Forcing Chains

Level 12 · Black Belt

Strategy:

Requires:

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

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

Example

Starting position

Look for Forcing Chains.

Sudoku grid showing the initial puzzle state for Look for Forcing Chains.

1

Start from bivalue cell r1c3

Find a bivalue cell r1c3 with candidates { 1, 5 }.

Sudoku grid illustrating step 1: Start from bivalue cell r1c3 — Find a bivalue cell r1c3 with candidates { 1, 5 }.

2

First branch: r1c3 is 1

If r1c3 = 1, follow implications to see what gets eliminated.

Sudoku grid illustrating step 2: First branch: r1c3 is 1 — If r1c3 = 1, follow implications to see what gets eliminated.

3

Second branch: r1c3 is 5

If r1c3 = 5, follow implications to see what gets eliminated.

Sudoku grid illustrating step 3: Second branch: r1c3 is 5 — If r1c3 = 5, follow implications to see what gets eliminated.

4

Remove the shared eliminations from r2c4

Both assumptions eliminate the same candidates. Remove them from r2c4.

Sudoku grid illustrating step 4: Remove the shared eliminations from r2c4 — Both assumptions eliminate the same candidates. Remove them from r2c4.

5

Result

Result after applying the technique.

Sudoku grid showing the 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.