X-Cycles (Part 2)

So far, I been looking at X-Cycles which alternate perfectly all the way round. There are two interesting rules that lead to eliminations when we identify an imperfection in a loop which is called a discontinuity.

A discontinuity occurs when we find two strong links next to each other (that is, with no weak link between them) or two weak links next to each other (with no strong link dividing them). These rules work only if there is exactly one discontinuity, and such a loop will always have an odd number of nodes.

'Discontinuity' doesn't mean that the loop is broken or that it's not chain; it refers only to the imperfection that would otherwise make links alternate strong/weak/strong, and so on.

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:

X-CYCLE on 3 (Discontinuous Alternating Nice Loop, length 6):
-3[B4]+3[B9]-3[C7]+3[J7]-3[J4]+3[B4]
- Contradiction: When 3 is removed from B4 the chain implies it must be 3 
  - other candidates 4 can be removed

We have two strong links joined at B4; therefore, B4 is 3. One way to make sense of this logically is to trace round the alternative. If B4 was not a 3 B9 and J4 would have to be 3s. That would remove the candidate 3 from J7 and C7. But hang on - that would remove all 3s from column 7. A contradiction so the 3 must exist in B4.

Nice Loops Rule 3

Our third rule dictates what happens when two weak links form a discontinuity in a loop:
If the adjacent links are links with weak inference (broken line), a candidate can be eliminated from the cell at the discontinuity.

The orange cell B5 is the discontinuity based on two weak links that converge on this candidate to complete the loop. We can safely eliminate the 1 from this cell revealing 5 to be cell solution. 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 returns this message:

X-CYCLE on 1 (Discontinuous Alternating Nice Loop, length 6):
+1[B5]-1[B8]+1[E8] -1[F9]+1[F5]-1[B5]
- Contradiction: When B5 is set to 1 the chain implies it cannot be 1 - it can be removed

This puzzle also contains another Rule 3 elimination at this point. Can you spot it before 'taking step'?

In this puzzle we have a cluster of X-Cycles on 3 in the same area. The solver first gives us

X-CYCLE on 3 (Discontinuous Alternating Nice Loop, length 8):
+3[B4]-3[B5]+3[E5]-3[E9] +3[F7]-3[J7]+3[J4]-3[B4]
- Contradiction: When B4 is set to 3 the chain implies it cannot be 3 - it can be removed

But in the very next step it uses an almost identical loop to remove 3 from C4. Is there a useful connection between the two steps? Yes! The solver does not combine the eliminations but we can. We can consider B4 and C4 to be a Grouped Cell that share the same fate. We can do this because the weak link coming up from J4 spies them both. Likewise the +3[B5] we started the loop with also pins both B4 and C4.

If you are new to X-Cycles I wouldn't worry too much about this configuration but as Rule 3 is very common it can be profitable to consider the orientation of candidates in links to catch extras. However we will make use of Grouped Cells as link components in a big way when we start playing with Grouped X-Cycles

X-Cycles introduced the idea of Weak and Strong links but I want to make a more precise definition of terms since there are subtleties which will be useful in other chaining strategies. The rough and ready distinction between Strong and Weak links is to do with how many candidates are in a unit – namely, Strong links are formed when only two are present, while three or more imply a Weak link.

From a strong link we can infer that
   if not A, then B

From a weak link, we can infer only that
   if A then not B, C, D according to how many candidates there are in a unit

This implies that:

• Strong links are "links with strong inference"; and
  
• Weak links are "links with weak inference".
  

However, the following is also true that for a strong link:
 if A, then not B

So, some Strong links can be reversed to give us a "link with weak inference" - if the occasion calls for it. It is perfectly logical to assert on a unit with two candidates of X both:

• If Not A then B (!A =>B)
  
• If A then Not B (A => !B)
  

In Figure 5 we have an array of 6 candidates on a board. A number of strategies can show that the 6 on H9 can be eliminated. I have coloured some cells using Simple Colouring Rule 2 which link up some pairs on the board - either all of the yellow cells will be 6 or all of the cyan cells will be 6. Since H9 can see C9 (yellow) and H5 (cyan) it cannot be a 6 since it can see cells with both colours.

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.

X-Cycle Exemplars

These puzzles require the X-Cycle strategy at some point but are otherwise mostly easy except the last few. They make good practice puzzles.

• Exemplar 1, Rule 3 (score 5.8)
• Exemplar 2, Rule 2 (score 6.3)
• Exemplar 3, Rule 3 (score 6.4)
• Exemplar 4, Rule 3 (score 6.4)
• Exemplar 5, Rule 3 (score 6.4)
• Exemplar 6, Rule 3 (score 6.4)
• Exemplar 7, Rule 2 (score 6.5)
• Exemplar 8, Rule 2 (score 6.6)
• Exemplar 9, Rule 3 (score 6.8)
• Exemplar 10, Rule 2 (score 6.9)
• Exemplar 11, Rule 1 (score 7.4)