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Forcing Chains |
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A forcing chain is a series of alternating weak and strong connections that begins and ends on the same candidate, the first and last connection both being weak. In a forcing chain this candidate can be eliminated.
Consider the example on this page. The starting candidate and candidate to be eliminated is the 6 highlighted in blue. It is weakly connected to the 2 in the same cell. The 2 is strongly connected to the 2 up above, which is weakly connected to the 5 in the same cell, which is strongly connected to the pair of 5's in the same box, which are weakly connected to the 5 up above, which is strongly connected to the 5 to the left, which is weakly connected to the 7 in the same cell, which is strongly connected to the 7 to the right, which is weakly connected to the 6 in the same cell, which is stronly connected to the 6 at the bottom, which is weakly connected to the original 6.
Note that pairs of candidates can act as a single node in the chain if they both share a unit with both nodes on either end. In this example that is exemplified by the 5's in the middle box that together were strongly connected to the other 5 in the box, and weakly connected to the 5 above.
Note also that we always alternated between weak and strong connections.
Let's follow the simple logic of why this chain eliminates the 6 in blue. Consider the second node of the chain - the 2 that is in the same cell as the 6. If this candidate is true, then the 6 in blue must be false. However, if this candidate is false, then the 2 up above must be true, the 5 in that cell must be false, one of the 5's in that box must be true, the 5 above those must be false, the 5 to the left must be true, the 7 in that cell must be false, the 7 to the right must be true, the 6 in that cell must be false, the 6 down below must be true, and the 6 in blue must be false. Either way, the 6 in blue gets eliminated. Another way to look at it is if the 6 is false, then it's false. If it's true, then it's false (which happens to be a contradiction and can't be the case). Either way, it must be false.
To identify forcing chains it can help to circle and draw lines between all candidates that are strongly connected. Then try to link different strong connections with weak connections to identify chains. If any candidate shares a unit with both endpoints of a chain it can be eliminated.
Sudoku Snake gives Forcing Chains a skill value of 2800.
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