Box/line reduction

What box/line reduction is

Box/line reduction is the observation that if every remaining home for a digit in a row or column falls inside a single 3×3 box, that box must use the digit on that line — and so the digit can be eliminated from every other cell in the box.

It is also called a claiming pair, or line/box interaction. The names are unhelpfully many for a simple idea: a line has claimed the digit from the box.

How it differs from a pointing pair

The two techniques concern the same overlap — the three cells where a box and a line cross — and differ only in which one does the constraining.

In a pointing pair, the box is constrained and the eliminations land on the line, outside the box. In box/line reduction, the line is constrained and the eliminations land in the box, off the line. If you find yourself unsure which you are looking at, ask where the candidates you are about to cross out are: outside the box means pointing pair, inside it means box/line.

Finding them

Go line by line rather than box by box — that is the opposite sweep to pointing pairs, and doing both is what makes the pass complete. For each row and column, take the digits it is still missing and mark their possible cells. If they all fall within one box, you have one.

Lines with three or four empty cells are the productive ones. A line with seven empties is very unlikely to have a digit confined to three consecutive cells.

Where it fits

Box/line reduction sits with pointing pairs as the last technique before things get properly difficult. Together they solve most hard boards. When both have been exhausted and the grid still will not move, the next step is a pattern that spans the whole grid — the X-Wing — or one that chains cells together, the XY-Wing.