Pointing pairs and triples: the first technique of medium sudoku

The Sudokumania team ·

Guide cover: Pointing pairs and triples: the first technique of medium sudoku

In easy sudoku, scanning and singles carry you all the way to the last cell. In medium ones, sooner or later, comes the moment when they find nothing more: every number has at least two possible places, every cell at least two candidates. It's not the time to guess. It's the time for the first technique that doesn't write a number but removes one from some cells: the pointing pair, with its sibling, the pointing triple.

The idea

Every box crosses three rows and three columns. Suppose that in a box the 5 can only go in two cells, and that these two cells are in the same row. You don't know yet which of the two is the right one, but one thing you already know: the 5 of that row will be inside the box, in one of the two. So, in the same row, outside the box, the 5 can't go.

The two cells “point” along the row and remove the 5 from all the other cells they meet. If the possible cells are three, still in the same row, it's a pointing triple and the reasoning is identical. There can't be more, because a row crosses a box in three cells. And everything works the same way for columns.

These techniques are also called intersections, because everything happens in the three cells where a box and a row (or a column) cross.

A step-by-step example

9 6 4 7 5 2 6 9 6 4 1 2 5 2 5 6 7 9 3 7 9 2 3 4 5 6 6 1 8 9 7 3 8 4 5 9 1 6 6 9 8 3 6 1 3 1 1 1
In box 3 the number 1 can only go in the marked cells, all in row 1: so in row 1, outside the box, it can't go.

Here's how you get there:

  1. Pick a box and a number that isn't in it yet. It's the same start as scanning.
  2. Rule out cells as you do in scanning. The rows and columns that already contain that number cut across the box. In scanning you hoped that only one cell would be left; here two, or three, are left.
  3. See whether the cells left are lined up. If they're all in the same row, or all in the same column, you've found a pointing pair (or triple).
  4. Remove the number from the rest of the line. In the cells of that row outside the box the number can no longer go: if you use notes, erase it from there; if you don't, remember it when you scan the neighbouring boxes.

The elimination alone doesn't fill any cell. But often, soon after, scanning a nearby box or checking a cell finds a number that used to have two places and now has only one.

With pencil marks

With notes the pointing pair is easier to see. In a box, follow one number and look at where it appears in the notes: if all its notes are in the same row or in the same column, you've found your intersection.

4 8 3 6 3 6 2 9 1 9 6 3 8 7 8 3 2 1 6 5 9 4 9 6 1 4 5 2 3 5 4 2 3 9 6 1 2 3 6 7 8 7 6 3 2 6 3 7 2 9 5 1 1 1 1
In the notes of box 9 the number 1 appears only in column 8: it's removed from the notes of column 8 outside the box.

To spot it quickly:

  • follow one number at a time, as in scanning: it's easier to see where the 4 is in a box than to read all the notes together;
  • look for boxes where a number has few notes, two or three: they're the most promising;
  • after every elimination go back to the singles: that's usually where the new number turns up.

The most common mistakes

  • Forgetting a cell. Before removing the number, check the whole box: a single possible cell outside the row and the pair is gone. With notes, keep them up to date: a missing note makes you see a pair that isn't there, and the elimination that follows is wrong.
  • Removing it from the pair too. The number is removed only from the cells outside the box. In the cells of the pair it stays: one of the two will hold it.
  • Reading it backwards. Here the lined-up cells are the only places for the number in the box, and the number is removed from the rest of the row. If instead they're the only places for the number in the row, and they're all in one box, the number is removed from the rest of the box. That's the inverse technique, box-line reduction, which has a guide of its own.
  • Expecting a number right away. The elimination is a half step: don't write anything until a single tells you where it goes.

When you need it, on Sudokumania

Our solver tries the techniques from the simplest: first the singles, then the intersections, then the pairs, the triples and the others. A sudoku is medium when, to solve it, the solver uses an intersection at least once and nothing harder. So in every medium sudoku there comes a moment when the pointing pair, or its inverse, is the move that unlocks the grid. In hard ones you'll use it often, before getting to the more complicated techniques.

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