Pointing Pairs and Box/Line Reduction, Explained

A pointing pair happens when a digit's only candidates inside a box all sit in one row or column. That box will put the digit somewhere on that line, so you can remove it from the rest of the line outside the box. Box/line reduction is the reverse: when a digit's candidates in a row or column all fall inside one box, remove it from the rest of that box.

Both are called intersection techniques, because they use the overlap between a box and a line. They're usually the next thing to learn after naked and hidden singles, and they unlock a lot of Medium and Hard puzzles.

Pointing pairs (and triples)

Take the top-left box and the digit 4. The box doesn't have a 4 yet. Looking at your notes, 4 is only a candidate in two cells of the box, and both are in row 2.

The box must have a 4, and it can only be in row 2. So row 2's 4 is inside this box. That means no other cell in row 2 can be a 4: erase 4 from row 2 in the other two boxes along it.

The same works with three cells in a line (a pointing triple), and with columns as well as rows. The box is "pointing" along the line where its candidates sit.

Box/line reduction

Now flip it. Take row 5 and the digit 9. Row 5 still needs a 9, and in your notes 9 is only a candidate in two cells, both inside the middle-left box.

Row 5's 9 must land in that box. A box holds each digit once, so the box's 9 is in row 5. Erase 9 from the other cells of that box, the ones in rows 4 and 6.

Which is which?

• Pointing: start in a box, candidates line up, clean the line outside the box.
• Box/line: start in a line, candidates fit in one box, clean the box outside the line.

Mixing them up is the most common mistake. If you remember where you started looking, you clean up the other one.

How to find them

1. Go digit by digit. Select a placed digit so its candidates stand out, then look at each box that still needs it.
2. Ask: are this box's candidates all in one row or column? If yes, it's pointing.
3. Then scan rows and columns that are missing the digit. Are its candidates all in one box? If yes, it's box/line.
4. After each elimination, look for singles in the cells you touched.

You can often spot pointing pairs without full notes, just by scanning, which makes them one of the most useful techniques to learn early.

Practice

Do both lessons above, then Intersections: Mixed Practice, which makes you decide which one you're looking at. Next up are naked pairs and triples.