Box/line reduction

The same overlap argument as a pointing pair, run in the opposite direction: from the line into the box. Solvers who know one and not the other miss roughly half the eliminations available to them.

Needed from Hard onwards · practise on a hard puzzle

The idea

Take a digit and a line — a row or a column. If every square on that line which could hold the digit sits inside a single box, then the line's copy of that digit is somewhere in that box.

The box therefore has its digit accounted for by those squares, so the digit can be erased from the rest of the box — the six squares that are not on the line.

A worked example

Row 2 is filled from c1 to c6, so its missing 4 has to be in c7, c8 or c9 — all three of which are in the top-right box. Wherever it lands, that box's 4 is used up by row 2.

So the 4 comes out of every square in the box that is not in row 2, including r1c7 and r3c8. Again nothing has been placed, and again the eliminations are what let the next technique bite.

Telling the two directions apart

It is worth being deliberate about which one you are using, because the elimination lands somewhere different each time.

• Pointing pair — evidence is confined within a box; you erase along the line, outside the box.
• Box/line reduction — evidence is confined within a line; you erase inside the box, off the line.

Both come from the same fact: a box and a line meet in exactly three squares, and if a digit is pinned into that overlap, it is claimed by both units at once.

Common mistakes

The mirror of the pointing-pair error: erasing along the line instead of inside the box. Row 2 keeps its candidates — it is the box that loses them.

The other is forgetting that filled squares count. A line only needs its emptysquares confined to one box; a row with six givens like the one above reaches that state far sooner than it looks.

Where to go next

The next technique in the order is x-wing — Four squares in a rectangle that force a digit into two columns, clearing both of them elsewhere.

All nine techniques: the techniques index. New to the rules? Start here. Ready to use it? Open a board.