What Is Box-Line Reduction?
Box-line reduction (also called "claiming" or "line-box interaction") occurs when a candidate number in a row or column is restricted to cells within a single 3×3 box. Since the row or column must place that number somewhere, and all its options fall within one box, you know the number will end up in that box. Therefore, you can eliminate it from all other cells in the box that are not part of that row or column.
Think of it as the row (or column) "claiming" a number for itself within the box. The box must accommodate the claim, and other cells in the box lose out.
How It Relates to Pointing Pairs
Pointing pairs work from box to line: a candidate in a box is confined to one row or column, so you eliminate from the rest of that line outside the box.
Box-line reduction works from line to box: a candidate in a row or column is confined to one box, so you eliminate from the rest of that box outside the line.
Same logic, opposite direction. Together they cover all interactions between boxes and lines. Many solvers check for both simultaneously.
How to Spot Box-Line Reduction
Step 1: Pick a row and a candidate number. Look at where that candidate appears across all nine cells of the row.
Step 2: Do all instances of that candidate fall within a single 3×3 box? If the candidate appears in columns 1, 2, and/or 3 (all within Box 1, Box 4, or Box 7 depending on the row), then it is confined to that box.
Step 3: Eliminate that candidate from all other cells in the box that are not in the claiming row.
Repeat for columns: if a candidate in a column is confined to one box, eliminate from other cells in the box outside that column.
Quick Visual Test
Look at a row's candidates. If a number only appears in the left third, the middle third, or the right third of the row, it is confined to one box. That is your cue.
Worked Example
Examine Row 8. The candidate 5 appears in the following cells: R8C4, R8C5, R8C6. All three cells belong to Box 8 (rows 7–9, columns 4–6).
Since Row 8 must place a 5 somewhere, and its only options are in Box 8, the 5 for Box 8 will definitely be in Row 8.
Now look at Box 8's other cells: R7C4, R7C5, R7C6, R9C4, R9C5, R9C6. If any of these have 5 as a candidate, erase it. Row 8 has claimed the 5 for this box.
Suppose R9C5 had candidates {3, 5, 9}. After the elimination it becomes {3, 9}. That reduction might create a naked pair with a neighboring cell, cascading into more placements.
Scanning Systematically
To find box-line reductions efficiently, scan each row and each column for each candidate number. For any given line, a candidate appears in 2 to 7 cells. If all those cells share the same box, you have a claiming opportunity.
The most productive cases are when the candidate appears in just 2 or 3 cells in the line, because smaller groups are easier to spot visually and more likely to be confined to one box.
With practice, you develop a feel for this. When you see a candidate clustering in one region of a row or column, check the box boundaries.
Common Mistakes
Eliminating from the wrong place. You eliminate from the box, not from the row or column. The claiming line keeps its candidates. Only other cells in the box (outside the claiming line) lose the candidate.
Confusing with pointing pairs. If you start by looking at a box and notice candidates line up in a row, that is a pointing pair (eliminates from the row outside the box). If you start by looking at a row and notice candidates are confined to one box, that is box-line reduction (eliminates from the box outside the row). The starting point determines the direction.
Incomplete pencil marks. If your candidates are stale, you might think a number is confined to one box when it actually appears elsewhere in the row. Always update pencil marks before scanning for interaction techniques.
Not checking columns. Solvers often check rows but forget columns. Box-line reduction works equally on both. Check every column the same way you check every row.
Overlooking the elimination's ripple effects. After removing a candidate from box cells, re-scan for hidden singles, naked singles, and pairs. Eliminations frequently create new placements.
Frequently Asked Questions
What is box-line reduction?
When a candidate in a row or column is confined to one box, you eliminate it from other cells in that box. Also called "claiming."
How is it different from pointing pairs?
Opposite direction. Pointing pairs go from box to line. Box-line reduction goes from line to box. Same underlying logic.
When should I look for it?
After singles and before advanced techniques like X-Wing. Most useful in medium and hard puzzles.
Do I need pencil marks?
Yes. You need to see candidate positions within rows, columns, and boxes to identify the claiming pattern.