Innies and Outies
Innies and Outies is the technique that turns the rule of 45 from a neat trick into a grid-cracking tool. Instead of one unit, you apply the 45-per-unit total to a whole region — a block of rows, columns or boxes — and use the mismatch to pin down the odd cell that pokes in or out.
Innies and outies defined
- An innie is a cell inside your region that belongs to a cage reaching in from outside — the cage's other cells are elsewhere.
- An outie is a cell outside your region that belongs to a cage that is otherwise fully inside.
In both cases you know two numbers: the region's true total (45 per unit) and the sum of the cages you're counting. The difference is the value — or combined value — of the innie or outie cells.
Rule of 45 in one box. The three cages inside sum to 9 + 18 + 11 = 38, so the lone innie (pink) — a cell belonging to a cage that reaches in from the box below — is forced to 45 − 38 = 7.
How to use it
- Choose a clean region. One box, or a stack of two or three, whose cage boundaries are tidy.
- Work out the region's total. 45 per unit: 45 for one box, 90 for two, 135 for three.
- Add the cages that sit fully inside. Keep track of any cage that crosses the boundary.
- Take the difference. Region total minus inside cages gives the sum of the innies; a similar count the other way gives the outies.
- Place or constrain. A single innie is solved outright; two or three give you a small sum to work with.
A worked example
Take the leftmost stack of two boxes — total 90. The cages entirely within them add up to 83, and a single cell of a border cage reaches in (one innie). Then:
90 − 83 = 7
The innie is a 7. Had two cells reached in, you'd instead know they sum to 7 — enough to reduce them to {1,6}, {2,5} or {3,4}, which you then narrow with cage combinations.
Common mistakes
- Wrong region total. Count the units carefully — 90 for two boxes, not 45.
- Double-counting a border cage. A cage is either fully inside (count it) or crossing (it's your innie/outie) — never both.
- Stopping at the sum. Once you know two cells sum to, say, 7, feed that into cage combinations to finish the job.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.