Forcing Chains

Level 12 · Black Belt

Strategy:

Forcing

Requires:

X-Cycles

Learn after:

Forcing Net

Overview

Forcing Chains explore what happens when a cell is assumed to be each of its candidates. If all assumptions lead to the same conclusion, that conclusion must be true.

How It Works

  1. Find a cell with 2-3 candidates.
  2. Assume each candidate in turn and follow the logical chain.
  3. If all chains lead to the same elimination, that elimination is valid.
  4. If all chains place the same digit in another cell, that placement is valid.

Example

Look for Forcing Chains.

1

Forcing Chains

Find a bivalue cell r2c4 with candidates { 5, 8 }.

2

Forcing Chains

If r2c4 = 5, follow implications to see what gets eliminated.

3

Forcing Chains

If r2c4 = 8, follow implications to see what gets eliminated.

4

Forcing Chains

BOTH assumptions eliminate the same candidates. Eliminate from r4c5.

5

Result after applying the technique.

Tips & Patterns

  • Forcing Chains are powerful but time-consuming.
  • Start from cells with few candidates.
  • They're essentially "if-then" logic following implications.
  • Consider using pencil and paper to track chains.