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Dynamic Forcing Chains |
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A Dynamic Forcing Chain is much like a Forcing Chain, except that multiple different chain paths can contribute to turn a weak connection into a strong connection. Dynamic Forcing Chains are nearly synonymous with guessing, except that the chain must eventually return back to the starting candidate with a weak link (just like with Forcing Chains). This turns the seemingly Guess-and-Check method into a thorough elimination technique.
Consider the example on this page:
The 8 highlighted in blue is the starting point.
Path 1:
The 8 in blue is weakly connected to the 3 in the same cell, which is strongly connected to the 3 below, which is weakly connected to the 2 in the same cell, which is strongly connected to the pair of 2's in the same box, which are weakly connected to the 2 to the right, which is strongly connected to the 2 above, which is weakly connected to the 8 in the same cell. Thus, if the 8 in blue were true, this final 8 would be false.
Path 2:
Follow the same as Path 1. The 8 in blue is weakly connected to the 3, which is strongly connected to the 3 below, which is weakly connected to the 2, which is strongly connected to the pair of 2's, which are weakly connected to the 2 to the right, which is strongly connected to the 2 above. Now we change the direction of the path. This 2 is weakly connected to the 2 to the right, which is strongly connected to the 2 above, which is weakly connected to the 2 way off to the left (the one that shares a cell with the 5 and 8 in brown. Thus, if the 8 in blue were true, this final 2 would be false.
Path 3:
The 8 in blue is weakly connected to the 8 two cells to the right (the one that shares a cell with a brown 2 and brown 5). Now, because we've establish in Path 2 that the 2 in this cell gets eliminated, we can determine that this 8 is strongly connected to the 5 in the same cell. This 5 is weakly connected to the pair of 5's above, which are strongly connected to the 5 off to the right, which is weakly connected to the 8 in the same cell. Because we established in Path 1 that the 8 below (that shares a cell with the brown 2) has been eliminated, this 8 above is strongly connected to the 8 directly beneath. This 8 beneath is weakly connected to our original 8. Since the dynamic forcing chain has a weak connection back into the original 8, this 8 can be eliminated.
Dynamic Forcing Chains can easily be spotted by guessing the original 8 and seeing which eliminations come thereafter. In actuality, once you find any contradiction then you know that your initial guess was incorrect. However, the more purely logical technique is the dynamic forcing chain that weakly connects back on itself. Sudoku Snake recommends using the Bookmark tool before performing any guesses.
Consider the following logic behind this technique. The two candidates adjoining the original 8 in the dynamic forcing chain are the 3 in the same cell and the 8 two cells to the right. If either or both of these candidates were true, then the 8 in blue would be directly eliminated. The only other possibility, if both these candidates were false, then through the dynamic forcing chain we see that the 8 in blue is again eliminated. Thus, no matter which combination of the adjoining 3 and 8, the 8 in blue is eliminated.
Sudoku Snake gives Dynamic Forcing Chains a skill value of 8000.
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