful the previous two rules are, but this type of Nice Loop configuration – two weak loops – is actually the most common.
The solver would return this message:
X-CYCLE on 1 (Discontinuous Alternating Nice Loop, length 6):
- Contradiction: When C3 is set to 1 the chain implies it cannot be 1 - it can be removed
Figure 2: Nice Loop on 1: Load Example or : From the Start
|Just a little further on from we have some more AICs including this 8 elimination
X-CYCLE on 8 (Discontinuous Alternating Nice Loop, length 6):
- Contradiction: When B7 is set to 8 the chain implies it cannot be 8 - it can be removed
Figure 3: Nice Loop on 8: Load Example or : From the Start
|Now, we can also create a Nice Loop as I have done with blue lines. Our aim is to show that the circled 6 on H9 is eliminated because there are two weak links forming a discontinuity. That is all correct and invokes Nice Rule 3. But there seem to be three strong links joined up. What happened to the alternating nature of the X-Cycle?
If a strong link can have weak inference, then let’s just change the link from C4 to A5 to imply such. Simple. We get our pattern. If 6 is on C4, then it is not on A5 (weak inference), or if it is on A5, then it is not on C4 (also weak inference – and all very logical).
I have coloured the Strong link with weak inference red in Figure 5.
Figure 5: Colouring Example and Nice Loop