Box/Line Reduction
Box/Line Reduction is the Pointing Pair turned inside out: when a digit within a row or column can only go in one box, the digit can be removed from the rest of that box.
Learn the techniquePointing Pair
A Pointing Pair is a digit that within a box can only go in one row or one column. The box then points: the digit can be removed from the rest of the row or column outside the box.
See the technique in practice
Work through the examples step by step. Each step explains what you see on the puzzle and why the conclusion holds.
Example:
- We write candidate notes in all the empty cells on the puzzle. Now we look more closely at box 1 and at row 2.
- Look at digit 7 inside the box. It can only go in the highlighted cells, and they all lie in row 2.
- The box must contain digit 7 somewhere. No matter which of the highlighted cells gets it, the digit lands in row 2. The box thus claims the spot for 7 in the entire line.
- Outside the box, row 2 therefore cannot contain 7. We remove the digit from the notes in row 2, column 7 and row 2, column 9.
How to recognize the pattern
A Pointing Pair arises when a digit within a box can only go in one row or one column. The candidates in the box then line up, usually as two or three notes side by side. No matter which of the cells the digit ends up in, it takes up that line inside the box.
The box then points: the digit cannot go in the rest of the row or column outside the box, and those candidates can be removed. The pattern is easiest to see if you scan box by box and look for digits where the candidates line up.
Step-by-step procedure
- Choose a box and a digit that is missing from it.
- Mark the cells in the box where the digit can go.
- If all possibilities lie in the same row or column, the digit must land there inside the box.
- Remove the digit from the rest of the row or column outside the box.
Common mistakes
- Not removing inside the box. The eliminations apply to the line outside the box, while the candidates inside the box should remain untouched.
- Missing a third possibility. If one of the digit's possibilities lies outside the line, the box does not point, and no removals are allowed.
- Confusing the direction with Box/Line Reduction. The Pointing Pair goes from box to line, while Box/Line Reduction goes from line to box.
When do you need the technique?
At Medium level it is no longer enough to place digits directly. Now it is about removing candidates: when you can prove that a digit cannot go in a cell, the rest of the puzzle becomes easier. Write candidate notes, because that is the key to all techniques from here on.
Try it on your own puzzle
Enter your puzzle in the Sudoku Solver and it will find the next step and explain the technique behind it.
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