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Weak and Strong Links
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In this diagram I am showing two short chains. In fact, only the top one using 3's is a chain since all the component candidates are conjugate pairs. The problem with the bottom chain is that there are three 4s in row F. The red line indicates the potential link that fails because the 4s on the row are not bi-location 4s.
But, there is another way of thinking about chains and it directly relates to the ON/OFF states that all the candidates must have.
In the diagram I show the same two chains. The start cell is essentially the same. In the upper 3s chain the start cell D2 is ON which causes the 3 in B2 to go off. Now, since there are two remaining 3s in row B the consequence of the OFF state in B2 ends. We don't know which of B4 or B7 to turn on.
But if we change the start state to OFF, as in J3 this causes F4 to be ON. When a candidate is turned on it turns OFF ALL the other candidates in all units it can see - including the 4 in F8. So there is a consequence and the chain can continue to H8.
Figure 2: Two Chains on 4
|A fuller example: Figure 3 marks out a classic X-Wing. X-Wings always contain four numbers arranged in a rectangle such that two opposite sides contain just two remaining candidates of a certain number, in this case 9. Such pairs are minimal 1-link chains, or simply bi-location pairs. If our X-Wing was a 4-link loop containing just conjugate pairs then it wouldn't be interesting - there would be no eliminations.
In Figure 3 the two pairs are at [B3,B8] and [H3,H8]. From a pair it is possible to draw two separate inferences, called weak and strong:
Figure 3: Nice Loop on 9
|To take the Nice Loop example from X-Cycles, we can draw links 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 take a look at the red link [C4,A5]. It is a Strong Link with Weak Inference. It is a Strong Link because there are only two 6s in the box but we are using it to imply that if A5 is a 6 then C4 is not and if C4 is a 6 then A5 is not.|
Figure 4: Strong Link with Weak Inference