Forcing Chains

Forcing chains sit at the boundary between human technique and trial logic. They follow what would happen under each of two candidates in a bivalue cell, and if both branches force the same conclusion elsewhere, that conclusion is true regardless of the starting choice.

How it works

Find a bivalue cell with candidates {A, B}. Assume the cell takes A and follow the deductions: hidden singles, naked singles, pointing pairs and anything else that does not require a new assumption. Note the forced placements.

Reset, then assume the cell takes B and run the same forward chain. Again, note the forced placements.

Now compare the two branches. If both branches force the same digit in some other cell, that digit is correct no matter which value the starting cell holds. Place it. If one branch produces a contradiction (two of the same digit in a unit, or a cell with no candidates), the other candidate was the true one. eliminate it.

When to look for it

After coloring, XY-wing and X-wing have all failed. Evil puzzles routinely require forcing chains. Pick a bivalue cell with rich consequences for both candidates; cells where each choice triggers a sequence of forced moves are productive.

Step-by-step example

1. 1
   Pick a bivalue start

   A cell with two candidates A and B. The fewer empty cells around it, the faster the chain converges.
2. 2
   Branch A

   Tentatively place A. Apply every forced move you can: singles, pointing pairs, anything mechanical. Stop when no more forced moves remain.
3. 3
   Reset

   Undo every placement from branch A. Critical step. Stale trial marks corrupt the rest of the solve.
4. 4
   Branch B

   Place B and run the same forward chain.
5. 5
   Compare

   Same conclusion in both? Place it. Contradiction in one? The other branch is true.

Tips for spotting it

• Start with bivalue cells whose two candidates each trigger long sequences of forced moves.
• Use two pencil colours so trial placements are distinguishable from real ones.
• Track the chain on paper if your memory cannot hold it. Twenty placements is too many to remember.
• Look for convergence and contradiction simultaneously. Both yield useful information.

Common mistakes

• Forgetting to undo trial placements when the branch fails. Stale marks corrupt everything that follows.
• Drawing conclusions from a single branch. Both must converge or one must break.
• Short chains without convergence. Useless. Pick a longer starting cell next time.
• Reaching for forcing chains before exhausting coloring. Coloring is cheaper when it applies.

Practise it

Forcing chains shine on evil sudoku. Pick a bivalue cell and follow each branch carefully. After a few solves you will see which starts produce convergent chains and which fizzle. The skill carries directly to Nishio, which uses the same trial logic on a single candidate.