If you have read the article on Pointing Pairs, you are already halfway there. Box Reduction is exactly the same mechanism taken backwards.
Where the Pointing Pair reads a box and acts on a row, Box Reduction reads a row and acts on a box. Two techniques, one single idea: the interaction between a box and a line, taken in both directions.
How a row can lock down a whole box, the Domino33 method for spotting the pattern, and the simple trick that keeps you from ever mixing Box Reduction up with its mirror.
🎬 The Domino33 video on Box Reduction
Here is the full tutorial, with worked examples and the search method.
▶ English audio track and English subtitles: pick either from the ⚙️ icon.
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The row speaks to the box
The principle is the exact symmetric of the Pointing Pair:
If, inside a row, a digit can only go into cells that all sit in the same box, then that digit will necessarily occupy that box. It is therefore impossible in the rest of the box, outside that row.
The reasoning works identically with a column. That is why the technique is sometimes called Box/Line Reduction: the box is reduced by what the line imposes on it.
A worked example
Here are rows 4 to 6 of a grid. Look at where the 4 can go on the middle row:
The row under examination is highlighted · shaded cells are digits already placed
On that row the 4 has only two possible locations, and both of them sit in the centre box. The row's 4 will therefore be placed inside that box.
Consequence: the centre box's 4 is already reserved for that row. It becomes impossible in the other cells of the box.
The 4 is eliminated from the two other rows of the centre box
Three candidates vanish, and the box becomes far more readable. No digit has been placed, but the grid has moved forward, and this kind of elimination is often what unlocks a cell somewhere else.
Remember one sentence: you always eliminate in the unit you did not read. You read the box, you eliminate in the row, that is a Pointing Pair. You read the row, you eliminate in the box, that is a Box Reduction.
Part 2
Row or column, two or three cells
Like its mirror, the pattern comes in four equivalent variants:
- From a row: 2 cells of the row, both in the same box
- From a row: 3 cells of the row, all three in the same box
- From a column: 2 cells of the column, both in the same box
- From a column: 3 cells of the column, all three in the same box
A row crosses three boxes, and so does a column. The question to ask is therefore always the same: do all the possible locations fit inside a single one of those three boxes?
The digit must be impossible everywhere else on the row. If it keeps even one possible location in another box of the row, the pattern collapses and no elimination is allowed.
Part 3
The search method
Step 1, Take a well-filled row or column
Useful configurations turn up mostly in rows with only 3 or 4 empty cells left. With few digits still to place, the odds of them concentrating in a single box become high.
Step 2, Digit by digit, list the locations
For each digit still missing from the row, spot the cells where it is still a candidate. It is always the same counting work, the one already used for hidden pairs and hidden triples.
Step 3, See which boxes they fall into
A row is cut into three thirds of three cells, one per box. If all the locations of a digit fall in the same third, you have a Box Reduction.
Step 4, Eliminate in the box, then read again
Remove the digit from the six other cells of the box, the ones that are not on the row. Then go through the box again from the start: these eliminations very often create a hidden pair or a Pointing Pair.
Do not read the row cell by cell. Cut it mentally into three packets of three cells and look for a digit that survives in only one packet. That three-way split is the real reflex to acquire.
🎬 See the method in action
In the video several examples are worked through on real grids, with the search method laid out step by step.
Exercise
Your turn
Here is a map of the possible locations of the 4. It contains a Box Reduction.
Look for a row whose 4s all fit inside a single box.
Part 4
Practice quiz
Four questions to check what you have learned. Good luck!
❓ Frequently asked questions
Box Reduction, or Box/Line Reduction, happens when a digit can only go into cells of a row that all belong to the same box. That digit is then necessarily placed inside that box, which lets you eliminate it from the six other cells of the box, the ones that are not on the row.
The direction of reading. The Pointing Pair reads a box and eliminates in the row. Box Reduction reads a row and eliminates in the box. You always eliminate in the unit you did not read.
Up to six, the cells of the box that are not on the row under examination. In practice many of them are already filled or no longer hold that candidate, but the potential stays high.
Yes, in exactly the same way. A column also crosses three boxes. If all the possible locations of a digit in that column fall inside a single box, the elimination applies in the rest of the box.
📋 Summary
Box Reduction essentials
- Direction of reading: you read the row or column, you act on the box
- Setup: inside a row, a digit confined to the cells of a single box
- Action: eliminate that digit from the 6 other cells of the box
- Condition: the digit must be impossible everywhere else on the row
- The reflex: cut the row into three packets of three cells
- Mirror technique: Pointing Pairs, which read the box and act on the row
With Box Reduction you have just completed the six basic techniques: naked pairs, hidden pairs, naked triples, hidden triples, Box Reduction and Pointing Pairs. They are enough to solve the vast majority of grids up to level 5, and they are the mandatory step towards the advanced techniques.
📘 Go further with the Domino33 books
Box Reduction and every other technique on this site are covered step by step in the Domino33 memento, available in English.
