What is box/line reduction?
Box/line reduction happens when every place a number can still go in a row or column falls inside a single 3 × 3 box. The line must hold the number in one of those cells, so the number can be removed from all other cells of the box.
The technique is also called claiming, and it is the mirror image of the pointing pair: there a box points along a line, here a line claims a box. Together they are the two types of locked candidates. In Marble Sudoku you follow one marble color along a row or column.
Use one in 3 moves
- 1Track. Follow one number along a row or column.
- 2Check. Are all its remaining spots inside a single box?
- 3Clear. Remove that number from the rest of the box.
Then look for singles where you removed it.
TRY IT · GUIDED EXAMPLE
Let the line claim
Only Sky is shown in color. Faded Sky marbles mark every hole where Sky could still go. Follow Sky along the glowing row, find the box it claims, and clear the way.
This is the real board from Marble Sudoku. The nine marble colors stand for the numbers 1 to 9.
- Track
- Clear
- Place
STEP 1
Let the line claim
Read the example in words
In row 7, Sky can only go in columns 4 and 6. Both holes sit inside the bottom-middle box, so row 7’s Sky belongs to that box. That removes Sky from row 8, columns 4, 5 and 6, and from row 9, column 5 — four holes in the box. One of them, row 9 column 5, was down to Teal and Sky; with Sky gone only Teal remains. That is a naked single. With numbers turned on, Sky is 9 and Teal is 4.
YOUR TURN · PRACTICE
Find the claiming patterns
Each round shows the spots for one color. Tap all of that color’s spots in a line when they fall inside one box, and watch them clear the rest of the box. On Easy the line glows. On Hard you find it yourself.
Round color:
Why box/line reduction works
Every row needs each number exactly once. If a row’s only options for a 4 are two cells inside the same box, then the row’s 4 sits in one of those cells, whichever it turns out to be. Both cells are in the box, so the box’s 4 is already spoken for by that row. Any other 4 in the box would leave the row without a place for its own 4, so it can be removed.
The same argument holds for columns, and it is the exact mirror of a pointing pair: there the box claims the line, here the line claims the box.
How to spot box/line reductions faster
- Use one number at a time. Pick a number and follow it along each row and column, the same scan you use for hidden singles.
- Watch lines with two or three spots. When a row or column keeps only two or three holes for a number, check whether they share a box.
- Check both directions. Pointing pairs and box/line reduction show up on the same scan: when a number is tight in a box or on a line, test both patterns before moving on.
- Expect a single next. Clearing the box often leaves one hole with a single candidate left.
Pointing pair vs. box/line reduction vs. naked pair
| Pointing pair | Box/line reduction | Naked pair | |
|---|---|---|---|
| Focus | One number inside one box | One number along one line | Two cells and their candidates |
| Pattern | The number’s box spots share a row or column | The number’s line spots sit in one box | Two cells share the same two candidates |
| Removes | That number from the rest of the line | That number from the rest of the box | Both numbers from the rest of the group |
| Needs notes? | Helpful, but you can scan it | Helpful, but you can scan it | Yes, usually |
Common mistakes
- Clearing along the line. That is pointing pairs. Here the line claims the box, so you remove the number from the rest of the box.
- Spots spread over two boxes. Every spot the line has for the number must sit in the same box. One stray spot breaks the claim.
- Forgetting the 3 × 3 intersection. A line crosses three boxes. The claim only clears inside the one box where all the spots sit, not along the rest of the line.
- Skipping the follow-up single. The removal is the setup. Look for the single it creates.
Box/line reduction FAQ
What is box/line reduction in Sudoku?
Box/line reduction is when a number’s only remaining spots in a row or column all fall inside one 3 × 3 box. The line must hold the number in one of them, so it can be removed from every other cell of that box.
How is box/line reduction different from a pointing pair?
They mirror each other. A pointing pair starts inside a box and clears the rest of a line. Box/line reduction starts on a line and clears the rest of a box. Together they are the two types of locked candidates.
Is box/line reduction also called claiming?
Yes. Claiming is the common second name for it. It is the second type of locked candidates, the mirror image of pointing pairs.
How many spots can the line have?
Two or three. A single spot is just a hidden single, and spots spread across two boxes claim nothing.
Does this work in Marble Sudoku?
Yes. Follow one marble color along a row or column. If every hole where it could go sits inside one box, remove that color from the rest of the box.
When should I use box/line reduction?
Whenever a row or column keeps a number waiting on only two or three holes. It pairs naturally with pointing pairs, and the removal often reveals a naked single.
PRACTICE IN THE GAME
Use box/line reductions in a real puzzle
When a tough puzzle stalls, follow one color along each row and column. A line that keeps its color inside a single box is often the move that opens everything up.
Next techniqueNaked triples
