Sudoku Assistant - The Sudoku Trainer and Solver

Forcing Chains

Forcing Chains is a technique in which we draw a connection (the 'chain') between cells that have just two candidate values. There are a small number of variations on this technique, but the implementation we use is as follows:

  2   9 5 3 7 6  
  6 3 8 2 7 5 9  
7 9 5 4 1 6   8  
  3   27 67 9 1 4 56
  4   3 8 5 9 2 7
  7 9 12 46 14 8 35  
  1   57 9 2 6 35 8
  5   6   8 4 1  
  8           7  

We start the chain by picking a cell with two candidate values; in this example [4,4]. The two possible candidates are {2,7}. If we pick 7, we look for another cell in the influence region of this cell, with two candidates where 7 is one of the candidates. In this case, the cell immediately to the right at [5,4] will do... if [4,4] took the value 7, then [5,4] could not be 7, hence it would have to be 6:
The whole chain: [4,4]:7 -> [5,4]:6 -> [9,4]:5 -> [8,6]:3 -> [8,7]:5 -> [4,7]:7 -> [4,4]:2

And it's that last step that seals the deal for this technique... if we set cell [4,4] with the value 7, a chain occurs which suggests the same cell would have to take the value of 2. Clearly, that is an impossible situation!

Conversely, if we start with [4,4] as 2:
The alternative chain: [4,4]:2 -> [4,6]:1 -> [6,6]:4 -> [5,6]:6 -> [5,4]:7 -> [4,4]:2

The alternative chain results in a consistent end-result, so in summary we can be sure that it can't be 7, and can be 2.

When considering the two chains (resulting from each possible candidate value from the starting cell) it is sometimes possible that you will see two chains that have a shared cell and share the same forced value on that cell. In these instances, you can of course set the cell with the consistent value.

 

If you are unsure of any of the terminology we use, you may find it helpful to refer to our Glossary.

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