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|In the X-Wing structure in Figure 1, as an example, we can consider B3 to be OFF. This forces the the 9 in B8 to be ON since it's the only remaining 9 in the row. If B8 is ON then that removes the 9 in H8. Again, the consequence is to turn ON the 9 in H3 - and that closes the loop by forcing the 9 back in B3.
In this X-Wing example three things are apparent. I went round clock-wise, but I could easily make the same logical chain going round from B3 to H3 to H8 to B8 and back to B3. Also, could decide that the start cell was ON and follow the loop round. And lastly, B3 is arbitrary, I could start on any of those four cells.
Figure 1: Nice Loop on 9
|A "Cycle", as the name implies, is a loop or joined-up chain of single digits with alternating strong and weak links, as the X-Wing in Figure 1 shows.
In Figure 2, we have a 2-2-2-formation Sword-Fish re-drawn to show the strong and weak links. The loop characterises the X-Cycle, and the strong/weak links alternate.
Figure 2: Nice Loop on 4
|Here is the notation for the 8-Cycle in Figure 3:
The yellow cells are units where other 8s can be eliminated, which in this case correspond to the third column and boxes 2 and 5 because thats where the weak links are located.
Figure 3: Nice Loop on 8
|Figure 4 is a real-life example of an X-Cycle based on 8. The cells with links are in red and green. We can immediately see that C2/C7 is a weak link across the row because of the 8 in C3. G2/H3 is also a weak across the box because of the third 8 in G3. The last weak link is in box 9, J7/H9. Any other 8s in these units can be removed, which makes it a powerful technique. We end up with a loop containing only strong links a result identical to a Colouring (Singles Chains) solution.
The output from the solver will contain the following information:
X-CYCLE (Alternating Inference Chain):
- Off-chain candidate 8 taken off C3 - weak link: C2 to C7
- Off-chain candidate 8 taken off C9 - weak link: C2 to C7
- Off-chain candidate 8 taken off G9 - weak link: J7 to H9
- Off-chain candidate 8 taken off G3 - weak link: H3 to G2
Figure 4: Nice Loop on 8: Load Example or : From the Start