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 known as "double-implication chains".)

For example, in the following Sudoku puzzle:

Sudoku Forcing chain example

Consider r2c1. This has the two candidates, 1 and 2. We will consider the implications of each of these candidates in turn.

if r2c1 = 2, then r1c2 = 7

if r2c1 = 1, then r5c1 = 2, and so r6c2 = 1, and so r6c8 = 3, and so r1c8 = 2, and so r1c2 = 7

So whichever of the two possible values are placed into r2c1, we've deduced that r1c2 must hold a 7. In other words, whichever chain of cells we follow, a certain cell is forced to have a specific value.

Note: unless the Sudoku puzzle has multiple solutions, one of the considered candidates must be incorrect. This means it may 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 Sudoku puzzles that can be solved using forcing chains (or XY-wings or XY-chains): (What are .sdk files?)