Box-Line Reduction in Sudoku: The Complete Guide

What is Box-Line Reduction?

Box-Line Reduction (also called "Locked Candidates Type 2" or "Claiming") occurs when a candidate appears in a row or column but is confined to just one box within that line. Since the candidate must be placed somewhere in that box, you can eliminate it from other cells in the same box that are outside that row or column.

Example:

In Row 5, candidate 8 appears in columns 4, 5, and 6—all within Box 5 (middle-center box). No other cells in Row 5 contain candidate 8. Since 8 must go in Row 5 of Box 5, you can eliminate 8 from all other cells in Box 5 (specifically rows 4 and 6 of columns 4-6).

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Why Box-Line Reduction Matters

• It creates eliminations within boxes that other techniques miss.
• It works perfectly with pointing pairs to form a complete "locked candidates" toolkit.
• It frequently reveals hidden singles in boxes.
• It's essential for intermediate puzzle solving.
• It helps clean up complex candidate grids systematically.

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How to Spot Box-Line Reduction

1. Focus on a single row or column.
   Pick any row or column on the grid.
2. Check each candidate number (1-9).
   For each number, identify which cells in that row/column contain it as a candidate.
3. Look for confinement to one box.
   If all instances of a candidate fall within just one 3×3 box, you've found box-line reduction.
4. Eliminate from the rest of the box.
   Remove that candidate from all other cells in the same box, but only those outside the row/column you're examining.

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Box-Line Reduction vs Pointing Pairs

These complementary techniques both deal with locked candidates but work in opposite directions:

• Pointing Pairs (Type 1): Start with a box. If candidates are confined to one row/column within that box, eliminate from that row/column outside the box. (Box → Line)
• Box-Line Reduction (Type 2): Start with a row/column. If candidates are confined to one box within that line, eliminate from that box outside the row/column. (Line → Box)

Think of it this way: pointing pairs point outward from box to line, while box-line reduction claims inward from line to box.

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Step-by-Step Example

Let's examine Row 7:

Row 7 candidates:

• Col 1: [filled with 5]
• Col 2: {2, 6, 9}
• Col 3: {2, 6}
• Col 4: {1, 3, 4}
• Col 5: [filled with 7]
• Col 6: {1, 3, 4}
• Col 7: [filled with 8]
• Col 8: {2, 6, 9}
• Col 9: {2, 9}

Analysis: Let's check where candidate 4 appears in Row 7:

• 4 appears in: Col 4 and Col 6
• Both are in Box 8 (bottom-middle box: rows 7-9, columns 4-6)
• This is box-line reduction!

Conclusion: Since 4 must be somewhere in Row 7 within Box 8, we can eliminate 4 from all other cells in Box 8 that are NOT in Row 7 (specifically rows 8 and 9 of columns 4-6).

Result: Check Box 8, rows 8 and 9, and remove candidate 4 from any cells there. This might reveal a hidden single or simplify other cells.

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Visual Example

• Scenario: In Column 3, candidate 5 appears in rows 1, 2, and 3 only.
• Observation: All three instances are in Box 1 (top-left: rows 1-3, columns 1-3).
• Box-line reduction identified: Candidate 5 in Column 3 is locked to Box 1.
• Elimination: Remove candidate 5 from all other cells in Box 1 outside Column 3 (specifically columns 1 and 2 of rows 1-3).
• Impact: This might reveal that 5 can only go in one specific cell in Box 1, creating a hidden single.

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Strategies for Spotting Box-Line Reduction Quickly

1. Systematic line scanning
   Go through each row and column methodically. For each line, check all candidates 1-9.
2. Look for candidate clusters
   When a candidate appears 2-3 times in a row/column and they're all close together, check if they're in the same box.
3. Visual box awareness
   Train your eye to see box boundaries clearly. Notice when candidates don't cross those boundaries.
4. Use after pointing pairs
   These techniques work together. After finding pointing pairs, immediately check for box-line reduction in the affected areas.
5. Focus on partially filled lines
   Rows or columns with 5-7 cells filled are prime candidates for box-line reduction opportunities.

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Common Pitfalls

• Eliminating from the wrong area: Only eliminate from the box outside the row/column you're analyzing, never from within that row/column.
• Missing candidates in other boxes: Make sure the candidate truly appears ONLY in one box within your row/column. If it appears in two boxes, you can't use this technique.
• Confusing with pointing pairs: Remember: box-line reduction starts with a line (row/column), pointing pairs start with a box.
• Incomplete scanning: Check all 9 rows and all 9 columns systematically. It's easy to miss opportunities if you only spot-check.
• Forgetting to update candidates: After eliminations, new patterns emerge. Always recheck for naked singles and hidden singles.

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Practice: Find the Box-Line Reduction

Try this Column 8 scenario:

Column 8 candidates:

• Row 1: {3, 7, 9}
• Row 2: [filled with 4]
• Row 3: {3, 7}
• Row 4: {1, 2, 5}
• Row 5: [filled with 6]
• Row 6: {1, 2, 5, 8}
• Row 7: [filled with 9]
• Row 8: {1, 2, 5, 8}
• Row 9: {1, 2}

Question: Can you find a box-line reduction?

Solution: Look at candidate 3 in Column 8:

• 3 appears in: Row 1 and Row 3
• Both are in Box 3 (top-right: rows 1-3, columns 7-9)
• Box-line reduction found! Candidate 3 in Column 8 is confined to Box 3.

Action: Eliminate candidate 3 from all other cells in Box 3 outside Column 8 (specifically columns 7 and 9 of rows 1-3).

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Why Box-Line Reduction Sets the Stage

Box-Line Reduction completes your understanding of locked candidates—a fundamental intermediate concept. Once you master both box-line reduction and pointing pairs:

• You can systematically check all rows, columns, and boxes for locked candidates
• You'll dramatically reduce candidates on complex grids
• You'll be prepared for advanced techniques like X-Wing and Swordfish
• You'll understand how constraints propagate across multiple units
• You'll solve intermediate puzzles efficiently without guessing

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Quick Recap

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Final Thought

When scanning rows and columns, ask yourself: are these candidates confined to just one box? Box-line reduction might unlock your next move.

Frequently Asked Questions

What is Box-Line Reduction in Sudoku?

Box-Line Reduction occurs when a candidate appears in a row or column but is confined to just one box within that line. Since the candidate must be in that box, you can eliminate it from other cells in the same box that are outside that row or column. Also called Locked Candidates Type 2 or Claiming.

How do I spot Box-Line Reduction?

To spot box-line reduction: 1) Focus on a single row or column, 2) For each candidate number, check which boxes contain it within that row/column, 3) If all instances of a candidate are in just one box, you've found box-line reduction, 4) Eliminate that candidate from other cells in that box outside the row/column.

What's the difference between Box-Line Reduction and Pointing Pairs?

Box-Line Reduction (Locked Candidates Type 2) starts with a row/column: when candidates are confined to one box, eliminate from that box outside the line. Pointing Pairs (Locked Candidates Type 1) start with a box: when candidates are confined to one row/column, eliminate from that line outside the box. They're complementary—box-line draws inward to the box; pointing pairs point outward from the box.

Why is Box-Line Reduction important?

Box-Line Reduction is important because it creates eliminations within boxes that other techniques miss, it works well with pointing pairs to clean up candidate grids, it frequently reveals hidden singles in boxes, and it's essential for intermediate puzzle solving.

When should I use Box-Line Reduction?

Use box-line reduction after exhausting naked and hidden singles, and ideally in combination with pointing pairs. Systematically check each row and column, looking for candidates that cluster in just one box. This technique is especially powerful when rows or columns have many filled cells.

Ready to advance? Check out our complete strategy guide for more techniques.

Related Strategies

Once you've mastered box-line reduction, these techniques build naturally:

• Pointing Pairs - The complementary technique (Locked Candidates Type 1)
• Hidden Singles - Often revealed after box-line eliminations
• Naked Pairs - Another key intermediate technique
• Hidden Pairs - Works within single units
• X-Wing - Advanced technique building on constraint propagation

Practice Box-Line Reduction

• Printable Intermediate Sudoku
• Play Daily Sudoku Online

Next up: Try Naked Triples to extend your pair techniques (coming soon).

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