Pointing pairs and triples

Pointing pairs and triples first appear in Medium puzzles on this site.

Two groups talking to each other

The singles and pairs covered so far work inside one group at a time. Pointing pairs are the first technique that links two groups: a box and a line. A box has to contain every digit. If, for one digit, the box's remaining possible spots all fall in a single row or a single column, then the digit's place in the box is somewhere on that line. The digit's place in the line must therefore be inside the box too, and every cell of the line outside the box can lose the candidate.

The diagram above is a real position from this site's Medium pool, drawn by the technique engine. Its caption says that in box 5, the digit 3 can only go in r4c4 and r5c4, which are both in column 4, so 3 can be removed from the rest of column 4. Boxes are numbered 1 to 9 in reading order, so box 5 is the centre box, and the cells are given row first, then column.

Pair or triple

The same logic holds whether the digit has two possible cells in the box or three. A "pointing triple" is exactly the same pattern with all three cells of a box's row or column still open for the digit. The only requirement is that every remaining candidate for the digit inside the box lies on one line. If they are spread over two lines, there is no pointing, and you are looking at a different pattern.

How to find them

1. Fill in candidate notes for the grid.
2. Pick a box and a digit that is not yet placed in it.
3. Look at the cells of the box that still show that digit as a candidate. Are they all in the same row, or all in the same column?
4. If so, scan along that line outside the box and delete the digit from every cell that carries it.

With practice you will do this while scanning for hidden singles: when a digit has two or three spots in a box, check whether they line up. A good time to look is straight after placing a digit, because each new entry blocks cells and leaves the remaining spots in a box clustered.

Why it works

Suppose the digit were placed in the line outside the box. Then it would sit in a cell that sees the box's line cells, and the box would have no place left for the digit, because all of its remaining spots were on that same line. That would leave the box without the digit, breaking the rule. So the digit cannot be anywhere on the line outside the box. The argument is a plain counting argument and does not depend on guessing.

Common mistakes

• Removing the digit from the box cells themselves. The candidate stays in the pointing cells; only the cells of the line outside the box lose it.
• Counting a spot that is already blocked. Make sure every cell you treat as available really still lists the digit. A stale note produces a false pointing pair.
• Using two lines. If the box's spots for the digit are in two different rows, no elimination is possible along either row.
• Looking only for pairs. The triple is just as common; check both.

The reverse view

Pointing pairs work from a box to a line. The opposite direction, from a line to a box, is box/line reduction. Both rest on the same fact, that a digit confined to the overlap of a box and a line must live in that overlap, but they remove candidates from different places. Learn them as a pair; many solvers treat them together under the name "locked candidates". This site keeps them as two entries because the hints name them separately and because the direction you look from changes how you find them.

Where the site's grading needs it

Pointing pairs are second-tier, together with box/line reduction and the pair techniques. In the committed frequency data over 1,000 stored classic puzzles per tier, a pointing pair or triple was used in 0 Easy puzzles, 952 Medium, 881 Hard, 806 Expert and 893 Evil puzzles. That makes it by far the most common step beyond singles in the sample. Each puzzle is solved cheapest technique first and a technique counts if used at least once. See how difficulty is graded and the frequency guide for the method and its limits.

In the variants

On the irregular board the same idea works between a jigsaw region and a line, though the region may overlap a line in an odd number of cells. On the diagonal board a box can also point along a diagonal when its remaining spots for a digit are on one. On the 6×6 board boxes are 2×3, so a digit confined to one row of the box is a very short pair.

Check your understanding

• In the top-left box, the digit 7 can only go in r1c1 and r1c3. What can you remove, and from where?
• In a box, 4 can go in r4c4, r5c5 and r6c4. Does it point?
• Why do pointing pairs often produce a hidden single afterward?

The answers: remove 7 from the other cells of row 1 outside the box; no, since the cells are in two columns and two rows; and once the digit is cleared from a line, another box that crosses the line may be left with one spot for the digit.

A worked example

Read the diagram the way the engine does. The centre box, box 5, still needs a 3. Of its nine cells, most are either filled or blocked for 3 by a 3 somewhere in their row or column. Only r4c4 and r5c4 remain as places for it, and both are in column 4. So the box's 3 is in column 4, and column 4 gets its 3 from the box. Every other empty cell in column 4, above and below the box, cannot hold a 3, and its note is erased.

The useful follow-up is to look at what that erasure changes. If one of the cleared cells was the only place for some digit in a row or another box, a hidden single may now be exposed. Pointing pairs seldom end a solve alone, but they very often start a chain: erase, find a single, place it, and let the new digit block more cells.

When to look for them

There are two natural moments. The first is after you have exhausted singles and have a full set of notes in front of you; box by box, ask where each digit can go. The second is straight after placing a digit, because that placement removes candidates from a box and may leave a missing digit's spots lined up. Players who check the freshly changed box for a pointing pair after every move tend to find them before the pattern is buried in clutter.

Practice

Play a Medium puzzle on the main board with notes on. After each placement, check the box you just changed for a digit confined to one line. If you stall, ask for a level-two hint, which will name the technique when it applies.

Previous: naked triples. Next: box/line reduction. Back to the technique list.