Peak Sudoku

Forcing Chain

Both branches from a bivalue cell reach the same conclusion.
All techniques

Overview

Tier 4
Fish, wings, chains & uniqueness
Typical difficulty
Peak Score 90–100
Also known as
Bifurcation Chain, Cell Forcing Chain, Dual-Branch Implication

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.

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.

Worked examples

1 / 5
Reading the exampleCells forming the patternCells losing candidatesEliminated candidates

R8C4 is either 1 or 8, and both lead to the same removals.

Suppose R8C4 is 1.

  1. Then R8C6 is 8, since row 8 has nowhere else for 8.
  2. So R8C6 can't be 5 or 7.

Suppose R8C4 is 8.

  1. Then R8C6 is 1, since row 8 has nowhere else for 1.
  2. So R8C6 can't be 5 or 7.

Either way, 5 and 7 can be removed from R8C6.

Removes: 5 from R8C6 · 7 from R8C6

How to apply it

Spotting the pattern and making the deduction, in order.

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

Common mistakes

  • 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.
Peak Score 75–90Nice Loop (AIC)An alternating chain of strong and weak links – open or closed.Peak Score 80–95ALS ChainLinks built from almost-locked sets – N cells sharing N+1 candidates.Peak Score 70–85XY-ChainA chain of bivalue cells that forces a digit at both endpoints.

Put Forcing Chain into practice

Try a puzzle from the Diabolical tier (Peak Score 91–100).

Play a Diabolical puzzleAll 25 techniques