Box/line reduction

Starting from the line instead of the box

A pointing pair starts in a box and reaches out along a line. Box/line reduction starts in a line and reaches into a box. Take a row, column, and one digit. If every cell where the digit can still go lies inside one box, then the digit's position in that line is one of those cells, and they all belong to the box. The box already has the digit placed in the line's part of it, so the rest of the box cannot hold the digit. Delete it from the other cells of the box.

The diagram above is a real position from this site's Medium pool, drawn by the technique engine. Its caption reads: in column 1, the digit 4 can only go in r1c1 and r3c1, which are both in box 1, so 4 can be removed from the rest of box 1. Notation is row first, then column, counted from the top left.

How to find them

1. Write in candidate notes.
2. Choose a row or column and a digit not yet placed in it.
3. List the cells of that line where the digit is still a candidate.
4. If all of them are inside one box, look at the box's other cells (the six cells outside the line) and erase the digit there.

It helps to say the rule aloud: "the row needs a 4, the row's 4 must be in this box, so the box's 4 is on this row." Once phrased that way, the other six cells of the box clearly cannot hold a 4.

Why it works

The line needs one copy of the digit, and every place it could go is inside the box. So the box's copy of the digit is the line's copy. A box has exactly one of each digit, and the line's copy uses it up. The box's remaining six cells, those that are not on the line, cannot also hold the digit. Like all the techniques in this series, this is a counting argument with no guess.

Telling it apart from a pointing pair

Both techniques use the overlap of a box and a line, so they are easy to confuse. The test is which group is the smaller set of candidates:

• If the digit's spots in the box all lie on one line, you have a pointing pair, and the digit leaves the line outside the box.
• If the digit's spots in the line all lie in one box, you have box/line reduction, and the digit leaves the box outside the line.

Sometimes both are true at once: the digit is confined to the same two or three cells from both viewpoints. Then the digit is fully fixed within the overlap, and both eliminations apply.

Common mistakes

• Erasing from the line instead of the box. In this technique the line keeps its candidates and the box loses its other ones.
• Missing a spot outside the box. If even one candidate for the digit sits in the line outside the box, the argument fails. Double-check the whole line.
• Working from stale notes. A candidate you forgot to remove can hide a valid reduction or invent a false one. Refresh notes after each placement.
• Stopping after one box. A single row can have the digit confined to one box, and the same column can be confined too. Check both directions.

Where the site's grading needs it

Box/line reduction is second-tier. In the committed frequency data over 1,000 stored classic puzzles per tier, it was used in 0 Easy puzzles, 301 Medium, 512 Hard, 407 Expert and 558 Evil puzzles. Compare that with pointing pairs, used in 952 Medium puzzles: the engine reaches for pointing first because it is listed first among equal-tier techniques, and box/line only steps in when a pointing pair is not available. Each puzzle is solved cheapest technique first and a technique counts if used at least once, so the figures depend on that ordering. See how difficulty is graded and the frequency guide.

In the variants

On the irregular board, the "box" is a jigsaw region, and a line can cross several regions, so the digit's spots in a line may well sit inside a single region. On the hyper board, a window overlaps rows and columns in the same way a box does, which opens more of these reductions. On the diagonal board a diagonal is a line, and a digit confined to a diagonal's overlap with a box behaves the same way.

Check your understanding

• In row 5, the digit 9 can only go in r5c1 and r5c3. What can you remove?
• In row 5, 9 can go in r5c2, r5c3 and r5c4. Does the technique apply?
• If a digit in a column is confined to one box, how many cells of the box lose the candidate in the best case?

The answers: 9 can be removed from the six cells of box 4 that are not in row 5; no, because r5c4 is in a different box; and up to six, the cells of the box outside the column.

A worked example

Read the diagram in the engine's order. Column 1 needs a 4. Going down the column, most cells are either filled or have a 4 in their row or box, so they are blocked. The only open cells that can still take a 4 are r1c1 and r3c1, and both sit inside box 1. The column's 4 is therefore in the top-left box. Box 1 only gets one 4, and the column has just used it, so the remaining cells of the box, in columns 2 and 3, cannot hold a 4. Their notes lose that candidate.

As with pointing pairs, the value is in the knock-on effect. Clearing a 4 from the middle of a box can leave another digit with a single spot, or turn a cell into a naked single. After any reduction, rescan the box and the rows that cross it before moving on.

When it is easiest to spot

Look for lines that are nearly full. A row or column with only three or four open cells gives a short list of candidate spots for each missing digit, and a digit whose spots all sit inside the same three-cell stretch of one box jumps out. On a fresh board the lists are long and the pattern is rare; in the middle of a solve it is common. This is also why box/line reduction is a good habit for the late-middle part of a puzzle, after most digits have been placed but before the hardest steps.

How the hint explains it

When a hint falls on this technique, level one highlights the line, level two names box/line reduction, and level three states the digit, the line, the box and the cells that lose the candidate, using the same wording as the diagram caption. The hint is built from the digits on your current board, with candidates worked out by the engine rather than read from your notes, so it only ever shows a reduction that is valid for the position in front of you.

Practice

Play a Medium puzzle on the main board. Once the singles are done, pick a row with few open cells and look at one digit at a time. If you cannot see the pattern, the level-one hint points at the group and the level-two hint names it. The daily puzzle is another good source of mixed practice.

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