Killer sudoku strategy: the rule of 45 and beyond
The techniques that carry you from Medium to Expert — innies, outies, split 45s, and combination cross-referencing. No guessing required, ever.
The rule of 45
The digits 1–9 add up to 45. That means every row, every column and every 3×3 box sums to exactly 45 — and because cage sums are printed on the grid, you can do arithmetic on whole regions before placing a single digit.
Innies: one cell in
Look at a 3×3 box. Suppose four cages fit entirely inside it and total 39, and the box's one remaining cell belongs to a cage that extends outside. That leftover cell is an innie — and it must be 45 − 39 = 6. You've placed a digit without touching a single combination table.
Outies: one cell out
Now the reverse. Suppose cages covering a box total 48, with exactly one cell of one cage sticking outside the box. Everything inside sums to 45, so the poking-out cell — the outie — must be 48 − 45 = 3.
Stretch it across multiple regions
Nothing limits you to one box. Two adjacent columns sum to 90; the whole top band of three boxes sums to 135. Scanning bands and stacks for near-complete cage coverage regularly yields an innie or outie the single-box scan misses. On Expert puzzles this is often the opening move.
Split 45s: innies and outies in pairs
When a region has two innie cells, you don't get a digit — you get their sum. If cages inside a column total 38 and two cells belong to outside cages, those two cells sum to 7. Now treat them like a phantom 2-cell cage: 1+6, 2+5 or 3+4. Cross-reference against the column's other constraints and it often collapses to one option.
Combination cross-referencing
This is the engine of mid-game solving. Take every cage's possible combinations and intersect them with sudoku constraints:
- Shared-unit collisions. Two 2-cell cages in the same row, one summing 16 (7+9) and one summing 17 (8+9): both need a 9, but the row only has one. Something must give — and since 16 and 17 have no alternatives, those cages cannot both sit fully in that row's cells that remain; whichever arrangement avoids two 9s in the row is forced.
- Cage-driven naked pairs. A 2-cell cage summing 4 (1+3 only) inside a box means no other cell in that box can be 1 or 3. That's a naked pair you got for free from a sum.
- Remainder analysis. A 4-cell cage summing 20 containing a placed 9 becomes a 3-cell problem summing 11 — jump back to the tables with the smaller cage.
The 45 rule, weaponised: overlapping regions
Advanced puzzles yield to arithmetic on overlapping regions. If a cage lies half in box 1 and half in box 2, then (box 1 total) + (that cage's outside half) = 45 + (something computable). Setting up two equations across neighbouring regions and subtracting them cancels the unknowns you don't care about, leaving the cell you do. It feels like algebra because it is — two minutes of it beats twenty minutes of staring.
When you're truly stuck
- Re-run the 45 scan. Placements change cage remainders; a band that gave nothing earlier may give an innie now.
- Check your smallest cages' remainders. Partially filled cages become forced far earlier than empty ones.
- Trust the puzzle. Every puzzle on Zaney Sudoku is verified to have exactly one solution before it's dealt — there is always a logical next step. Guessing is never required, which means being stuck is information: it tells you a technique on this page applies somewhere you haven't looked.