Intersections

Pointing Pairs and Box/Line Reduction

Every box shares its cells with three rows and three columns, and that overlap is a tool. Two techniques — pointing pairs and box/line reduction — mine it to remove candidates that nothing else will.

The idea behind intersections

A box and a line cross in three cells. Anything true of a candidate in that shared strip is true in both units at once, and that double membership is what makes the eliminations work. Neither technique places a number directly; both strip candidates, and those removals routinely expose the singles and pairs that finish the job. Think of them as the crowbars that pry a stuck board open. What makes them so reliable is that they ask for no guesswork at all: you are simply reading a constraint the grid already contains but has not made obvious. Every elimination they produce is a certainty, which is exactly why they compose so neatly with the singles and pairs that follow them.

Pointing pairs (and triples)

Look inside a single box at one candidate — say the 4. If every cell where the 4 can still go in that box lies in the same row, then wherever the 4 ends up, it will be somewhere along that row inside the box. That means the 4 cannot appear in that same row anywhere outside the box, so you strike it from the rest of the line. The candidate "points" out of the box along the row (or column) and clears it. Two such cells make a pointing pair; three in a line make a pointing triple, working identically.

Box/line reduction

Box/line reduction runs the same logic in reverse. Now look along a single row or column at one candidate. If every cell where that candidate can still go in the line falls inside one box, then the candidate is locked into that box — so you remove it from the box's other cells, the ones outside the line. Where a pointing pair clears a line using the box, box/line reduction clears a box using the line. Two sides of one coin, and it pays to know both directions.

Telling the two apart

The names blur together until you fix the direction in mind. Pointing pair: the constraint lives in the box, and you eliminate along the line. Box/line reduction: the constraint lives in the line, and you eliminate inside the box. A quick way to keep them straight is to ask where the candidate is trapped, then eliminate in the other unit. Get the direction wrong and you will "eliminate" a candidate that is perfectly legal, which quietly breaks the grid.

A worked pointing pair

Take the top-left box and the candidate 7. Suppose the only cells in that box where a 7 can still go both sit in the box's top row. You do not know which of the two holds the 7, but you know it is one of them — and both are in that row. Therefore the 7 cannot live anywhere else along that row, in the two boxes to the right. Sweep across and delete the 7 from those cells' notes. Frequently one of them was clinging to the 7 as its second-to-last candidate, and removing it leaves a naked single that drops straight in.

When to reach for them

These are the natural next step once pairs stop yielding and before you go hunting X-Wings. They need a full candidate map, so pencil the grid completely first, then walk each box and each line asking whether a candidate is confined to a single shared strip. Practice the eye on Medium Sudoku, where intersections appear often, and lean on them on Expert Sudoku. The classic board is always at Sudoku AZ.