Box/Line Reduction

Interactive Tutorial

Box/Line Reduction is the complement of Pointing Pair/Triple. When a digit in a row or column can only appear in cells that are all within the same box, the digit can be eliminated from other cells in that box outside the row/column.

How It Works

The Pattern

Look for a digit in a row (or column) where:

The Logic

If digit 2 in Row 1 can only appear in R1C4 and R1C5 (both in Box 2), then:

The line "claims" the digit within that box, eliminating it from other cells.

How to Spot Box/Line Reduction

Single-Digit Focus

The key to spotting intersection techniques is focusing on one digit at a time:

Approach Question What to Look For
Pointing Pair "Where can digit X go in this box?" All candidates fall into one line within a box
Box/Line Reduction "Where can digit X go in this row/column?" All candidates fall into one box within a line

Step-by-Step Scanning

  1. Pick a digit — Choose any digit 1-9
  2. Pick a row or column — Focus on one line at a time
  3. Ask: "Where can this digit go in this row/column?"
  4. Check confinement — Do all candidates fall within a single box?
  5. Eliminate — If yes, remove that digit from the rest of that box outside the line

Using Focus Mode

Focus Mode makes box/line reduction much easier to spot:

  1. Tap a digit to highlight all cells containing it
  2. Look at each row/column — are the highlighted cells all in one box?
  3. If confined, check if there are candidates to eliminate in the rest of that box

Example

Look at Row 1 and ask: where can digit 2 go?

Row 1 Analysis:

Since digit 2 in Row 1 must be in Box 2, no other cell in Box 2 can contain 2.

Eliminations: Remove 2 from R2C5, R3C4, and R3C5 (Box 2 cells outside Row 1)

puzzle: S9B160106188c07087u031a0i140828262l2i2z0h07141c8g010s068a2d04082t3p1u039e9f060e2b2n2n0i2j0h022d0c092l654a06052j2v062v096b4i2i02622r0h2r2z2z020i0c0f090b04065y4605015u
mode: guided
technique: Box/Line Reduction
initial:
  layers:
    hints: true
steps:
  - text: >
      Use Focus Mode to highlight digit 2. Where can 2 go in Row 1?
    hint: subtle
    technique: BL
    state:
      focus:
        enabled: true
        digits: [2]

  - text: >
      Digit 2 in Row 1 only appears in R1C4 and R1C5 — both in Box 2!
    hint: obvious
    technique: BL
    state:
      selection:
        cells: [R1C4, R1C5]
      focus:
        enabled: true
        digits: [2]

  - text: >
      This is Box/Line Reduction. Row 1 "claims" digit 2 for Box 2.
    hint: obvious
    technique: BL
    state:
      selection:
        cells: [R1C4, R1C5]
      focus:
        enabled: true
        digits: [2]

  - text: >
      Eliminate 2 from other Box 2 cells outside Row 1: R2C5~2, R3C4~2, R3C5~2.
    hint: detailed
    technique: BL
    state:
      selection:
        cells: [R2C5, R3C4, R3C5]
      focus:
        enabled: true
        digits: [2]
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered

Second Example in Same Puzzle

Also in this puzzle, look at Column 8:

Pointing vs Box/Line

These techniques are inverse operations on the same geometric intersection:

           Box 2
       +---------+
       | .  .  . |   Row 1 "claims" 2 for Box 2
-------+-2--2--.-+-------
       | x  x  . |   <- Eliminate 2 from rest of box
       | x  x  . |
       +---------+
Start With Find Eliminate From
Digit in Box aligned in row/col Pointing Pair Rest of row/column
Digit in Row/Col confined to box Box/Line Reduction Rest of box

Tips

  1. Check intersections systematically — Each box has 6 intersection zones (3 rows × 2 + 3 columns × 2)
  2. Look for confined candidates — When a row/column's digit options all fall in one box
  3. Common in medium puzzles — Often unlocks progress after basic techniques stall
  4. Can chain with pointing pairs — One elimination may create the other

More Puzzles

Related Techniques