Pointing Pairs: A Deep Dive

Pointing pairs are the technique that turns hard sudoku from impossible into solvable. The pattern is the first one that uses the box rule to constrain rows and columns, instead of the other way around. It is the workhorse of every hard puzzle and the bridge to most expert puzzles.

Most stuck moments on a hard sudoku resolve themselves once you have looked for a pointing pair in every box. The technique is easy to learn and surprisingly satisfying to use.

Why the elimination works

The box must contain the digit somewhere. The candidate cells in the box all live on the same row, so the digit must end up on that row within the box. The rule of sudoku also says the row contains the digit exactly once. Therefore the digit is on that row inside the box and nowhere else on the row. Strike it from the rest of the row.

The same logic works with columns. If a digit's box candidates all sit on the same column, the digit is committed to that column within the box, and the rest of the column can have the digit eliminated.

How to scan for them

Walk each box one at a time. Inside the box, count the candidate cells for each digit. Digits with two or three candidates inside a single box are your pointing-pair candidates. If those two or three candidates all sit on the same row or column, you have the pattern.

The first three or four boxes on a hard puzzle often yield at least one pointing pair each. They cascade fast: each elimination usually surfaces a hidden single in the row or column.

The reciprocal technique

Box-line reduction is pointing pair's mirror twin. Where pointing pair starts from the box and eliminates from a row, box-line reduction starts from the row and eliminates from a box. The two cover the same shape from opposite directions, and any sudoku that has one has the other lurking somewhere.

Scan both at once. After every placement, ask: did this placement create a pointing pair or a box-line reduction in any adjacent unit? The discipline pays off across the rest of the solve.

The mistake that wastes time

Eliminating inside the box. The pointing-pair logic does not say anything about the rest of the box. It says the digit cannot go in the rest of the row (or column) outside the box. Inside the box, the digit is still a candidate in the cells you identified.

Beginners erase the candidates from the wrong place and accidentally make the puzzle unsolvable. Read the rule twice the first few times. Once it sinks in, the direction is automatic.

Where pointing pairs fail

If the box has only one candidate cell for the digit, that is a hidden single, place it. If the box has candidate cells across two rows and two columns, no pointing pair applies; you need a different technique. Pointing pairs are specific to the case of cells aligned on one row or one column.

Most hard puzzles include three to five pointing pairs across the solve. Most expert puzzles include one or two; once you are at expert level, the eliminations have already been done by pairs and triples earlier.

For the formal definition see the pointing pairs strategy page. The mirror technique is at box-line reduction. Try one on a hard puzzle after the easy singles run out.