Digit Forcing Chain

A Digit Forcing Chain assumes a candidate is either ON (true) or OFF (false), traces both chains of implications, and looks for convergent conclusions. If both assumptions lead to the same result, that result must be true.

How It Works

The Two Chains

Starting from any candidate:

ON Chain: "If this candidate IS the solution..."

Trace all logical consequences
What other candidates are eliminated?
What values are forced?

OFF Chain: "If this candidate is NOT the solution..."

Trace all logical consequences
What candidates become forced?
What eliminations occur?

Convergence

The technique succeeds when both chains reach the same conclusion:

Both chains place the same digit in a cell → That digit IS the solution
Both chains eliminate the same candidate → That candidate IS eliminated
Both chains force the same cell → Apply the forcing result

Convergence Rules

Rule 1: Same Digit Placed

Both chains place digit X in cell Y. Result: Cell Y = X

ON chain:  ... → R3C5 = 7
OFF chain: ... → R3C5 = 7
─────────────────────────
Conclusion: R3C5 = 7

Rule 2: Same Digit Eliminated

Both chains eliminate digit X from cell Y. Result: Eliminate X from Y

ON chain:  ... → R3C5 ≠ 7
OFF chain: ... → R3C5 ≠ 7
─────────────────────────
Conclusion: Eliminate 7 from R3C5

Rule 3: Same Cell Affected

Both chains force the same cell (possibly different conclusions). Result: Analyse the intersection

Rule 4: Unit Constraint

Both chains create the same unit constraint. Result: Apply that constraint

Example

Starting candidate: 5 in R1C1

ON Chain (R1C1 = 5):

R1C1 = 5
R1C4 ≠ 5 (same row)
R2C4 = 5 (only place in column 4)
R2C7 ≠ 5 (same row)
R5C7 = 5 (only place in box)

OFF Chain (R1C1 ≠ 5):

R1C1 ≠ 5
R3C1 = 5 (only place in column 1)
R3C9 ≠ 5 (same row)
R5C9 ≠ 5 (chain continues...)
R5C7 = 5 (forced by elimination)

Convergence: Both chains conclude R5C7 = 5 Result: R5C7 = 5

The Logic

The forcing chain argument is:

Either the starting candidate is true OR it's false
There's no third option
If both possibilities lead to the same conclusion...
That conclusion MUST be true (regardless of the starting assumption)

This is a form of proof by exhaustion — we've checked all cases.

Complexity

Digit Forcing Chains are expert-level because:

Exponential complexity: Chains can branch extensively
Deep analysis required: Must trace implications through many steps
Multiple chain types: ON and OFF chains behave differently
Verification needed: Must confirm both chains are valid

Relationship to Other Techniques

Technique	Chain Type	Digits
X-Cycles	Alternating loop	Single digit
3D Medusa	Colouring network	Multiple digits
Digit Forcing Chain	Dual hypothesis	Single digit

Digit Forcing Chains are more direct than cycles — they explicitly test both possibilities.