Lesson 8 of 9
Box/line reduction
The same overlap argument run the other way — from the line into the box.
What this grid does
- Starting digits
- 25 of 81
- Deductions to solve
- 66
- Uses of Box/line reduction
- 2
- Digits placed before it is needed
- 12
- First needed at
- deduction 20, in box 9 (R7C7 and R7C9, on the 3)
- Also calls for
- last free cell, cross-hatching, naked single, hidden single, naked pairs and pointing pairs
Measured on this grid by the same solver the hint button uses — how these puzzles are chosen.
Why this one is worth drilling
Only about 1 puzzle in 73 needs it. That is the whole reason this lesson exists: you cannot learn a deduction a random grid almost never asks you for, so it has to be set deliberately. That figure is measured rather than guessed: 1.4% of solvable 22-clue grids over 4,000 digs — about 1 in 73.
This one and pointing pairs are mirror images, and that is precisely what makes it hard to apply: having learnt to look from the box outwards, you have to look from the line inwards instead. Solvers who know both still routinely check only one direction.
Where it sits in the nine
Eighth, immediately after pointing pairs, and deliberately so: the two are mirror images, and the only reliable way to learn the pair is to practise them back to back. Separated, solvers reliably internalise one direction and forget to check the other — which looks like a stuck grid rather than a missed deduction.
Working it
Having swept the boxes on the previous lesson, sweep lines here. Take a row or column, take a digit it still needs, and check whether every square that could hold it falls inside one box. If so, the digit is confined to that overlap and can go from the rest of the box. Doing the two lessons back to back is the point: the pair of them is one habit, not two. Lines with the fewest remaining squares are worth checking first: a row missing three digits is far likelier to confine one of them to a single box than a row missing seven. And when you find one, note that the eliminations land inside the box, away from the line you reasoned about — the opposite direction to the previous lesson, which is exactly the thing that needs drilling.
The full explanation of box/line reduction covers the reasoning with diagrams.
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