sudokly
Intermediate technique

Box-Line Reduction

Box-line reduction is the mirror of pointing pairs. When all the candidate cells for a digit within a row or column lie inside a single 3x3 box, the digit must go in that box, so it can be eliminated from the rest of the box.
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Box-line reduction is the mirror of pointing pairs. Where a pointing pair uses a box to constrain a line, box-line reduction uses a line to constrain a box. Same shape, opposite direction.

How it works

Pick a row (or column) and a digit that has not been placed in it. List the empty cells in that row where the digit can still go. If every one of those cells sits inside the same 3x3 box, the digit has to land in that box on that row.

Since the digit is committed to that box-row intersection, the digit cannot appear in the rest of the box. Erase it from the other six cells of the box. That cleanup nearly always exposes a single elsewhere.

44✗✗✗✗✗✗4 IN ROW 5 LIVES ONLY IN THE LEFT BOX
In row 5 the digit 4 can only go in the leftmost box. So 4 is eliminated from the box's other cells.

When to look for it

Alongside pointing pairs. After pencil marks are complete, walk each row and column one digit at a time. If the digit's candidate cells in a line all share a single box, you have a box-line reduction.

The pattern shows up most often in puzzles where boxes are sparsely populated but lines are dense. Hard and expert puzzles fit that profile.

Step-by-step example

  1. Pick a row that still needs the digit 4. Find every cell in the row where 4 is a candidate.
  2. Check which box each candidate cell belongs to. If all candidates fall inside the same box, you have a box-line reduction.
  3. The digit must land somewhere in those candidate cells.
  4. Erase the digit from every other cell of that box (the six cells outside the row).
  5. Re-scan the box. Box-line reductions frequently produce a hidden or naked single inside the cleaned-up cells.

Box-line reduction vs pointing pair

Pointing pairBox-line reduction
Start fromA boxA row or column
RestrictionBox restricted to one lineLine restricted to one box
Eliminations targetRest of the line outside the boxRest of the box outside the line
Scan styleBox-firstLine-first
Two views of the same box-line overlap.

Tips for spotting it

  • Scan each row and column one digit at a time. The pattern jumps out when candidates are pre-marked.
  • Pair this with pointing pairs so you sweep every box-line overlap.
  • If a row has two or three candidate cells for a digit, check whether they sit in the same box. That is the trigger.
  • Apply eliminations immediately. New singles often appear in the cleaned-up box.

Common mistakes

  • Eliminating from the row instead of the box. The technique sweeps the box.
  • Missing the pattern when only two candidate cells share a row but they are in different boxes.
  • Stopping at the find. Apply the eliminations and re-scan for singles.
  • Skipping pencil marks. The pattern is invisible without them.

Practise it

Play hard sudoku and start every puzzle with a full pencil-mark pass. Scan rows for box-restricted digits before reaching for X-wing. Most hard puzzles yield to pointing pairs and box-line reduction alone.

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