Pointing Pairs and Box-Line Reduction Explained

The pointing pair rule

A pointing pair needs three things:

1. Choose one box and one digit that is not yet placed in that box.
2. Find every cell of the box where the digit is still a candidate.
3. Check that all of these cells are in the same row or in the same column.

Every box must contain each digit once. If the only cells of the box that can hold the digit are in one line, then the digit of that box will be in that line.

A row or a column can also contain each digit only once. The box already provides the digit for this line, so no cell of the line outside the box can hold it. Remove the digit from all of those cells. The cells inside the box keep their candidates.

The rule in one sentence

One box, one digit, all its cells in one line: remove the digit from the rest of that line.

A real example, step by step

This position comes from a real puzzle. Every empty cell shows its candidates, the digits that its row, its column and its box still allow.

Find the pattern

Look at box 3, the top right box, and only at the digit 7. The box has four empty cells: R1C7, R1C8, R2C9 and R3C9. R1C7 and R1C8 do not have 7 as a candidate, because row 1 already contains a 7. So a 7 is possible in only two cells of box 3: R2C9 and R3C9. Both cells are in column 9. This is a pointing pair on 7.

Remove the candidates

The 7 of box 3 will be in R2C9 or in R3C9. In both cases it is in column 9, so column 9 gets its 7 inside box 3. No cell of column 9 below box 3 can be a 7. Two of those cells have 7 as a candidate: R5C9 and R8C9. Remove 7 from both.

Use the result

A pointing pair places no digit by itself, but its eliminations often open the next move. Look at row 5. Before the pointing pair, a 7 was possible in two cells of row 5: R5C1 and R5C9. Now R5C9 has lost its 7, so R5C1 is the only cell of row 5 that can hold a 7. That is a hidden single: place 7 in R5C1.

Pointing triples

The rule does not depend on the number of cells. If a digit has three possible cells in a box and all three are in the same row or column, the pattern is called a pointing triple. The logic and the elimination are the same as for a pointing pair.

A box and a line share exactly three cells, so a pointing pattern has two or three cells, never more. The pattern can point along a row or along a column. Check both directions.

Box-line reduction: the same idea in reverse

Box-line reduction, also called claiming, uses the logic of a pointing pair in the opposite direction. Pointing starts in a box and removes candidates in a line. Claiming starts in a line and removes candidates in a box. When all cells of a row or column that can hold a digit are inside one box, the digit of that line will be in that box. You can then remove the digit from the other cells of the box, the ones that are not in the line. Together, the two techniques are called locked candidates. The guide to claiming explains the rule with its own real example.

How to find pointing pairs

Pointing pairs are much easier to see with pencil marks, the small candidate digits written in the empty cells. Snyder notation, which marks a digit only when it has two possible cells in a box, shows many pointing pairs directly. Search in this order:

1. Take one box and one digit. Start with boxes that have few empty cells.
2. Count the cells for that digit. If the digit has two or three possible cells in the box, continue. With four or more cells, they cannot all be in one line.
3. Check the line. If all the cells are in the same row or the same column, you have a pointing pair or a pointing triple.
4. Remove the candidates. Remove the digit from every cell of that line outside the box.
5. Look for singles again. As in the example, one elimination can leave a digit with only one possible cell in a row, a column or a box.

Common mistakes

• Missing a third cell. If the digit is also possible in a cell of the box that is not in the line, there is no pointing pair and you cannot remove anything.
• Removing inside the box. The two cells of the pair keep the digit. Only cells of the line outside the box lose it.
• Mixing it up with a naked pair. A naked pair is two cells that contain the same two candidates and nothing else. A pointing pair is about one digit, and its two cells may contain many other candidates.
• Incomplete pencil marks. One missing candidate can make a digit look as if it has two cells in the box when it really has three.

When to use pointing pairs, and what comes next

Easy puzzles usually do not need pointing pairs. Look for them in medium and hard puzzles when no single is left, and always before an advanced technique. Some positions that seem to need an X-Wing can be solved with a pointing pair.

The next techniques to learn are the naked pair and the hidden pair, which work with two digits in two cells. To see where each technique belongs, read the full list of sudoku techniques.