Multi-Cell Innies and Outies
Multi-cell innies and outies is the grown-up version of the 45 rule: instead of one leftover cell, you work out the combined total of two or more, then treat that group as a cage that was never drawn on the grid.
What the pattern is
Every row, column and box contains 1 to 9 once each, so every unit totals 45. Two rows (or two columns) total 90. Three total 135. A box and a row that do not overlap total 90 as well.
Pick a region built from whole units. Add the sums of the cages lying entirely inside it. The difference from the region's total belongs to the cells left over, which belong to straddling cages. When exactly one cell is left over you get a digit. When two or more are left over you get a pseudo-cage: a group of cells with a known sum but no printed outline. Enumerate it exactly as you would a real cage, and strike the candidates that appear in no valid split.
When to look for it
- A row, a column, a box, or a band of two or three of them is covered by whole cages except for two or three stray cells.
- Two or more cages straddle the boundary, so single-cell innies and outies do not apply.
- The leftover sum is extreme for the number of cells: a total of 7 across two cells, or 23 across three, cuts candidates hard.
How to apply it, step by step
- Choose your region and write down its total: 45, 90 or 135.
- Add every cage that lies completely inside the region, plus placed digits no such cage covers.
- Subtract. The remainder is the pseudo-cage total for the leftover cells.
- Decide whether those cells can repeat a digit. Cells that share a row, a column, a box or a cage must differ; cells that share nothing may hold the same digit.
- List every combination that reaches the total under that rule, then remove any candidate that survives in none of them.
A worked example
Look at columns 1 and 2 together.
Each column totals 45, so the two together total 90. Three cages lie completely inside them: the 29 cage (B1, B2, C1, D1, E1, F1), the 22 cage (D2, E2, F2, G2, H2) and the 20 cage (G1, H1, J1, J2). They cover 15 cells and add up to 29 + 22 + 20 = 71.
Three cells are left over: A1 and A2 (in the 26 cage along row A) and C2 (in the 25 cage B3, C2, C3, D3). Together they make 90 - 71 = 19, a pseudo-cage with no printed outline. C2 can only be 6, 8 or 9, so A1 + A2 is 13, 11 or 10. A 1 in A1 or A2 would need the other to be 12, 10 or 9, and neither cell holds a 9. So 1 leaves both A1 and A2.
Here the three cells all see each other (row A, column 2 and box 1), so they must differ. Leftovers that share no unit may repeat a digit, and that case needs extra care when you list the combinations.
Single-cell versus multi-cell
| Single-cell innies and outies | Multi-cell version | |
|---|---|---|
| Straddling cages | One | Two or more |
| Leftover cells | One | Two or three, usually |
| Result | An exact digit | A sum you enumerate |
| Digits may repeat | Not applicable | Yes, when the cells do not see each other |
| Effort | Arithmetic only | Arithmetic plus a combination list |
Common mistakes
- Assuming the leftovers must differ. Two innies in different rows, columns and boxes are free to share a digit. Forbidding the repeat throws away real solutions.
- Mixing overlapping units. A row and a box that share three cells do not total 90. Only disjoint units add cleanly.
- Chasing large groups. Four or five leftover cells rarely cut anything. Two or three is the sweet spot.
- Forgetting to recheck. Each placement changes the inner total, so a region that yielded nothing an hour ago may yield now.
Where you meet it
This is tough-tier logic. Easier Killer grids are laid out so that single leftovers appear early and often, but from hard Killer Sudoku upwards the setters deliberately arrange two straddling cages per box so the simple form fails. Learning to read a band of two rows as one 90-point region is the step that keeps those puzzles moving.