Box-Line Reduction: Claiming and Intersection Removal

Box-Line Reduction is the mirror image of pointing pairs. Also known as "claiming" or "intersection removal," this technique uses row and column constraints to eliminate candidates from boxes.

What is Box-Line Reduction?

When a candidate number in a row or column can only appear within one box, you can eliminate that number from all other cells in that box (outside the row/column).

The Logic

If a number in a row can only fit in cells that are all within the same box, then:

• The number must be somewhere in that box-row intersection
• It cannot be elsewhere in the box
• You can eliminate it from other cells in the box

Example

Imagine row 4 where the number 8 can only appear in:

• Columns 4, 5, or 6 (which are all in box 5)

Since 8 must be in this intersection within row 4, you can eliminate 8 from:

• The rest of box 5 (rows 5 and 6, columns 4-6)

Why It's Called "Claiming"

The row or column "claims" certain positions for a number within a box, preventing that number from appearing elsewhere in the box.

The Two Types

Row Claims Box

All candidates for a number in a row are confined to one box → Eliminate from the rest of that box

Column Claims Box

All candidates for a number in a column are confined to one box → Eliminate from the rest of that box

Relationship to Pointing Pairs

These techniques are opposites:

• Pointing Pairs: Box restricts a line → eliminate from line
• Box-Line Reduction: Line restricts a box → eliminate from box

Both exploit the intersection between boxes and lines!

How to Find Box-Line Reductions

1. Pick a row or column
2. Choose a missing number
3. Mark where it could go in that line
4. Check if all candidates are in one box
5. If yes: Eliminate from the rest of that box

Common Mistakes

• Eliminating from the wrong place: Remove from the box, not the row/column
• Not checking all positions: Ensure ALL possible positions are in one box
• Missing the technique entirely: Many solvers forget to look for this pattern
• Confusing with pointing pairs: Remember which way the elimination goes

Scanning Strategy

By Row:

1. For each row, scan for numbers with 2-3 possible positions
2. Check if they're all in the same box
3. If yes, eliminate from that box

By Column: Same process, but scanning columns instead

Quick Method:

• Focus on rows/columns with many filled cells
• These are more likely to have box-line reductions

Impact on Solving

Box-line reduction is valuable because:

• Complements pointing pairs perfectly
• Often creates hidden singles in boxes
• Works when basic techniques stall
• Relatively easy to spot with practice

Visual Recognition

Look for this pattern:

• A row/column intersects three boxes
• A number can only go in one of those three boxes
• That's your cue to eliminate from the box

Practice Tips

1. After using pointing pairs, immediately check for box-line reductions
2. These two techniques work hand-in-hand
3. Draw lines to visualize row-box and column-box intersections
4. Start with nearly-complete rows or columns

Next Steps

You've now learned both intersection techniques! Continue with:

• Hidden Pairs - Finding non-obvious candidate pairs
• Naked Triples - Extending naked pairs to three cells
• X-Wing - Your first advanced pattern-based technique