Box-Line Reduction
The reverse of Pointing Pairs: a digit in a row or column that is confined to one box — eliminate it from the rest of the box.
What is Box-Line Reduction?
Box-Line Reduction (also called Claiming, or Line-Box Interaction) is the complementary intersection technique to Pointing Pairs. Instead of starting from a box and projecting outward into a line, you start from a line (row or column) and project inward into a box. If a digit's remaining candidate cells in an entire row or column all happen to fall within the same 3×3 box, then that line "claims" those box positions for the digit. Since the digit must go in the row or column (by box constraint), and it can only do so within one specific box, it cannot appear in any other cell in that box outside the line. This means you can eliminate the digit from all other cells in that box that are not part of the selected row or column. Like Pointing Pairs, this technique requires no guessing and produces logically certain eliminations. It is especially valuable in the middle game of Hard puzzles where simple pairs have been exhausted but X-Wing and more complex patterns are not yet visible.
When to Use Box-Line Reduction
Apply Box-Line Reduction after Pointing Pairs, or alternatively, check both simultaneously as part of an "intersection pass." For each row, cycle through unplaced digits and check whether all candidates for that digit in the row fall within a single box. Repeat for each column. This technique complements Pointing Pairs perfectly — Pointing Pairs work from box to line, Box-Line Reduction works from line to box. Together they form a complete analysis of all box-line intersections on the board. When either technique yields an elimination, immediately update the candidate grid and re-scan for singles and other patterns before continuing.
How to Apply Box-Line Reduction — Step by Step
- 1
Choose a row or column to analyze
Pick a row or column that has several placed digits (the more placed, the fewer candidates remain, making alignment more likely). Select a digit that has not been placed in that row or column yet.
- 2
Find all candidate positions in the line
Within your selected row or column, identify every empty cell that could contain the chosen digit. These are the cells where the digit's row/column constraint is still satisfied. Mark or note these positions.
- 3
Check for single-box confinement
Now examine where these candidate cells are located. Do they all fall within the same 3×3 box? If yes, the technique applies — the row or column is "claiming" that digit for cells exclusively within that one box. If the candidates span more than one box, move on to the next digit or line.
- 4
Eliminate within the box outside the line
Since the digit must appear in the selected row/column, and can only do so within the identified box, the digit cannot appear elsewhere in that box. Remove it from every cell in the box that is not part of the selected row or column. After eliminating, scan for new singles or intersection patterns.
💡 Pro Tip
Box-Line Reduction and Pointing Pairs are two sides of the same coin. Experienced solvers apply both simultaneously by doing a comprehensive "intersection scan" — examining every row-band and column-stack and checking both directions (box-to-line and line-to-box). A helpful mental rule: if all occurrences of a digit in a row are clustered in one box, those box-cells are safe to treat as "belonging" to the row, and the digit can be cleared from the remaining box-cells. Apply the same logic for columns. This bilateral thinking greatly accelerates your ability to spot both patterns.
Practice Box-Line Reduction Now
Put this technique to the test on a live puzzle. The Practice Mode lets you work through real examples with candidate marking.