Killer Sudoku

Innies & Outies

Innies and outies are the rule of 45 at work in Killer Sudoku: every row, column and box holds 1 to 9 once, so it totals 45. Add up the cages lying fully inside one of them; the cells left over inside it — the innies — make 45 minus that total, and the cells that crossing cages push outside — the outies — make the sum of every cage touching the house minus 45. A lone innie or outie is solved outright, and two innies keep only the pairs of digits that make their sum.

What you need to see on the grid

Reach for the rule of 45 whenever a row, column or box is almost covered by cages that stay inside it. In Killer it is often the very first move of a puzzle, and it keeps paying off to the end. Four terms carry the method:

The rule of 45: Each row, column and box holds the digits 1 to 9 once, so its nine cells add up to 1 + 2 + … + 9 = 45. Here it is applied to one of them at a time; the same count also works across two or three neighbouring rows or columns.
Inside cage: A cage whose every cell lies in the row, column or box you are counting. Its whole sum belongs to that house, even when some of its digits are already placed. A cage with even one cell outside does not count.
Innie: A cell of the house that no inside cage covers: its cage crosses the edge. The innies together make 45 minus the inside cages. Subtract any innie that already holds a digit; the rest belongs to the empty ones.
Outie: A cell outside the house that belongs to a cage crossing into it. All the cages touching the house cover its 45 plus the outies, so the outies make that total minus 45.

How to apply it, step by step

One Killer grid, part-way through a puzzle, runs through all five steps. Every diagram shows the whole grid; the cages a step uses are outlined with their sum in the corner. The cells a step needs are filled in; the others are left blank and hatched, so you know you can ignore them.

Step 1Add up the cages that lie fully inside one box

Pick a row, column or box that is nearly covered by cages, and sort the cages touching it into two groups: those lying entirely inside it, and those crossing its edge. Only the first group counts. In box 7, the 15, 16 and 9 cages stay inside the box, and together they make 15 + 16 + 9 = 40.

The 7-cage is different: R9C3 is in the box, but its partner R9C4 is in box 8, so the 7 tells you nothing about box 7 on its own. That leaves R9C3 as the box's one innie — the only cell that no inside cage covers.

Technique: Rule of 45 (innies/outies)

Box 7 and every cage that touches it. The 15, 16 and 9 cages stay inside the box; the 7-cage crosses into box 8, leaving R9C3 as the innie.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5
R6
R7	15		16
R8	9
R9			7

Box 7: its nine cells make 45
The innie: its cage leaves the box
Cage, sum in its corner
Not needed for this step

Filled in: box 7; the 4 cages. The other 71 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 2One innie left: it takes what the 45 still needs

The nine cells of box 7 make 45, and the inside cages account for 40 of it, so the innie must be 45 − 40 = 5: R9C3 = 5. Its candidates were 1, 2, 3 and 5, and no other cell had to be looked at to settle it.

Before you write the digit, check two things: every cage you added really stays inside the box, and the box has only one empty innie. Had an innie already held a digit, you would subtract that too; whatever is left belongs to the empty ones.

Technique: Rule of 45 (innies/outies)

45 − (15 + 16 + 9) = 5, so R9C3 = 5.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5
R6
R7	15		16
R8	9
R9			75

Box 7: its nine cells make 45
The innie: 45 − 40
Digit placed
Cage, sum in its corner
Not needed for this step

Filled in: box 7; the 4 cages. The other 71 cells are left blank: they are not needed for this step. Placed: R9C3 = 5.

Step 3Two innies: keep only the pairs that make the sum

Box 2 has two inside cages, 22 and 16, which make 38. The 11-cage crosses the edge: two of its cells, R1C4 and R2C4, are in the box, and the third, R1C3, is not. So box 2 has two innies, and together they make 45 − 38 = 7.

Two cells of one box always hold different digits, so 7 can only be 1 + 6, 2 + 5 or 3 + 4. Hold that against the candidates: R1C4 lists 1, 4, 6 and 7, R2C4 lists 1, 6 and 7. A 4 in R1C4 would need a 3 in R2C4, which is not there — box 2 already has its 3 — and a 7 in either cell would need a 0. Only 1 + 6 fits, in either order: remove 4 and 7 from R1C4, and 7 from R2C4.

Technique: Rule of 45 (innies/outies)

Box 2 and every cage that touches it. The 22 and 16 cages make 38, so the innies R1C4 and R2C4 make 7; only 1 + 6 fits their candidates.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1			11		22
R2					16	5
R3				3		2
R4
R5
R6
R7
R8
R9

Digit already on the grid
Box 2: its nine cells make 45
An innie: the two make 7
Candidate removed
Cage, sum in its corner
Not needed for this step

Filled in: box 2; the 3 cages. The other 71 cells are left blank: they are not needed for this step. Removed: 4 from R1C4; 7 from R1C4 and R2C4.

Step 4Read the same box from outside: the outie

Now add up every cage that touches box 2, the crossing one included: 22 + 16 + 11 = 49. Those three cages cover the nine cells of the box, which make 45, plus one cell outside it, R1C3. That cell is the outie, and it must be 49 − 45 = 4. The innies agree: the 11-cage holds R1C4 + R2C4 = 7, so its third cell is 11 − 7 = 4.

On paper you can write the 4 in straight away. The app's hint, however, never counts the outies of a single row, column or box: it reaches the same digit through the innies of step 3, as step 5 shows. That is why this step only reads the grid.

The 22, 16 and 11 cages touch box 2 and add up to 49. The box takes 45 of it, so the outie R1C3 is 4.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1			11		22
R2					16	5
R3				3		2
R4
R5
R6
R7
R8
R9

Digit already on the grid
Box 2: its nine cells make 45
The outie: 49 − 45
Cage, sum in its corner
Not needed for this step

Filled in: box 2; the 3 cages. The other 71 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 5Collect what the rule of 45 unlocks

Step 3 took 4 out of R1C4, and that was the only other place for 4 in row 1. The row still needs its 4, so R1C3 = 4: a Hidden Single, and exactly the digit the outie predicted.

When two routes give the same digit, your arithmetic is right. After every rule-of-45 move, look again at the rows, columns and boxes of the cells you just narrowed — the next single is usually there. In box 2, for instance, R3C5 is now the only cell left for 4 (look back at step 4).

Technique: Hidden Single

Row 1 after step 3, with the 11-cage. 4 now fits only R1C3 — the outie.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1			114
R2
R3
R4
R5
R6
R7
R8
R9

Row 1: one place left for 4
The cell that must be 4
Candidate 4, the digit studied
Digit placed
Cage, sum in its corner
Not needed for this step

Filled in: row 1; the cage. The other 71 cells are left blank: they are not needed for this step. Placed: R1C3 = 4.

Common mistakes

Counting a cage that crosses the edge

A cage counts only when every one of its cells is in the house. Add the full sum of a crossing cage and you have counted cells outside the house, so the innie you work out is wrong — sometimes impossible, sometimes a perfectly believable wrong digit. Follow each dashed outline all the way round before you add its sum.

Subtracting in the wrong direction

Innies are 45 minus the cages inside; outies are the cages touching the house minus 45. Mixing them up gives answers like −4 or 13 — easy to catch — or a wrong digit between 1 and 9, which is not. Before you subtract, say which side of the edge the cells are on.

Letting two innies share a digit

Two innies of one house can never hold the same digit, so 8 is 1 + 7, 2 + 6 or 3 + 5, never 4 + 4 — forgetting this keeps candidates that should go. Outies are different: two outies in different rows, columns, boxes and cages may repeat a digit, so check whether they share one before you rule out a double.

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 rule of 45 is what opens the next cell, the hint in the app names it — Rule of 45 (innies/outies) — and shows the cell it unlocks, like the Hidden Single of step 5.

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