Killer Sudoku
Innie–Outie Difference
The innie–outie difference is the rule of 45 with one unknown on each side. In a house, add up the cages wholly inside; one more cage sticks out by a single cell — the outie — and the house has a single cell no counted cage covers — the innie. The 45 then fixes outie minus innie. Know one of the two cells and the other follows; know neither and both lose candidates that cannot keep that gap.
What you need to see on the grid
Look for it when a house is covered by cages except for a cage that pokes one cell out and another that pokes one cell in. Four terms carry the method:
- Outie
- The one cell, outside the house, of a cage that otherwise lies inside it.
- Innie
- The one cell of the house that none of the counted cages covers — here a cell of a cage that lies mostly outside.
- The difference
- Outie minus innie = (cages inside) + (the outie's cage) − 45. It can be positive, negative or zero.
- Pairing up
- With both cells open, each keeps only candidates that have a partner at the right distance in the other cell.
How to apply it, step by step
One example runs through all three steps, early in an expert Killer puzzle from the app, at the exact point where its hint needs the innie–outie difference. Every diagram shows the whole grid; the cells a step needs are drawn, the others hatched. The dashed outlines are cages, their sums in the corner. No pencil marks are needed: the arithmetic does all the work.
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Step 1Total the cages that sit wholly inside the house
Pick a house and add up the cages entirely inside it. In box 1 that is the 18-cage on R1C3, R2C2, R2C3 and R3C3.
Box 1 and the one cage wholly inside it: 18. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 9 18 R2 R3 6 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Box 1
- Cage, sum in its corner
- Not needed for this step
Filled in: box 1; the cage. The other 72 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 2Find the one cell that sticks out and the one cell left in
The 23-cage covers four cells of box 1 (R1C1, R1C2, R2C1, R3C1) and one cell below it, R4C1: that cell is the outie.
The 14-cage covers R3C2 and R4C2. Only R3C2 is in box 1, and no counted cage covers it: R3C2 is the innie. It already holds 6.
The 23-cage pokes out at R4C1 (outie); the 14-cage pokes in at R3C2 (innie). C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 23 9 18 R2 R3 146 R4 8 R5 R6 R7 R8 R9 - Digit already on the grid
- Box 1
- The outie R4C1 (the innie R3C2 holds 6)
- Cage, sum in its corner
- Not needed for this step
Filled in: box 1; the 3 cages. The other 70 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 3Work out outie minus innie from the rule of 45
Box 1 adds up to 45, and it is made of the 18-cage, the 23-cage without its outie, and the innie: 18 + (23 − R4C1) + R3C2 = 45, so R4C1 − R3C2 = 18 + 23 − 45 = −4. The outie is always 4 less than the innie.
The innie R3C2 is 6, so the outie R4C1 is 6 − 4 = 2 — the app's next hint in this puzzle. Had neither cell been filled, you would instead keep only pairs 4 apart: R3C2 could not be 1 to 4, and R4C1 could not be 6 to 9.
Technique: Innie–outie difference
18 + 23 − 45 = −4, and R3C2 = 6: R4C1 = 6 − 4 = 2. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 23 9 18 R2 R3 146 R4 2 8 R5 R6 R7 R8 R9 - Digit already on the grid
- Box 1
- The outie R4C1 (the innie R3C2 holds 6)
- Digit placed
- Cage, sum in its corner
- Not needed for this step
Filled in: box 1; the 3 cages. The other 70 cells are left blank: they are not needed for this step. Placed: R4C1 = 2.
Common mistakes
Getting the sign the wrong way round
The outie is added through its cage and taken back out; the innie is added on its own. Write the equation out — 18 + (23 − outie) + innie = 45 — rather than guessing which side is bigger. Here a wrong sign gives R4C1 = 10, which cannot be.
Two cells sticking out, or two left in
The method needs exactly one outie and one innie. With two of either, you only learn their sum, not a single cell — that is a different, weaker deduction.
Using 45 for a region of two houses
The same idea works over two neighbouring rows or columns, where the cages can reach further — but that region adds up to 90, not 45. Use 45 for each house the region spans.
Keep going
- Innies & OutiesA row, column or box totals 45: take away the cages inside it, and the cells left over must make the rest.
- Cage SplitA cage crossing a house still counts toward its 45 when the part inside has a fixed sum.
Every cage combination, for every size and sum: Killer Sudoku Cage Sum Chart — all combinations
Learn the move, not just the digit
In a real game, when the innie–outie difference is what opens the next cell, the hint in the app names it and shows the cell it unlocks — the same move as step 3 above.