Box-Line Reduction
A digit confined to one line within a box clears the box.
Box/line reduction — also called claiming or line-box interaction — is the mirror image of a pointing pair: instead of a box confining a digit to a line, a line confines a digit to a box. It fires when, within a single row or column, a candidate digit can only be placed in the cells that happen to fall inside one 3×3 box. The box/line reduction sudoku move then clears that digit from the rest of the box, because the line has already claimed one of the box's cells for it.
Worked example
In this board row 1 already carries the givens 1, 2, 3, 5, 6 and 7 across columns 4 to 9, so its three missing digits — 4, 8 and 9 — can only go in the empty cells R1C1, R1C2 and R1C3, all of which sit inside box 1. Take the 4: since row 1 can place it nowhere except those three cells, the 4 that box 1 needs is forced into its top row, wherever exactly it lands. That lets you delete 4 as a candidate from every other cell of box 1 — the whole of its second and third rows: R2C1, R2C2, R2C3, R3C1, R3C2 and R3C3.
When to use it
Reach for box/line reduction once naked and hidden singles have dried up and you are trimming candidates between houses — it works the same intersection as a pointing pair, so it pays to scan for both together. To spot it, run along each row and column and, for every digit still missing, ask whether all the cells that could take it lie inside one box; when they do — as the 4's only homes in row 1 sit entirely in box 1 above — that box loses the digit everywhere else. A row or column already crowded with givens is the easiest place to catch it, because its few gaps often collapse into a single box.
How to use it
- Find a line-locked digit. In a row or column, find a digit whose only candidate cells all fall inside a single 3×3 box.
- Reduce the box. The digit must lie on that line, so within the box it can only be in those line cells.
- Eliminate elsewhere in the box. Remove the digit from the other cells of that box.
Frequently asked questions
What is box/line reduction in Sudoku?
Box/line reduction is an intersection technique — often called claiming — where a candidate digit within a single row or column can only be placed in the cells that lie inside one 3×3 box. Because the line must put that digit somewhere and every remaining option is in the same box, the box is committed to holding the digit in one of those shared cells, so it can be removed from the rest of the box. In the example, 4 can only go at R1C1, R1C2 or R1C3 within row 1, so 4 is stripped from the other six cells of box 1.
How do I spot box/line reduction on the grid?
Work along one row or column at a time with pencil marks in place, and for each digit that line still needs, note every cell that could hold it. If all of those cells fall inside a single box, you have a box/line reduction and can clear the digit from the box's other cells. Lines packed with givens are the best hunting ground — here columns 4 to 9 of row 1 are all filled, which forces 4, 8 and 9 into the three box-1 cells R1C1, R1C2 and R1C3.
How is box/line reduction different from a pointing pair?
They use the same box-line intersection but run in opposite directions. A pointing pair starts inside a box — a digit confined to one of the box's rows or columns — and eliminates along that line where it crosses the neighbouring boxes. Box/line reduction starts with a line — a digit confined within a whole row or column to the cells of one box — and eliminates inside that box instead. Ask which region does the confining: in this board row 1 confines the 4, so the eliminations land in box 1.