The Killer Sudoku forty-five rule: innies and outies explained

The forty-five rule works because every complete Sudoku row, column or box contains 1–9, whose sum is forty-five. Compare that fixed total with cage totals to calculate cells left inside the region, called innies, or cells extending outside it, called outies.

Begin with one complete region

Choose a row, column or box. Its cells must contain every digit once, so the sum is fixed even when all cells are empty. A cage has a different fixed total. Comparing the two can reveal a digit without resolving any cage's internal arrangement.

Start with a region almost covered by whole cages. Whole means every cell of the cage lies inside your chosen region. A cage crossing its boundary cannot be treated as if its full total belongs inside.

Part of the calculation	Meaning
Region total	Forty-five for one full row, column or box
Whole interior cages	Totals that can be subtracted directly
Innie	A leftover cell inside the region
Outie	A cell outside the region included by selected cages

Draw the region boundary mentally before adding anything. The arithmetic is usually short; choosing exactly which cells are being counted is the real technique.

Worked innie example

Imagine a box in which complete interior cages cover eight cells and sum to forty. The ninth cell, X, belongs to a cage crossing into the next box. The box equation is forty plus X equals forty-five, so X is 5.

Notice what you did not need: the total of X's crossing cage. Only the eight covered cells and the fixed box total mattered. Once X is filled, subtract 5 from its cage target and reconsider the other cells in that cage.

If instead two inside cells are uncovered, the subtraction gives their combined sum. It does not automatically give either digit. A two-cell remainder totaling 7 permits 1+6, 2+5 and 3+4 before positional restrictions. Keep the result as a useful equation.

Worked outie example

Now select whole cages whose union covers every cell of a box and exactly one cell Y outside it. Suppose their targets add to fifty. The selected cells total fifty, while the box contributes forty-five. Y must therefore be 5.

Write the relationship as “selected cage total minus region total equals outside total.” This direction matters. Reversing the subtraction produces a negative answer and signals that the region and cage coverage have been mixed up.

The same idea works for a row or column. Geography does not change the equation: everything counted beyond the chosen complete region must be accounted for as an outie.

When both sides have leftovers

A general comparison may leave cells both inside and outside. Let I be the sum of uncovered interior cells, and O the sum of selected exterior cells. Then selected cage totals minus forty-five equals O minus I.

That is a relationship between sums, not a direct placement. If O minus I is 2 and I later becomes 3, O becomes 5. Until that additional information exists, do not fill either side by intuition.

Also check whether leftover cells share a constraint before forbidding repeats. Two innies in the same box cannot repeat. Two distant leftover cells may repeat if they share neither a row, column, box nor cage. A derived sum is not automatically a new cage.

Extend the region carefully

Two non-overlapping boxes total ninety. Several complete rows can also be added. Larger regions are useful when their cage boundaries are simpler than those of individual boxes, but they bring more opportunities to count a cell twice.

Do not add a row and a crossing box as if they were disjoint: their shared cells would be included twice. Beginners can avoid this by choosing only separate boxes or separate rows until they are comfortable with the accounting.

A short checking routine

1. Name the complete region and its fixed total.
2. List each selected cage exactly once.
3. Identify every uncovered inside cell and included outside cell.
4. Subtract in the direction matching your equation.
5. Check any resulting digit against its row, column, box and cage.

Practise first on one-cell innies and outies. Then try two-cell sums. A small deduction you can explain precisely is more useful than a large region with an uncertain boundary.

Where to play

Kudoku offers five difficulty tracks and cage-aware pencil notes for practising these deductions. Its interactive introduction covers cage sums alongside ordinary Sudoku constraints. Use manual notes to keep a derived sum distinct from a confirmed digit as you work through a puzzle.