TWO SYSTEMS, ONE PUZZLE
Killer Sudoku is two systems bolted together: arithmetic cage logic and standard Sudoku scanning. They're not alternatives — you use both, in sequence.
The sequence matters: cage arithmetic first to narrow the candidate pool, then Sudoku scanning to make placements. Beginners who jump straight to Sudoku scanning on an empty grid get nowhere. The cages are the door.
UNIQUE-SUM CAGES — YOUR FREE MOVES
Some cage size + target combinations have exactly one possible digit set. Spot these and you get candidates for free, no scanning needed.
2-cell cages:
3 → {1, 2}
4 → {1, 3}
16 → {7, 9}
17 → {8, 9}
3-cell cages:
6 → {1, 2, 3}
7 → {1, 2, 4}
23 → {6, 8, 9}
24 → {7, 8, 9}
4-cell cages:
10 → {1, 2, 3, 4}
30 → {6, 7, 8, 9}
Pattern: extreme-low and extreme-high targets lock the digit set. Mid-range targets have several possible sets — save those for later.
THE 45-RULE
Digits 1–9 always sum to 45. That means any complete row, column, or 3×3 box also sums to 45.
When cages don't line up neatly with a row or box, some cage cells "spill out" of the region. The cells inside the region sum to 45 minus the total of the cages fully contained within it.
Example: a row contains two complete cages (sums 8 and 14) plus one cell that belongs to a cage whose other cell is in another row. That single spilling cell = 45 − 8 − 14 = 23. That's the digit.
The 45-rule is Killer Sudoku's most powerful tool. Use it on every row, column, and box before reaching for pencil marks.
CAGE AS A CONSTRAINT UNIT
Digits cannot repeat within a cage — the same rule that applies to rows, columns, and boxes. A 3-cell cage summing to 7 can only be {1, 2, 4}, which means 3, 5, 6, 7, 8, and 9 are impossible in ANY of those three cells.
Cross-reference this with the row and box the cage lives in. If the box already has a 1 somewhere, the 1 inside this cage is placed. Often that's the move that unlocks the whole region.
WHEN TO SWITCH TO SUDOKU SCANNING
Once cage arithmetic and the 45-rule have placed 4–6 digits or eliminated most candidates from several cells, switch to standard Sudoku technique: naked singles, hidden singles, pointing pairs.
The transition happens naturally. You'll be working a cage, use the digit set it implies to eliminate something in a box, and a naked single appears. Follow it — that's the switching signal.
Alternate back: each placed digit unlocks new 45-rule calculations and cage intersections. Mid-solve you'll blend both systems in the same pass.
THE BEGINNER MISTAKE
Trying to solve from Sudoku scanning alone. On an empty grid there are no starting digits, so naked and hidden singles don't exist yet. The cages must create them first.
If you're stuck, the answer is always one of: an unused unique-sum cage, an unapplied 45-rule on some row/box, or a cage-intersection elimination you skipped. Killer puzzles don't require guessing — the arithmetic route is always there.