Digit Forcing Chains

Extreme

Run inference chains from each possible position of a digit; if all chains reach the same conclusion, it must be true.

How It Works

For a digit confined to a few cells in a unit, run inference chains from each possible position. If every chain leads to the same conclusion (a placement or elimination), that conclusion is certain regardless of where the digit actually goes.

Based on the "all paths lead to the same result" principle.

Example

68
37
13
2
15
49
57
25
3
47
46
4
67
57
28
1
1
49
24
89
45
25
63
3
17
6
29
17
48
15
25
47
12
9
36
5
26
13
14
27
48
15
25
14
83
16
79
9
26
12
5
4
17
26
3
47
27
46
8
26
16
19
25
14
56
57
14
14
29
78
89
15
26
Key cells of the techniqueCells where elimination is appliedCandidate to be placedEliminated candidate (crossed out)

Digit 5 in column 3 at R1C3 or R7C3. If R1C3=5 → ... → R4C6≠8. If R7C3=5 → ... → R4C6≠8. Both paths agree → Eliminate 8 from R4C6.

Practice with a Real Puzzle

This 9×9 puzzle is solver-verified to require this technique on its solution path.

Expert25 givens