Killer Sudoku
Cage Split
Cage Split extends the rule of 45. Every row, column and box adds up to 45; cages wholly inside a house tell you most of that. A cage that crosses the house can still count if the part of it inside always adds up to the same number — for instance because its cells outside are already filled. Subtract everything you know from 45, and the cells left over hold the rest.
What you need to see on the grid
Reach for Cage Split when a house is almost covered by whole cages and the cage that spoils it has its outside part settled. Four terms carry the method:
- The rule of 45
- The digits 1 to 9 add up to 45, so every row, column and box does too.
- Split cage
- A cage with cells on both sides of the house's border. Its part inside is a virtual cage when its sum is fixed.
- Fixed inside sum
- The part inside always adds up to the same number, whatever the cage's final digits — most simply because the cells outside are already filled.
- Loose cells
- Cells of the house covered by no counted cage. Together they hold 45 minus everything counted; one loose empty cell is forced.
How to apply it, step by step
One example runs through all three steps, early in an expert Killer puzzle from the app, at the exact point where its hint needs Cage Split. Every diagram shows the whole grid; the cells a step needs are drawn, the others hatched. The dashed outlines are cages, their sums in the corner. No pencil marks are needed: the arithmetic does all the work.
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Step 1Total the cages that sit wholly inside the house
Pick a house and add up the cages that lie entirely inside it. In box 8, the 22-cage (R7C5, R7C6, R8C5, R8C6, R9C5) is the only one: 22 of the box's 45.
Four cells are left: R7C4, R8C4, R9C4 and R9C6. R7C4 already holds 6.
Box 8 and its one whole cage, the 22-cage. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 6 22 2 R8 R9 - Digit already on the grid
- Box 8
- Cage, sum in its corner
- Not needed for this step
Filled in: box 8; the cage. The other 72 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 2Find a crossing cage whose inside part has a fixed sum
The 18-cage covers R8C3, R8C4, R9C3 and R9C4: two cells in box 7, two in box 8. Its box 7 half is already filled — R8C3 = 3 and R9C3 = 6 — so its box 8 half, R8C4 + R9C4, is always 18 − 3 − 6 = 9, whichever digits end up there.
That half acts like a small cage of sum 9 inside box 8.
The 18-cage crosses into box 8. With 3 and 6 outside, its two cells inside add up to 9. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 6 22 2 R8 183 R9 6 - Digit already on the grid
- Box 8
- Inside part of the 18-cage: always 9
- Cage, sum in its corner
- Not needed for this step
Filled in: box 8; the 2 cages. The other 70 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 3Apply the rule of 45 to the cells left over
Box 8 adds up to 45. The 22-cage gives 22, the split cage's half gives 9, and R7C4 holds 6. That leaves 45 − 22 − 9 − 6 = 8 for the only cell not yet counted, R9C6. So R9C6 = 8 — and that is the app's next hint in this puzzle.
With two cells left over instead of one, the same sum would keep only the candidate pairs that reach it.
Technique: Cage split
45 − 22 − 9 − 6 = 8: R9C6 is 8. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 6 22 2 R8 183 R9 6 8 - Digit already on the grid
- Box 8
- The one loose empty cell
- Digit placed
- Cage, sum in its corner
- Not needed for this step
Filled in: box 8; the 2 cages. The other 70 cells are left blank: they are not needed for this step. Placed: R9C6 = 8.
Common mistakes
Counting a crossing cage whose inside sum can vary
Only a fixed inside sum can be subtracted. If the 18-cage's box 7 half were still empty, its box 8 half could be 9, 10 or more — and the subtraction would be a guess. Check that every way of filling the cage gives the same inside sum.
Forgetting digits already in the house
R7C4 belongs to no counted cage, but its 6 is part of box 8's 45. Leave it out and R9C6 comes out as 14, which is impossible — a sign you dropped something.
Counting a cell twice
Each cell of the house must be counted exactly once: in a whole cage, in the inside half of a split cage, or as a loose cell. Mark the cells as you go so none falls into two groups.
Keep going
- Killer Sudoku & the Rule of 45The rules of Killer Sudoku, the first easy cage deductions, and the rule of 45 that turns a box's cages into a digit.
- Innies & OutiesA row, column or box totals 45: take away the cages inside it, and the cells left over must make the rest.
Every cage combination, for every size and sum: Killer Sudoku Cage Sum Chart — all combinations
Learn the move, not just the digit
In a real game, when Cage Split is what opens the next cell, the hint in the app names it and shows the cell it unlocks — the same move as step 3 above.