Box/Line Reduction
When all candidates for a digit in a row or column fall within the same block, that digit can be eliminated from other cells in that block.
Box/Line Reduction applies when all candidates for a specific digit in a row or column are confined to a single block. The digit can then be eliminated from other cells in that block (those not in the same row/column). This is the reverse operation of Pointing Pair.
Relationship to Pointing Pair
Pointing Pair eliminates in the "block to row/column" direction, while Box/Line Reduction eliminates in the "row/column to block" direction. The logical basis is the same - constraint intersection - but the starting point of the scan differs. Being aware of both directions ensures you never miss an elimination opportunity.
Two Directions
| Direction | Where the candidates are confined | Where you can eliminate |
|---|---|---|
| Pointing direction | To one row or column inside a single block | Cells on that row or column that sit outside the block |
| Claiming direction | To one block inside a single row or column | Cells in that block that sit off that row or column |
Both rest on the same fact: the digit can only go where the line and the block overlap.
The technique is sometimes split into two named directions. Pointing direction: candidates in a block confined to one row/column allow elimination in other blocks along that line. Claiming direction: candidates in a row/column confined to one block allow elimination from other rows/columns within that block. Both use the same logical principle of constraint intersection.
Usage at Hard Level
Box/Line Reduction is frequently needed at Hard difficulty. Along with Naked Pairs, it is one of the key techniques for breaking through the Hard-level barrier. With accurate pencil marks, it is relatively easy to spot. It places no digit directly; by reducing candidates it indirectly recreates the conditions for single-type techniques to fire again. In the board below, the only cells of row 5 that can take a 6 both sit inside the middle-right block. One of those two cells has to be the 6, so every other empty cell of that block can drop the 6 from its candidates.
- The row being scanned (row 5)
- The block the 6 candidates are confined to
- The only two cells in row 5 that can hold the 6
- Cells that lose the 6 as a candidate
Once the 6 is gone, r4c7 and r4c8 are left with a single candidate each: 3 and 1.