By setting one cell to a given number, you may force another cell to be set to
another number. This in turn may force another cell to only have one possible value,
and so on. This creates a chain of events, one cell forcing another one to a given
value, and so on along the chain, hence the name Forcing Chains. It works by moving
around the puzzle between cells that only have two candidates.
The technique can start from any cell, although some variations start from a cell
that only has two candidates. This will be called cell A, and is the starting cell
for the whole chain. Picking one of the candidate numbers in cell A, think
about how it affects the cells in the same row, column and block. If any of these
affected cells can then have only one possible number (which we'll call cell B), then
that cell has been 'forced' to that value.
Considering that cell B now can only have one value, look and see which cells in
the same row, column and block can only have one possible number. If such a cell is
found (cell C), the we repeat the process from there (to find cells D, E, etc). By
checking each possible chain, you may find a chain that comes back round to the
starting cell, cell A. If the last cell in the chain forced cell A to be a different
number than the one it started at, then cell A can't contain the starting number. If
it did, the chain of consequences means that it has to contain a different number, in
order to make a valid solution the puzzle.
3 5
1 8
3 5
1 7 8
1 7 8
4
9
2
6
4
7
9
2 6
2 6
1 8
3 5
1 8
3 5
6
2
1 8
3
9
5
1 4 8
4 7
1 4 7 8
5 7
3
1
2 5 8
4 8
2 5
5 6 7
4 6 7
9
2 5 7
2 4 8
5 7 8
9
3
6
1 5 7
1 4
1 5 7
4 5
6
9
1 4 5
4 7
1 4 7
2
8
3
2 3
5
2 3
4
1 6
8
1 6
9
7
8
9
4
3 7
1 6
3 7
1 6
5
2
7
1
6
9
5
2
4 8
3
4 8
Example 1: Forcing Chains Start and end cell, chain cells,
This chain begins at [r2,c2], and follows what would happen if that cell was
set to 1. This forces [r2,c6] to be 8 (same row), then [r3,c5] to be 1 (same block),
then [r8,c5] to be 6 (same column), then [r8,c2] to be 1 (same row), which finally
forces [r2,c2] to be 7 (same column, starting cell). Because this contradicts the start of the chain,
which said that [r2,c2] was 1, we know that we can't put the 1 in this cell, so we can
remove the 1 from [r2,c2].
A little later on in this puzzle, there is another forcing chain, which eventually
leads to the full solution for the puzzle.
3 5
1 8
3 5
1 7 8
7 8
4
9
2
6
4
7
9
2 6
2 6
1 8
3 5
1 8
3 5
6
2
1 8
3
9
5
1 4
4 7
1 7 8
5 7
3
1
2 5 8
4 8
2 5
5 6 7
4 6 7
9
2 5 7
2 4 8
5 7 8
9
3
6
1 5 7
1 4
1 5 7
4 5
6
9
1 4 5
4 7
1 7
2
8
3
2 3
5
2 3
4
1 6
8
1 6
9
7
8
9
4
3 7
1 6
3 7
1 6
5
2
7
1
6
9
5
2
8
3
4
Example 2: Forcing Chains Start and end cell, chain cells,
This chain begins at [r1,c2], and follows what would happen if that cell was
set to 8. This forces [r5,c2] to be 4 (same column), then [r5,c8] to be 7 (same row),
then [r5,c9] to be 1 (same row), then [r1,c9] to be 8 (same column), which finally forces
[r1,c2] to be 1 (same row, starting cell). Because this contradicts the start of the chain,
which said that [r1,c2] was 8, we know that we can't put the 8 in this cell, so we can remove
the 8 from [r1,c2].
