Innies and Outies in Killer Sudoku

Innies and outies turn cage arithmetic straight into a placed digit, which makes them the fastest way into a Killer grid. They rest on one fact: every row, column and box holds the digits 1 to 9 exactly once, so every one of them totals 45. That is the 45 rule, and it is the engine behind almost all Killer logic.

What the pattern is

A cage is a dotted group of cells with a target sum and no repeated digit inside it. Pick one unit, say a box, and add up the cages that lie completely inside it. If those cages cover eight of the nine cells, the ninth cell is an innie, and its value is 45 minus the cage total.

An outie is the same sum read backwards. The cages that cover the unit spill over by exactly one cell, so the total of those cages counts nine cells plus one extra. That extra cell is the outie, and its value is the cage total minus 45.

Both produce a placement, not an elimination. That is why they belong at the very start of a solve.

When to look for it

• On the opening pass, before you write any candidate at all. Innies need nothing but the printed sums.
• Along the outer frame of the grid. Edge rows, edge columns and corner boxes have fewer cages crossing them, so leftovers of exactly one cell are common there.
• Every time you finish a cage. A completed cage counts as fully inside its unit, which can leave a single cell bare that was not bare a move ago.

How to apply it, step by step

1. Choose one row, column or box.
2. Mark every cage that lies entirely inside it and add their sums. Digits already placed outside those cages count too, so add them in.
3. Count the cells covered. Eight covered and one bare means you have an innie.
4. Innie value equals 45 minus the total you just added.
5. If instead the cages covering the unit reach exactly one cell beyond it, that cell is the outie: subtract 45 from the total of those cages.
6. Place the digit, then re-check the neighbouring units. One placement frequently uncovers the next innie.

A worked example

Box 4, cage layout (letters mark cages, * marks the odd cell):

   D  D  *        D = 2 cells, sum 9
   E  E  E        E = 3 cells, sum 15
   F  F  F        F = 3 cells, sum 12

  D, E and F lie completely inside box 4 and cover 8 cells.
  9 + 15 + 12 = 36
  The box must total 45, so  *  = 45 - 36 = 9.

An outie uses the same arithmetic in reverse. Suppose box 1 holds two complete cages of 16 and 20 covering six cells, plus a four-cell cage of 15 whose last cell sits over in box 2. The three cells of that cage inside box 1 must supply 45 minus 36, which is 9. The cage totals 15, so the cell outside the box is 15 minus 9, which is 6.

Common mistakes

• Counting a cage that only partly fits. A cage helps only if every one of its cells is inside the unit, unless it is the single straddler you are measuring.
• Ignoring placed digits. A solved cell that belongs to no complete inside cage still contributes to the 45. Add it.
• Two leftover cells. With two bare cells you get a sum, not a digit. That is the multi-cell version of the technique, and it is a step harder.
• Arithmetic slips. Recount the covered cells before you trust the answer. Nine cells in a unit, no more and no less.

Where you meet it

Innies and outies are the first move on nearly every Killer grid, including easy Killer Sudoku. Learn the single-cell version cold: the multi-cell variants further up the ladder are the same subtraction applied to bigger leftovers.