Box-line reduction (claiming)

What is box-line reduction?

Box-line reduction, also called claiming, is the pointing pair turned around. Look along a row or column instead of inside a box. If every possible place for a digit on that line falls inside one box, the line’s copy of the digit is in that box.

That copy is also the box’s only copy, so the digit can be removed from every other cell of the box — the six cells that are not on the line.

How to spot box-line reduction

Pick a row or column and a digit it is still missing, and look at where the digit can go along the line. If all of the candidate cells sit in one of the three boxes the line passes through, the line has claimed the digit for that box. Now look at the rest of that box and cross the digit out wherever it appears as a candidate.

Nearly full lines are the best place to look, because a line with two or three gaps often has them bunched into one box.

A worked example

Cells are named by row and column: r4c1 is row 4, column 1. Rows count from the top, columns from the left, and boxes are numbered 1 to 9 reading like a book, so box 1 is top-left and box 9 is bottom-right.

In this position from a Hard puzzle in our game, neither kind of single is available. Look at column 7 and the digit 4. Column 7 has three empty cells — r5c7, r7c7 and r9c7 — and a 4 can only go in r7c7 or r9c7. Both are in box 9, the bottom-right box.

So column 7’s 4 is in box 9, and it is the only 4 that box 9 gets. Every other cell in the box loses 4 as a candidate: r7c8, r7c9, r9c8 and r9c9. That turns r7c9, which held 4 and 7, into a naked single: r7c9 is 7.

Why it works

Column 7 must contain a 4, and both places it can go are inside box 9. Box 9 can contain only one 4. So the 4 that column 7 needs is the 4 that box 9 gets, and a 4 in any of the box’s other cells would make two. As with a pointing pair, you never learn which of the two cells is right — only that the answer is on the line.

Which difficulty levels need it

Hard and Expert. Hard puzzles are graded so that singles run out before the grid is full, and this, a pointing pair or a naked pair has to break the deadlock.

It is the one our solver reaches for most: it used box-line reduction in 194 of the 199 Hard puzzles we sampled, because it checks rows and columns before boxes. Most of the time a pointing pair would have done the same job, which is why only 11 of those puzzles could not be finished without it. Expert puzzles need this and more.