LeafyOrb

Sudoku techniques · Hard level

Box/line reduction: when a line claims a box

Box/line reduction happens when a digit's only places in a row or column all lie inside one box. The line must put the digit in that box, so the digit can be removed from the rest of the box.

How it works

Take row 8 and the digit 7. If the only cells in row 8 that can hold a 7 are all in box 9, then row 8's 7 is in box 9. Box 9 can only have one 7, so its other cells, off row 8, cannot have it.

It is the mirror image of the pointing pair, which starts from the box and clears the line. Some books call it "claiming" or a "line/box interaction". You need pencil marks to see it, and it often turns up on Hard puzzles once the pairs run out.

  • The row, column or box the move uses
  • The cells that make the pattern
  • Candidates the move removes
Box/line reduction example on a Sudoku board: In this line, 7 can only go in R8C7, R8C8 and R8C9, all in box 9. So 7 cannot appear anywhere else in box 9.

Example 1 · Row · from Hard puzzle #1308

In row 8, the 7 can only go in R8C7, R8C8 and R8C9, and they are all in box 9. Row 8 needs a 7, so it must come from box 9. That uses up the 7 for box 9, so no other cell in box 9 can have it. Remove 7 from R7C7, R7C8, R9C7 and R9C8.

Play this position in Ninebloom: Home › Your own puzzle, then paste the code. NG-16IR52-C5DTKM-LO3JZ1-CK615S-AN7JF3-BOL0UH-2XU9MC-KA1MDS-JIBDG

Cells are named by row and column: R4C3 is row 4, column 3. Boxes are numbered 1 to 9, left to right and top to bottom.

How to spot it

  1. Pick a row or column and a digit it still needs.
  2. Mark every cell on the line where that digit could go.
  3. If those cells all fall inside one box, the line claims that box for the digit.
  4. Remove the digit from the other cells of the box, the ones not on the line.

More examples

Box/line reduction example on a Sudoku board: In this line, 1 can only go in R4C1 and R6C1, all in box 4. So 1 cannot appear anywhere else in box 4.

Example 2 · Column · from Hard puzzle #264

In column 1, the 1 can only go in R4C1 and R6C1, and they are all in box 4. Column 1 needs a 1, so it must come from box 4. That uses up the 1 for box 4, so no other cell in box 4 can have it. Remove 1 from R4C2, R6C2 and R6C3.

Play this position in Ninebloom: Home › Your own puzzle, then paste the code. NG-24JMYL-M9358F-DM9WA8-TV6ZDW-Y90EQU-28YUW6-34W63Y-23ZJGS-DJDDS
Box/line reduction example on a Sudoku board: In this line, 8 can only go in R8C5 and R8C6, all in box 8. So 8 cannot appear anywhere else in box 8.

Example 3 · Row · from Hard puzzle #1424

In row 8, the 8 can only go in R8C5 and R8C6, and they are all in box 8. Row 8 needs an 8, so it must come from box 8. That uses up the 8 for box 8, so no other cell in box 8 can have it. Remove 8 from R7C5 and R9C6.

Play this position in Ninebloom: Home › Your own puzzle, then paste the code. NG-1UG2V1-02QDQI-TYSMHO-OIUS8P-3T41QO-8UN3LV-5AR1X9-12LF0H-HLBB4

Common mistakes

  • Removing the digit along the line. The elimination is inside the box, off the line.
  • Counting a line whose candidates are in two boxes. Every one of them must be in the same box.
  • Confusing it with a pointing pair. Box/line starts from the row or column; pointing starts from the box.

Questions

What is the difference between box/line reduction and a pointing pair?

A pointing pair starts in a box: the digit's spots in the box line up, so the line loses it outside the box. Box/line reduction starts in a row or column: the digit's spots on the line are all in one box, so the box loses it off the line.

Is box/line reduction the same as claiming?

Yes. "Claiming", "line/box interaction" and "box/line reduction" are names for the same move.

Which Ninebloom level uses box/line reduction?

Hard, together with naked triples.