Solving Tips

Sudoku Box-Line Reduction: Using Intersections of Boxes and Lines

2025-01-24 · 7 min read

Box-Line Reduction (also known as Pointing & Claiming) is a very practical intermediate Sudoku technique. This technique uses the intersection relationship between boxes (3×3 regions) and lines (rows/columns) to eliminate candidates, and comes in two types: Pointing and Claiming.

                 Core Principle:

                Each box in Sudoku intersects with three rows and three columns. If a candidate appears in a box only in the same row (or column), then that number cannot appear in other boxes on that row (or column). Conversely, if a candidate appears in a row (or column) only within a certain box, then that number cannot appear in other positions of that box.

Before reading this article, we recommend understanding Sudoku row, column, and box naming conventions, which will help you follow the analysis examples below.

Type 1: Pointing

Pointing means: when a candidate appears in a box only in the same row or column, you can eliminate that candidate from other boxes on that row/column.

Pointing Rule

If a candidate appears in a box only in the same row (or column),
Then that candidate can be eliminated from other boxes on that row (or column).

Let's look at an example:

Figure 1: Box 5's candidate 1 only appears in Row 6, so Row 6 cells outside Box 5 cannot be 1

Analysis Process

1 Observe box distribution: Check Box 5 (the middle 3×3 region) and find that candidate 1 only appears in cells of Row 6.

2 Understand the principle: Since Box 5's number 1 must be placed in some position on Row 6 (no other rows in the box can place 1), Row 6 positions in other boxes cannot contain 1 (otherwise Box 5 would have nowhere to place 1).

3 Execute elimination: From all cells on Row 6 that are not in Box 5, delete candidate 1. This includes cells in Box 4 and Box 6 on Row 6.

Conclusion:
Box 5's candidate 1 "points to" Row 6, therefore candidate 1 can be deleted from other boxes on Row 6 (Box 4 and Box 6).

Type 2: Claiming

Claiming is the reverse application of Pointing: when a candidate appears in a row or column only within a certain box, you can eliminate that candidate from other rows/columns in that box.

Claiming Rule

If a candidate appears in a row (or column) only within a certain box,
Then that candidate can be eliminated from other rows (or columns) in that box.

Let's look at another example:

Figure 2: Column C's candidate 2 only appears in C1, C2, C3 (all in Box 1), so Box 1 cells outside Column C cannot be 2

Analysis Process

1 Observe row/column distribution: Check Column C (3rd column) and find that candidate 2 only appears in cells of Box 1 (C1, C2, C3 are all within Box 1).

2 Understand the principle: Since Column C's number 2 must be placed in some position within Box 1 (no other boxes in the column can place 2), Box 1 positions in other columns cannot contain 2 (otherwise Column C would have nowhere to place 2).

3 Execute elimination: From all cells in Box 1 that are not in Column C, delete candidate 2. This includes cells in Column A and Column B within Box 1.

Conclusion:
Column C "claims" Box 1's candidate 2, therefore candidate 2 can be deleted from other columns in Box 1 (Column A and Column B).

Pointing vs Claiming Comparison

These two types are essentially the same principle viewed from different perspectives:

Comparison	Pointing	Claiming
Starting Point	Start from box	Start from row/column
Condition	Candidate appears in box only in same row/column	Candidate appears in row/column only within same box
Elimination Range	Other boxes on that row/column	Other rows/columns in that box
Metaphor	Box's candidate "points to" a row/column	Row/column "claims" space in the box

                 Memory Tip:

                Pointing: Box → Row/Column, imagine the box's candidate "pointing" to an external line
Claiming: Row/Column → Box, imagine the line "claiming" space within the box

Practical Application Steps

When solving, follow these steps to find Box-Line Reduction opportunities:

Mark candidates: Ensure all cell candidates are marked
Check each box: Examine each box to see if any candidate is concentrated in the same row or column
Check each row and column: Examine each row and column to see if any candidate is concentrated in the same box
Execute elimination: When conditions are met, immediately delete candidates
Chain reaction: Elimination may create new naked singles or elimination opportunities, continue solving

Common Mistakes:

Confusing elimination direction: Pointing eliminates from box to lines, Claiming eliminates from lines to box
Wrong elimination range: Can only eliminate cells outside the intersection area
Ignoring candidates: Must ensure candidates are accurately marked, or opportunities may be missed

Technique Summary

Core points of Box-Line Reduction:

Use intersections: Cleverly use intersection relationships between boxes and lines for elimination
Bidirectional observation: Must view lines from box perspective and boxes from line perspective
Concentration principle: Candidates must be "concentrated" in the intersection area to apply this technique
Immediate elimination: Execute immediately when opportunities are found, don't accumulate too many steps

                 Why It's Important:

                Box-Line Reduction is the bridge between beginner and advanced techniques. After mastering this technique, you'll find many "stuck" puzzles can find breakthroughs through box-line interactions. It's also the foundation for understanding more advanced techniques (like X-Wing).

Practice Suggestions

To master Box-Line Reduction, we recommend:

Systematically check each box and line relationship when solving, don't skip based on intuition
Use different colors to mark candidates, helping visual identification of concentration areas
When encountering medium difficulty puzzles, use basic techniques first, then actively seek Box-Line Reduction opportunities
Understanding the principle is more important than memorizing terminology, understand "why elimination works"

Practice Now:
Start a medium difficulty Sudoku game, specifically looking for and applying Box-Line Reduction!

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