Claiming in Sudoku: Box-Line Reduction Explained

The claiming rule

Claiming needs three things:

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

Every row and every column must contain each digit once. If the only cells of the line that can hold the digit are inside one box, then the digit of that line will be in that box. You do not know yet which of the cells gets it, but you know the box.

A box can also contain each digit only once. The line already provides the digit for this box, so no other cell of the box can hold it. Remove the digit from all cells of the box that are not in the line. The line claims the digit for its own cells, and this gives the technique its name.

The rule in one sentence

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

Claiming, box-line reduction and locked candidates

Claiming and box-line reduction are two names for the same technique. Books and solving programs use both. A third name covers two techniques together: locked candidates. Pointing is locked candidates type 1, and claiming is locked candidates type 2. In both, a digit is locked inside the three cells where a box and a line cross.

The two types differ only in direction. Pointing starts in a box and removes candidates in a line. Claiming starts in a line and removes candidates in a box. If you know one of them, you already know the logic of the other.

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 row 7 and only at the digit 8. Row 7 has three empty cells: R7C4, R7C8 and R7C9. R7C4 does not have 8 as a candidate, because column 4 already contains an 8. So an 8 is possible in only two cells of row 7: R7C8 and R7C9. Both cells are in box 9, the bottom right box. This is claiming on 8.

Remove the candidates

The 8 of row 7 will be in R7C8 or in R7C9. In both cases it is in box 9, so box 9 gets its 8 from row 7. No other cell of box 9 can be an 8. Five of those cells have 8 as a candidate: R8C7, R8C9, R9C7, R9C8 and R9C9. Remove 8 from all five. R7C8 and R7C9 keep their 8.

Use the result

Claiming places no digit by itself, but its eliminations often open the next move. Look at R8C7. Before the move it had two candidates, 7 and 8. Now the 8 is removed and only 7 is left. A cell with one candidate is a naked single: place 7 in R8C7.

Rows, columns, two cells or three

The example uses a row, but the rule is the same for a column. If all cells of a column that can hold a digit are inside one box, remove the digit from the cells of that box in its other two columns.

The number of cells can be two or three. A line and a box share exactly three cells, so the digit can have at most three possible cells in the pattern. With two cells or with three, the elimination is the same.

How to find claiming while solving

Claiming is harder to see than pointing, because you must check a line of nine cells and not a compact box. Complete pencil marks help a lot. Search in this order:

1. Take one row or column and one digit. Start with lines that have few empty cells. In the example, row 7 had only three.
2. Count the cells for that digit. If the digit has two or three possible cells in the line, continue. With four or more cells, they cannot all be in one box.
3. Check the box. If all the cells are inside the same box, you have found the pattern.
4. Remove the candidates. Remove the digit from every cell of that box outside the line.
5. Look for singles again. Check the cells that lost a candidate first. One of them may now have only one candidate left.

A quick check

A row or column passes through three boxes. If the digit is already placed in two of these boxes, the digit of the line must be in the third box. That is a claiming pattern.

Common mistakes

• Removing in the wrong place. Claiming removes candidates from the box, not from the line. If you remove from the line, you are using the pointing rule on a pattern that is not a pointing pair.
• Missing a cell outside the box. If the digit is also possible in a cell of the line that is in another box, there is no claiming and you cannot remove anything.
• Removing from the pattern cells. The cells where the line and the box cross keep the digit. Only the other cells of the box lose it.
• Incomplete pencil marks. One missing candidate in the line can make the pattern appear when it does not exist.

When to use claiming, and what comes next

Easy puzzles usually do not need claiming. Look for it in medium and hard puzzles when no single is left and no pointing pair helps. It is simple logic, so always try it before an advanced technique such as the X-Wing.

After each elimination, look for singles again. Then learn 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.