Box Line Reduction

Box line reduction is the mirror image of pointing pairs, and solvers who know one but not the other miss half the eliminations available to them. Here the line does the work: a digit trapped inside one box by a row or column is banned from the rest of that box.

What the pattern is

Pick a row and a digit still missing from it. Mark every cell of the row where that digit is a candidate. If all of those cells sit inside a single box, then the row's copy of the digit must land in that box. The box only gets one copy of any digit, so it is spoken for, and the digit can be deleted from the box's cells that lie outside the row.

Columns work the same way. The pattern is sometimes called claiming, because the line claims the digit from the box.

	Box line reduction	Pointing pair
What is confined	a line's copy of the digit, inside one box	a box's copy of the digit, inside one line
Cells you clear	the box cells off that line	the line cells outside that box
You scan	rows and columns	boxes
Same family	locked candidates	locked candidates

Scanning for one does not find the other, so make both sweeps part of the same routine.

When to look for it

After a pointing-pair sweep, using the same unit list but reading it the other way round.
On rows and columns that already have five or six digits placed, where the survivors bunch up.
On digits that appear in only two or three cells of a line.
Immediately after any elimination on a line, which can pull the remaining cells into one box.

How to apply it, step by step

Pick a row or column and list the digits it still needs.
For one digit, mark every cell of that line where it is still legal.
Check whether all those cells fall inside a single box.
If they do, erase the digit from the six cells of that box that are not on the line.
Repeat for the next digit, then the next line, covering all nine rows and all nine columns.
Re-run a singles sweep. Clearing a box this way very often reveals a hidden single in it.

A worked example

Example

Row 3.  x marks a cell where 4 is still a candidate:

    box 1      |    box 2      |    box 3
   c1  c2  c3  |  c4  c5  c6   |  c7  c8  c9
    -   -   -  |   -   -   -   |   x   -   x

  Row 3 needs a 4 and can only place it in box 3.
  So box 3 takes its 4 from row 3.
  ->  remove 4 from r1c7, r1c8, r1c9, r2c7, r2c8 and r2c9

Six cells lose a candidate from one observation, and none of the six is in row 3. That is the whole point: the eliminations happen off the line you were looking at.

Common mistakes

Erasing on the line. The cells of row 3 keep their candidates. It is the rest of box 3 that loses the digit.
Overlooking a stray cell. If the digit is still legal anywhere else in the line, in another box, the pattern does not hold.
Mixing up the two directions. Ask yourself which unit the digit is confined by, and clear the other one. Confined by a box means clear the line, confined by a line means clear the box.
Only sweeping rows. Columns produce just as many reductions and are easier to overlook on a wide screen.

Where you meet it

Box line reduction becomes necessary around medium classic Sudoku and appears constantly from Hard upwards in both classic and Killer grids. Together with pointing pairs it is the last basic technique that works on raw candidates alone, so master it before moving on to patterns that span three or more units.