Forcing Chains
Follow the consequences of a candidate being true or false until every path forces the same conclusion.
Forcing chains are the heavy artillery of logical solving. Pick a cell and follow the implications of each of its candidates through the grid. If every candidate leads to the same placement or elimination somewhere, that conclusion is certain — no guessing required.
Most "trial-and-error" feelings late in a hard puzzle are really forcing chains waiting to be written down rigorously.
How to spot it
Choose a bi-value cell. Assume its first candidate and propagate forced placements; note the result. Reset, assume the second candidate, propagate again. Wherever both assumptions force the same cell to the same value (or the same elimination), that result holds unconditionally.
- Start from a cell’s candidates.
- Propagate the forced consequences of each.
- A shared conclusion across all branches is proven.
Discipline over guessing
Forcing chains are logic, not bifurcation: you keep both branches in mind and only act on what they agree on. Writing the chain out avoids the errors that come from blindly trying a number.
Worked example
- A bi-value cell shows {1,4}.
- Assume 1: a sequence of singles forces cell Q to 8.
- Reset, assume 4: a different sequence also forces cell Q to 8.
- Both branches agree, so Q is 8 regardless.
- Place 8 in Q without resolving the start cell.
Try it yourself
Tap a cell, then a number, to practise.
Frequently asked questions
- Is this just guessing?
- No. You evaluate all branches and act only on conclusions every branch shares, so the result is fully logical.
- When should I reach for forcing chains?
- Last, after singles, locked candidates, fish, wings and coloring have been exhausted.
Related techniques
Practice: Forcing Chains
Put the Forcing Chains to work on a live board — free puzzles with notes, hints and four difficulty levels.
Try it on a live board