Cage and Unit Overlap
Cage and unit overlap is what happens when a cage and a row, column or box share cells: the cage's arithmetic tells you a digit must live in that shared area, so the digit leaves everywhere else in the unit. It is the Killer version of locked candidates, with a sum doing the work of a box.
What the pattern is
Every cage sum can be reached only by certain digit sets. If a digit appears in every one of the sets still available, the cage must contain it somewhere. That is a must-have digit.
Now look at where it can sit. If all of its possible cells lie inside one unit, then that unit's copy of the digit is spoken for, and the digit is a dead candidate in the rest of the unit, even in cells the cage never touches.
The logic also runs backwards. If a digit's only remaining places in a unit all fall inside one cage, that cage must contain it, and every combination that lacks the digit can be thrown away.
When to look for it
- A cage of two, three or four cells whose combination list has shrunk to two or three options.
- A cage that spills across a box edge but keeps most of its cells in one line, or vice versa.
- A cage sum near the top or bottom of its range, since extreme sums force high or low digits into every combination.
- A unit where one digit is already down to two or three places and they all belong to the same cage.
How to apply it, step by step
- List the combinations the cage can still make, given its sum, its size and its current candidates.
- Find a digit present in every surviving combination.
- Mark the cells of the cage that can still hold that digit.
- Check whether those cells share a unit. All in one box, all in one row, or all in one column will do.
- Remove the digit from every other cell of that unit.
- Then run it the other way: for each digit in the unit, see whether its remaining places are all inside a single cage, and prune that cage's combinations accordingly.
A worked example: the cage tells the unit
Cage 13 covers E1, E2, F2 and G2. G2 is already 1, so E1, E2 and F2 must add up to 12. Watch which digit every remaining set shares.
Three different digits without a 1 reach 12 as 2+3+7, 2+4+6 or 3+4+5. No cell of the cage can be 6, which leaves 2+3+7 and 3+4+5. Both contain a 3, so the cage must hold a 3 in E1, E2 or F2. Those three cells all lie in box 4 (G2 is outside, in box 7), so box 4's 3 is spoken for. The 3 leaves D3, F1 and F3.
A worked example: the unit tells the cage
Here box 4 can place its 9 only in E3 or F3, and both cells belong to cage 17 (D3, E3, F3). Follow what that does to the cage's combinations.
Three digits reach 17 in seven ways, but only 1+7+9, 2+6+9 and 3+5+9 contain a 9, and none of them has an 8. So the 8 leaves D3, E3 and F3. In 1+7+9 the 9 sits in E3 or F3 and D3 cannot be 1, so D3 takes the 7. That means 7 can never be in E3 or F3.
Which direction are you using?
| Cage tells the unit | Unit tells the cage | |
|---|---|---|
| Starting point | The cage's combination list | A digit's places in the unit |
| Condition | A must-have digit, all its cells in one unit | All the digit's places lie in one cage |
| Result | Digit removed outside the cage | Combinations removed inside the cage |
| Demo above | Cage 13 and the 3 of box 4 | Cage 17 and the 9 of box 4 |
| Feels like | Pointing pairs | Box line reduction |
Common mistakes
- Using a digit that is only common to some combinations. It must appear in every surviving one, not most of them.
- Forgetting current candidates. A combination that the sum allows may already be impossible because a cell lost the digit it needed.
- Sloppy geometry. Two cells in the same box and a third in another box means the digit is not locked. Every possible cell must share the unit.
- Eliminating inside the cage. The digit is removed from the rest of the unit, never from the cage cells that carry it.
Where you meet it
The simple form, where a small cage sits entirely inside one box, turns up in easy Killer grids. The overlap version, where you must notice that only part of the cage can carry the must-have digit, belongs to the tough tier and starts to matter at hard Killer Sudoku. Once cages routinely cross box edges, this is the technique that keeps the grid from stalling.