Pointing Pairs

Every technique you have met so far works inside one house — a single row, column, or box. A pointing pair is different. It uses two houses at the same time, and the work happens where they cross.

Pick a box, and pick a digit that has not yet been placed in it. That digit has a handful of candidate cells inside the box — the empty cells that could still take it. Normally those cells are scattered around the box, and there is nothing to say. But every so often they all sit in one line: the same row, or the same column. When that happens, you can draw a firm conclusion.

The digit has to go somewhere in the box, and the only places left for it are on that one line. So the box will place that digit on the line — you do not yet know which cell, but you know the line. And a digit can appear only once in a line. Everywhere else along that line, outside the box, the digit is now impossible. Those are your eliminations.

The box, in effect, points the digit down the line. That is where the name comes from.

Work box by box. For each box, take the digits that are still missing from it, one at a time, and ask a single question: do all of this digit's candidate cells share a row, or share a column?

If they do, you have a pointing pair. Look along that line, outside the box, and remove the digit from every cell that still holds it as a candidate. If they do not — if the candidates spread across two or more rows and two or more columns — move on. There is nothing here for this digit.

It helps to have your pencil marks in place first, because the technique reads directly off them. A box with only two or three candidates for a digit is the most promising, since those are the ones most likely to fall on a single line. Do not overthink it. You are scanning for a box where a digit's options happen to line up, and that is a quick, mechanical thing to check.

Look at the grid above. We are working in box 1, the top-left box, and the digit we care about is 8.

Inside that box, 8 has been narrowed down to just two cells, both outlined in blue: r1c3 and r3c3. Notice what they have in common — they are both in column 3. Every place 8 could still go in box 1 lies in that one column. There is nowhere else in the box for it.

So box 1 is going to place its 8 in column 3, either at r1c3 or at r3c3. We cannot yet say which, and we do not need to. What matters is that the 8 for this whole column is being used up here, at the top.

Now run your eye down column 3, past the bottom of the box. Any 8 pencilled in further down cannot survive, because column 3 already has its 8 committed to box 1. In the grid that cell is shaded red with the 8 struck through — for instance r9c3. That candidate is gone. You have not solved a cell, but you have removed a real possibility, and removals like this are what open the next move.

It is worth being clear about why this is safe, because it can feel like you are asserting more than you know. You never claim that 8 goes in r1c3, nor in r3c3. Either is still open.

The certainty is one step back. Box 1 must contain an 8 somewhere — every box holds each digit exactly once. The only cells in box 1 that can still take an 8 are in column 3. Therefore the 8 in box 1, wherever it lands, is in column 3. And a column holds each digit exactly once too. So the column's single 8 is spoken for by box 1, and no other cell in column 3 may have it.

That chain does not depend on guessing. It rests only on facts you already trust: one of each digit per box, one of each digit per line. The eliminations follow with the same force as a placement, even though nothing was placed.

A pointing pair need not be exactly a pair. If a digit's candidates in a box fill three cells and all three sit in one line, the logic is identical — that is a pointing triple. Two cells or three, the box still points the digit along the line, and you still clear the rest of that line outside the box. The word pair is just the common case; do not let the count distract you.

This technique also has a mirror. A pointing pair reasons from a box out to a line. Run the same idea the other way — from a line into a box — and you have box/line reduction. There, a digit confined within a line to the cells of a single box lets you remove it from the rest of that box. The two are a natural pair, and once you are comfortable pointing a digit out of a box, the reverse reading comes quickly.

In everyday solving, reach for pointing pairs once naked and hidden singles have dried up. They rarely finish a puzzle on their own, but they chip away at candidates, and a puzzle that felt locked will often loosen after two or three of these. Scan each box, check each missing digit against its line, and take the removals as they come.