Box/Line Reduction
Volodymyr Sakhan ·
Box/Line Reduction is an intermediate sudoku technique that lets you eliminate candidates from a box using information from a single row or column. When every remaining position for a digit in a row (or column) falls inside the same 3×3 box, that digit is "claimed" by the line — it must land inside that box, so it can be removed from all other cells in the box.
This technique is also called Locked Candidates Type 2 or Claiming. It is the mirror of Pointing Pairs: Pointing Pairs goes box → line, while Box/Line Reduction goes line → box. Both exploit the same intersection logic.
What Is Box/Line Reduction?
The rule is straightforward: scan a row or column for a specific digit. If every empty cell that can hold that digit belongs to the same 3×3 box, the digit is locked to that intersection. Because the digit must appear somewhere in the row or column, and all options are inside one box, it cannot occupy any other cell in that box.
You may see this technique listed under different names in books and software: Box/Line Reduction (SudokuWiki), Locked Candidates Type 2 (most textbooks), or Claiming (HoDoKu). All three names describe exactly the same move.
When to Use Box/Line Reduction
Apply Box/Line Reduction after you have exhausted naked and hidden singles:
- Medium and hard puzzles routinely require this step before further progress is possible.
- Pencil marks (candidate notation) must be filled in — you cannot apply the technique by eye alone.
- Run a Box/Line scan immediately after any Pointing Pairs scan — they cover the same intersection zones and often appear together in the same pass.
Step-by-Step Example
The two examples below use the same source puzzle transformed in different ways, so you can see both column-claiming and row-claiming in action.
Column-claiming — digit 5 in column B
Look at column B (the second column from the left) and focus on digit 5.
- Scan column B for digit 5: check every empty cell in the column.
- Only B1 (candidates 1, 5) and B2 (candidates 2, 5) can hold digit 5 in column B. Every other cell in the column already contains a digit or has 5 excluded by its row or box constraints.
- B1 is in column B, row 1 — that is box 1 (top-left, rows 1–3, columns A–C). B2 is in column B, row 2 — also box 1.
- Both options for digit 5 in column B are inside box 1. Column B has claimed digit 5 within box 1.
- Digit 5 must land somewhere in column B, and both options are in box 1 — so digit 5 cannot appear anywhere else in box 1.
- Find all empty cells in box 1 that are not in column B: A2 [2,5,6,7], A3 [1,5,7], C2 [2,4,5,6], C3 [4,5]. All four carry 5 as a candidate.
- Eliminate 5 from A2, A3, C2, and C3.
- C3 had only [4, 5] — after removing 5 it is left with [4] only, so C3 is immediately solved to 4.
Because digit 5 must land somewhere in column B — and both options sit inside box 1 — no other cell in that box can hold 5, netting an immediate solve in C3.
Row-claiming — digit 3 in row 2
Now look at row 2 (the second row from the top) and focus on digit 3.
- Scan row 2 for digit 3: check every empty cell in the row.
- Only D2 [3,6,7,8] and E2 [3,6,7,8] can hold digit 3 in row 2. All other row-2 cells are given or have 3 excluded.
- D2 is in column D, row 2 — that is box 2 (top-centre, rows 1–3, columns D–F). E2 is in column E, row 2 — also box 2.
- Both options for digit 3 in row 2 are inside box 2. Row 2 has claimed digit 3 within box 2.
- Digit 3 must land somewhere in row 2, and both options are in box 2 — so digit 3 cannot appear anywhere else in box 2.
- Find empty cells in box 2 not in row 2 that carry 3: E3 [3,4,5,6,7,8] and F3 [3,4,5,6].
- Eliminate 3 from E3 and F3.
The same logic as column-claiming, just rotated: when a row claims a digit within a box, every other row in that box loses that digit.
Pointing Pairs vs. Box/Line Reduction
Both techniques exploit the intersection between a box and a line. The only difference is where you start scanning. Pointing Pairs starts from a box; Box/Line Reduction starts from a row or column.
| Pointing Pairs (Type 1) | Box/Line Reduction (Type 2) | |
|---|---|---|
| Where you look | A 3×3 box | A row or column |
| What you find | Digit confined to one row/column within the box | Digit confined to one box within the row/column |
| What you eliminate | Digit from the rest of that row/column outside the box | Digit from the rest of that box outside the row/column |
| Direction | Box → Line | Line → Box |
Whenever you finish a Pointing Pairs scan, immediately run a Box/Line Reduction scan — they share the same intersection zones and often appear together in the same solving pass.
How to Scan for Box/Line Reduction
A systematic pass covers all 18 lines in a single sweep:
- Work through each digit 1–9 one at a time.
- For each digit, mentally (or visually) highlight all remaining candidate cells on the board.
- Scan each row: do all highlighted cells in this row fall within a single box? If yes, eliminate that digit from other cells in that box.
- Scan each column: do all highlighted cells in this column fall within a single box? If yes, same elimination.
- There are 18 lines to check (9 rows + 9 columns), so a thorough pass is fast and finds all instances.
Eliminations can trigger hidden singles or open up further intersection moves. After each elimination, check whether any cell in the affected box now has only one candidate remaining.
Practice Box/Line Reduction Online
Box/Line Reduction appears regularly in medium and hard puzzles once the simpler techniques are exhausted. Enable pencil marks on the board and try spotting a line that claims a digit inside a single box.
Once you can apply Box/Line Reduction reliably, the natural next steps are X-Wing — a two-row/two-column fish pattern — and Hidden Pairs, where two cells in a unit share two exclusive candidates.
Frequently Asked Questions
Ready to practice Box/Line Reduction? Play medium sudoku and try spotting a line that claims a digit inside a single box — or create a free account to track your progress.