Sudoku

Sudoku technique

Pointing pairs and box-line reduction

Every box overlaps each row and column that crosses it in three cells. When a digit is confined to one of those overlaps, you can clear it from somewhere else. There are two versions of the idea, and together they are called locked candidates. They need no notes beyond a single digit, and they unlock a lot of medium puzzles.

Pointing pairs

Scan a box for one digit and you may find two cells left for it, side by side in the same row. You don’t know which cell gets the digit, but you know it will be in that row, inside that box.

So the digit can’t appear anywhere else in that row. Erase it from the row’s cells outside the box. The two cells “point” along the row.

In this example the bottom-right box needs a 9. Its empty cell in row 7 can’t take one, because row 7 has a 9 already. The two empty cells in row 9 are the only places left. Whichever holds the 9, row 9 gets its 9 inside this box, so the cell in column 5 loses the 9 from its notes.

In the shaded box, 9 can only go in the two outlined cells, both in row 9. The red cell in the same row can’t be 9.

Pointing triples

The cells don’t have to be a pair. If a box has three places for a digit and all three are in one row or column, the same deduction holds.

Here the bottom-middle box needs a 4. Row 9 has a 4 already, so the box’s empty cell in that row is out, and the three cells in row 7 are all that remain. The 4 for row 7 will be inside this box. The cell in column 7, which had the notes 3, 4 and 5, drops the 4.

The 4 in the shaded box must be somewhere in row 7, so the red cell further along the row loses its 4.

Box-line reduction

Now run the idea the other way. Scan a row or a column for one digit. If every cell that can take it lies inside one box, then that box gets the digit from this line, and the rest of the box can’t have it.

In this position, column 4 needs a 2. Row 4 already has a 2, which rules out the cell there, and the only places left are in rows 1 and 3. Both are in the top-middle box. So the 2 in that box will be in column 4, and the box’s two other empty cells lose the 2 from their notes.

This is box-line reduction, also called claiming: the line claims the digit for itself inside the box.

In shaded column 4, the 2 can only go in the outlined cells, both in the top-middle box. The red cells in that box can’t be 2.

Telling the two apart

Both are about the three cells where a box and a line overlap. The difference is where you looked and where you erase.

  • Pointing: you scanned a box. The digit is stuck in one line, so erase it from the rest of the line.
  • Box-line reduction: you scanned a line. The digit is stuck in one box, so erase it from the rest of the box.

Pointing pairs are the easier of the two to see, because scanning boxes is what you were doing anyway. When a puzzle stalls, run through the rows and columns looking for box-line reductions.

How to spot them

  • While scanning, notice any box where a digit is down to two or three cells in a line. You can make the elimination on the spot.
  • It pays to mark those cells with a small note of the digit, so the line stays blocked when you scan again.
  • An elimination on its own places nothing. Check straight away whether it has left a hidden single in the line or box you just cleared.

When the same digit is locked this way in two different rows at once, the pattern has a name of its own: the X-Wing.