Innies and Outies Sudoku: The 45 Rule for Killer Grids

The 45 rule explained

Innies and outies sudoku technique rests on one number: 45. Because every row, column, and 3 by 3 box in a standard grid contains the digits 1 to 9 exactly once, the cells in any such region always add up to 45. Killer sudoku hands you cage sums, so you can compare the cages covering a region against that fixed total and solve for the leftover cells. This turns arithmetic into a kind of geometry, because you are no longer guessing at combinations but subtracting a known total from a fixed one to reveal what is missing. It is the moment many players fall in love with killer variants.

What innies and outies are

• An innie is a cell inside a region that is not fully covered by cages contained in that region.
• An outie is a cage cell that pokes outside the region you are studying.
• If the cages fully inside a box sum to 40, the remaining innie cells must sum to 5.
• If a cage straddles the box edge, subtract the known inside portion to expose the outie value.
• A single leftover cell is the jackpot, because its value is then completely determined.

A worked example

Take a column whose cages entirely within it sum to 38. The column must total 45, so the uncovered innie cells sum to 7. If only two innie cells remain, they are a distinct pair adding to 7, which can be 1 and 6, 2 and 5, or 3 and 4. Crossing information from the intersecting boxes then usually picks the correct pair, and sometimes a lone innie is simply forced to 7 outright with no ambiguity at all. That single subtraction unlocks a cell you could never reach by listing cage combinations alone, which is why the method feels almost like a trick the first few times it works.

Working across several regions

1. Choose a region where most cells are covered by clean, fully contained cages.
2. Add those contained cage sums and subtract from 45 to find the innie total.
3. For cages crossing the border, isolate the outie by subtracting the part you already know.
4. Chain regions together, using one solved outie as the known part of the next region.

Why innies and outies sudoku feels like magic

The habit behind innies and outies sudoku, breaking a whole into a known part and an unknown part, is the same disciplined thinking that classic Sudoku rewards, and setting up a clean region teaches patience because the payoff arrives only after careful counting. Shapedoku gives that same muscle a calm, beautiful workout by keeping the rules while replacing digits with nine glowing shapes, so you reason about identity and placement without the arithmetic getting in the way. Notes with smart cleanup handle the bookkeeping, and the first hint is free when you cannot find the next step. Learn the 45 rule for killer boards, then unwind with pure shape logic at app.shapedoku.com whenever you want deduction without the sums.