CageDokuKiller Sudoku 45 Rule: Innies and Outies Explained
By Andrew Boyko, developer of CageDoku · Updated
The 45 rule is the single most useful trick in Killer Sudoku, and it is easier than its reputation. It rests on one fact: every row, column and 3×3 box contains the digits 1 to 9 exactly once, so each of them adds up to 45.
Why it works
Cages come with sums. If you add up the sums of several cages that together cover a whole region (a row, a column or a box), the total must be 45. When the cages cover a little more or a little less than the region, the difference tells you the value of the cells that stick in or out.
Innies: one cell inside the region
An innie is a cell inside a region whose cage reaches outside it. Worked example: take the top-left box. Three cages sit completely inside it: a 4-cell cage with sum 22, a 2-cell cage with sum 11 and another 2-cell cage with sum 7. Together they cover 8 cells and add up to 22 + 11 + 7 = 40. The ninth cell is the innie, and it must be 45 − 40 = 5.
Outies: one cell outside the region
An outie is a cell outside the region that belongs to a cage reaching into it. Worked example: row 1 is covered by three cages. Cage A (r1c1–r1c3) has sum 20, cage B (r1c4–r1c6) has sum 11 and cage C (r1c7–r1c9 plus r2c9) has sum 21. Together they cover the row plus one extra cell, r2c9, and total 20 + 11 + 21 = 52. The row alone is 45, so the extra cell is 52 − 45 = 7.
Two innies or two outies
If exactly two cells stick in or out, you cannot name them yet, but you know their sum: 45 minus the full cages for two innies, or the cage total minus 45 for two outies. A pair sum of 3 or 17 is already a unique pair (1+2 or 8+9), and 4 or 16 leaves only 1+3 or 7+9. These pairs are often the key to the next step.
How to hunt for them
Go region by region. In each row, column and box, count the cells covered by cages that lie entirely inside it. If that count is eight, you have a single innie. If a few cages cover the region plus exactly one more cell, you have a single outie. Check again after every placement: a region that had two unknown cells yesterday may have one today.
Bigger regions
The same idea works for several regions at once. Two adjacent rows add up to 90, three boxes in a band add up to 135. Larger regions give you more cages to add, but also more chances for a stray cell. See advanced Killer Sudoku techniques.
Practice
In CageDoku, the 45-rule hint highlights the exact cages and shows the subtraction, so you can see the trick work on a real puzzle. It is free in your browser and on Android, with no ads.
Frequently asked questions
Does the 45 rule work in 6×6 puzzles?
Yes, with 21 instead of 45 (1+2+3+4+5+6).
Is the 45 rule cheating?
No. It is plain arithmetic on the puzzle’s own rules.
What if no region has exactly one innie or outie?
Look at pairs, at must-have digits in cages, or at bigger regions.
Practice it in CageDoku
Free, ad-free Killer Sudoku and seven more variants. Hints name the technique and highlight the cells, so you learn why a digit goes where it goes.