Intermediate Technique

Box/Line Reduction

Eliminate candidates from boxes when all occurrences of a number in a row or column are confined to a single box.

What is Box/Line Reduction?

Box/Line Reduction (also called "Claiming") is the reverse of Pointing Pairs. It occurs when all the possible positions for a specific number within a row or column are contained entirely within one 3x3 box. This means the number must appear somewhere in that box, so you can eliminate it from other rows or columns within the same box.

This technique leverages the constraint that each number must appear exactly once in every row, column, and box.

How Box/Line Reduction Works

1

Examine a row or column

Look for a candidate number that can only appear in cells within one specific 3x3 box.

2

Verify confinement

Confirm that this number cannot appear anywhere else in that row or column.

3

Eliminate from the box

Remove the candidate from all other rows or columns within that box.

Example: Row Claims a Box

In this example, all possible positions for the number 6 in the middle row are confined to the center box:

Before: Identifying the Claiming Line

1
2
3
4
5
6
7
8
9
4
5
6
7
8
9
1
2
3
7
8
9
1
2
3
4
5
6
23 5 89
13 79
12 7
6
3 6
2 46
23 5 89
13 4 79
12 45 78
3 45 789
13 4 79
12 45 78
3 56 9
12 6 9
12 56
23 5 89
13 4 79
12 45 78
3 5 89
13 79
12 7
56 89
123 46
12 46
23 5 89
13 4 79
12 45 78

Analysis: In row 4 (highlighted in blue), the number 6 can only appear in the three cells within the center box. Since the 6 must be placed somewhere in this row within the center box, it cannot appear in any other rows of the center box.

Elimination: All 6s in the center box outside of row 4 can be eliminated (shown with strikethrough in the green highlighted cells).

🔄 Box/Line Reduction vs. Pointing Pairs

Pointing Pairs: Two cells in a box point to eliminate candidates in the rest of the line.

Box/Line Reduction: All candidates in a line are confined to one box, eliminating candidates from the rest of the box.

These techniques are complementary - one works from box to line, the other from line to box.

Column Claiming Example

Box/Line Reduction also works with columns. When all instances of a number in a column are confined to one box, you can eliminate that number from other columns within the same box.

Scenario: The number 3 in column 7 can only appear in cells within the top-right box. Therefore, 3 cannot appear in columns 8 or 9 of the top-right box.

Application: Remove 3 from all candidates in columns 8 and 9 within the top-right box.

Step-by-Step Application

  1. Choose a line: Select a row or column to analyze
  2. Find confined candidates: Look for numbers that can only appear within one 3x3 box
  3. Verify confinement: Ensure the number truly cannot appear elsewhere in the line
  4. Identify the target box: Note which box contains all the candidate positions
  5. Eliminate from other lines: Remove the candidate from other rows/columns within the box
  6. Continue solving: Look for new opportunities revealed by the eliminations

⚠️ Common Mistakes

  • Incomplete analysis: Make sure you've checked all possible positions in the line
  • Wrong elimination direction: Eliminate from other lines in the box, not from the claiming line itself
  • Missing opportunities: Check both rows and columns systematically
  • Confusing with pointing pairs: Remember this technique goes from line to box, not box to line

🎯 Practice Exercise

Look for Box/Line Reduction opportunities:

  1. Choose a row or column with several filled cells
  2. For each missing number, identify all possible positions
  3. Check if all positions fall within a single 3x3 box
  4. If yes, eliminate that number from other lines within the box
  5. Repeat for all rows and columns

Tip: This technique is particularly effective in partially filled puzzles where many constraints have already been established.

When to Use Box/Line Reduction

  • After basic techniques and pointing pairs have been applied
  • When you have rows or columns with multiple candidates confined to boxes
  • As part of systematic candidate elimination
  • Before attempting more complex techniques like X-Wing

Related Techniques

  • Pointing Pairs - The reverse technique working from box to line
  • Hidden Pairs - Finding restricted pairs within regions
  • Obvious Pairs - Eliminating candidates using visible pairs
  • X-Wing - Advanced rectangular elimination patterns
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