The 45 Rule
A powerful technique for Killer Sudoku based on a simple fact: every row, column, and 3×3 box contains the digits 1-9 exactly once, which always sums to 45.
The Core Principle
In any completed Sudoku grid, each row, each column, and each 3×3 box contains the digits 1 through 9 exactly once. The sum of these digits is:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
This fact is always true, even while you're solving the puzzle. Every partial row, column, or box is on its way to summing to 45.
The 45 rule exploits this by analyzing how cages overlap regions. When cages partially cover a row, column, or box, you can deduce the sum of the "extra" cells—the innies or outies.
Innies: Cells Inside the Region
An "innie" is a cell (or cells) inside a row, column, or box that is not fully covered by the cages you're analyzing.
Example: Row with one innie cell
Row 3 has 9 cells total. 8 cells are covered by cages A and B. 1 cell (the "?") is the innie.
Calculating the Innie
The row must sum to 45. Cages A and B contribute 15 + 22 = 37 to the row.
Innie value = 45 − (Cage A + Cage B) = 45 − 37 = 8
The unknown cell must be 8. This is immediate progress!
The 45 rule turned a seemingly ambiguous cell into a definite value, using only arithmetic and the structure of the grid.
Outies: Cells Outside the Region
An "outie" is a cell (or cells) that belongs to a cage but extends outside the row, column, or box you're analyzing.
Example: Box with one outie cell
Box 1 should sum to 45, but one cage cell extends beyond the box boundary.
Calculating the Outie
Suppose all other cells in Box 1 (excluding Cage C) are covered by other cages that sum to 26 (within the box).
Cage C has 5 cells total: 4 inside Box 1, 1 outside (the outie). The 4 inside cells contribute some amount X to Box 1's sum of 45.
Box 1 sum = 45 = (other cages) + (Cage C inside) = 26 + X
X = 45 − 26 = 19
Cage C total = 20 = X + outie = 19 + outie
Outie = 20 − 19 = 1
The outie cell (the one outside Box 1) must be 1.
Example 1: Single Innie in a Row
Let's walk through a complete example with real cages:
Row 5 (9 cells):
8 cells covered by cages, 1 innie cell
Step 1: Sum the known cages
Cage D + Cage E + Cage F = 18 + 9 + 10 = 37
Step 2: Apply the 45 rule
Row must sum to 45. Innie = 45 − 37 = 8.
Result:
The unknown cell is 8. Mark it immediately and use this to constrain intersecting columns/boxes.
Example 2: Two Innies in a Box
The 45 rule works even when multiple cells are innies, though you won't get individual values—you'll get constraints on their sum.
Box 2 (middle-top 3×3, 9 cells):
Calculation:
?A + ?B = 45 − 38 = 7
What this tells us:
The two cells sum to 7. Possible pairs (all different):
{1, 6}
{2, 5}
{3, 4}
You can now mark these as candidate pairs and use Sudoku rules to narrow down which pair (and which order).
When to Use the 45 Rule
The 45 rule is most useful when:
1.
A region is nearly full
If a row, column, or box has only 1-2 cells not covered by known cages, the 45 rule often solves them directly.
2.
Cages cross region boundaries
When a cage spans multiple boxes or regions, the 45 rule helps isolate the contribution from each region.
3.
You're stuck on other techniques
If cage combinations and Sudoku logic aren't progressing, scan for 45-rule opportunities.
Common Mistakes to Avoid
❌ Forgetting to account for all cages
When summing cage values for a region, make sure you include every cage that overlaps the region, even partially. Missing one cage invalidates the calculation.
❌ Applying the rule to incomplete cages
The 45 rule works with cage sums (the targets), not the current partial fills. Use the cage's target value, not the sum of digits you've placed so far.
❌ Confusing innies and outies
An innie is inside the region but outside the cages you're summing. An outie is inside a cage but outside the region. They're opposites—don't mix them up.
Advanced: Multi-Region 45 Rule
You can apply the 45 rule to combinations of regions. For example, if a cage covers parts of two adjacent boxes, you can analyze both boxes simultaneously.
Advanced pattern: Two boxes sum to 90 (45 + 45). If cages covering both boxes sum to 85, the remaining cells (innies or outies) must sum to 5.
This technique is powerful but requires careful tracking of which cells belong to which regions and cages. Use it when single-region 45-rule applications don't yield progress.
Why the 45 Rule Matters
The 45 rule often provides breakthroughs in mid-to-late puzzle solving. Early on, cage combinations dominate. But as the grid fills and cages become more interconnected, the 45 rule finds cells that cage logic alone cannot determine.
It's especially powerful in harder puzzles where the generator intentionally designs cages to create innies and outies. Mastering this technique is essential for solving Expert-level Killer Sudoku.