Peak Sudoku
Fish, wings, chains & uniquenessPeak Score 90–100

Forcing Chain

Both branches from a bivalue cell reach the same conclusion.

Also known as: Bifurcation Chain, Cell Forcing Chain, Dual-Branch Implication

Quick Summary

Choose a two-candidate cell. Follow the logical implications of assuming the first number is true, and follow the implications of assuming the second number is true. If both assumptions lead to the exact same deduction, that deduction is guaranteed true.

Understanding the Forcing Chain Technique

A Forcing Chain is the ultimate deductive tool for puzzles that resist all standard fish, wings, and simple loops. It is not trial-and-error: rather than guessing and hoping for a contradiction down the road, a forcing chain systematically investigates both branches of a binary split. If assuming option A forces a target cell to be 7, and assuming option B also forces that same target cell to be 7, then that cell is undeniably 7, because one of A or B must be true.

Visual Examples

Green cells highlight the Forcing Chain pattern, amber cells denote where candidates are affected, and struck-through pencil marks show the eliminations. Use the buttons or arrow keys to browse examples.

Example 1 of 5
R8C4 is either 1 or 8. Either way the same eliminations follow: If R8C4 = 1 (chain depth 1): 1. assume R8C4 = 1 2. R8C6 = 8 (only place for 8 in row 8) → therefore R8C6 cannot be 5 or 7 If R8C4 = 8 (chain depth 1): 1. assume R8C4 = 8 2. R8C6 = 1 (only place for 1 in row 8) → therefore R8C6 cannot be 5 or 7
Removes: 5 from R8C6 · 7 from R8C6

How to Spot & Solve a Forcing Chain (Step-by-Step Guide)

The complete visual scanning and logical deduction process to execute this technique:

  • 1Select a pivotal two-candidate cell (for example, with candidates 3 and 7).
  • 2Trace branch 1: Assume the cell is 3 and follow the chain of forced deductions to see how it affects a target cell.
  • 3Trace branch 2: Assume the cell is 7 and follow the deductions along that branch.
  • 4Compare results: if both branches independently force the exact same placement or candidate elimination at the target cell, the result is guaranteed.
  • 5Confidently apply that placement or elimination to the board.

Common Traps & Mistakes to Avoid

  • Making an invalid inference along one of the branches, creating a false convergence.
  • Branching too deep without clear direction; keep forcing chains shallow and focused on a target cell.
  • Reaching for forcing chains prematurely when a simpler pattern (like a Skyscraper or XY-Wing) was present.

Frequently Asked Questions

Is a Forcing Chain considered guessing?

No. Guessing means placing a digit without knowing the outcome and having to backtrack later. A Forcing Chain is a constructive proof: it shows that regardless of which binary state is true, the conclusion is inevitable.

When should I use Forcing Chains?

Only as a last resort on the hardest possible puzzles (Peak Score 90+) when all fish, wings, and loops have been checked.

Expand your repertoire by learning prerequisites and advanced counterparts:

Nice Loop (AIC)Peak Score 75–90

An alternating chain of strong and weak links – open or closed.

Read guide →
ALS ChainPeak Score 80–95

Links built from almost-locked sets – N cells sharing N+1 candidates.

Read guide →
XY-ChainPeak Score 70–85

A chain of bivalue cells that forces a digit at both endpoints.

Read guide →

Ready to Practice Forcing Chain?

Try a puzzle in the Diabolical tier (Peak Score 91–100), or browse all 25 techniques.

Play Diabolical puzzleAll 25 techniques