Cage-Line Reduction
When a compulsory cage digit can occupy only cells on one row, column or box, remove it from the rest of that unit.
What it is
First use the cage sum to find a digit that occurs in every compatible combination. That digit is unavoidable somewhere in the cage. Then check its actual cage-aware positions: if they all lie in one row, column or box, the digit is locked to that unit segment. The rest of the shared unit cannot contain it.
How to spot it
Look for a cage with several remaining combinations that all share one digit. Next, inspect where that shared digit can still go inside the cage. The pattern appears only when those possible cage cells all share one row, column or box.
Worked example
Step through the example. First the position, then the cage the argument rests on, then the candidates, and finally what follows from it.
Step 1 of 4: Starting position: the highlighted three-cell cage totals 22 and still has more than one compatible digit set.
- Pattern cells
- 8/8, 8/9, 9/9
- Digits
- 9
- Follows from it
- 8/2, 8/3, 8/7
Press a marked square to point it out on the board above.
Why it matters
The cage proves that the digit must occur; the candidate positions prove where it can occur. A Sudoku unit may contain each digit only once. Once every possible cage location is inside that unit, no cell outside those locations can also hold the digit.
Practise now
The practice puzzle is picked so that this technique is what gets the solve moving again.
Frequently asked questions
Must the digit appear in every cage combination?
Yes. A digit is locked only when every remaining compatible combination contains it. If one valid combination omits it, the cage does not yet guarantee that digit.
Why can a 22-cage force 9 without fixing every cell?
A three-cell 22-cage still has more than one digit set, but its compatible sets all contain 9. The other digits may vary; 9 cannot be absent.
Where may I eliminate the digit?
Only from the rest of the shared row, column or box. The cage establishes the digit and its possible positions; the ordinary Sudoku unit rule makes the outside candidates impossible.
Put it to use
Where you will meet it
Expert grids are the only place this pattern regularly decides the solve. Bring your notes.
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