Forcing Chains

Forcing chains is a technique that allows you to deduce with certainty the content of a cell from considering the implications resulting from the placement of each of another cell's candidates. (This technique is also know as "double-implication chains".)

For example, in the following puzzle:

8 {2,7} 6
{1,2} 3  
5 9  
4 9 5
8 7 6
     
{3,7}   1
5 9  
8 6  
4 5 3
{1,2} 6 9
7   8
1 6 8
  4  
5   9
2 7 9
{1,3} 5 8
4   6
3 4  
6 8  
9   5
6    
9 5 3
  8 4
9 8 5
  4  
6   3

 

(The numbers is curly brackets { } are the candidates for the cell.)

Consider r1c2. This has the two candidates, 2 and 7. We will consider the implications of each of these candidates in turn.

if r1c2 = 2, then r2c1 = 1, and r5c1 = 2

if r1c2 = 7, then r1c7 = 3, and r5c7 = 1, and r5c1 = 2

So whichever of the two possible values are placed into (1, 2), we've deduced that (5, 1) must hold a 2. In other words, whichever chain of cells we follow, a certain cell is forced to have a specific value.

Note: unless the puzzle has multiple solutions, one of the considered candidates must be incorrect. This means it must eventually lead to either a contradiction or a dead end. If, when considering a single candidate, you reach a dead end, or find two chains that lead to different conclusions, you can eliminate that candidate from the starting cell. This is verging onto trial-and-error, and SadMan Software Sudoku doesn't do this as part of the forcing chain strategy. However, it can be useful when solving manually.

Here are some puzzles that can be solved using forcing chains:

forcingchain1.sdk forcingchain2.sdk forcingchain3.sdk
forcingchain4.sdk forcingchain5.sdk

 

Last Update
2008-05-02