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 adjacent rows 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 — the cells of 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

1. Choose your region and write down its total: 45, 90 or 135.
2. Add every cage that lies completely inside the region, plus placed digits no such cage covers.
3. Subtract. The remainder is the pseudo-cage total for the leftover cells.
4. 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.
5. List every combination that reaches the total under that rule, then remove any candidate that survives in none of them.

A worked example

Rows 1 + 2  ->  2 x 45 = 90

  cages completely inside rows 1-2 .......... 76   (16 cells)
  cells left over ........................... r1c9, r2c1
  r1c9 + r2c1 = 90 - 76 = 14

  r1c9 and r2c1 share no row, column, box or cage, so a repeat is legal

  ways to make 14:  5+9  6+8  7+7  8+6  9+5
  -> both cells are limited to {5,6,7,8,9}
  -> 1, 2, 3 and 4 removed from r1c9 and from r2c1

The two rows hold eighteen cells and 90 points. Sixteen of those cells are covered by complete cages worth 76 together, so the last two must make up the missing 14. Neither cell can be 4 or less, because the partner would have to be 10 or more. Four candidates leave each cell, and no cage sum had to be enumerated at all.

Single-cell versus multi-cell

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.