Killer Sudoku: How to Solve Cages by Their Sums
Killer Sudoku draws dashed outlines, called cages, around groups of cells on an ordinary Sudoku grid. The small number in the corner of each cage is the sum of the digits inside it, and no digit may appear twice in the same cage. In Sudoku Fog the variant is called Killer Cage.
The rule at a glance:
- The digits in a cage add up to the small number in its corner.
- A digit never repeats inside a cage, even when the cage crosses into another box.
- A cage can take any shape, as long as its cells connect side to side.
- Every row, column and box still holds 1 to 9 once each, so each of them adds up to 45.
In Sudoku Fog the sum is written in the cage’s top-left cell: the first cell of its top row. Cages that touch are drawn in different colours so you can tell them apart.
Subtract what you already know
The quickest cage deduction is arithmetic. Take the sum, subtract every digit already in the cage, and what is left has to be shared by the empty cells.
This cage adds up to 15 and already has a 4, so its two empty cells make 11. Sudoku alone leaves the top-left cell with 1 or 3. A 1 would need a 10 next to it, so the cell is 3.
Some sums have only one combination
Most sums can be made in several ways. The two smallest and two largest sums for each cage size cannot, and spotting them tells you exactly which digits the cage holds, even before you know where each one goes.
| Cells | Small sums | Large sums |
|---|---|---|
| 2 | 3 = 1+2, 4 = 1+3 | 16 = 7+9, 17 = 8+9 |
| 3 | 6 = 1+2+3, 7 = 1+2+4 | 23 = 6+8+9, 24 = 7+8+9 |
| 4 | 10 = 1+2+3+4, 11 = 1+2+3+5 | 29 = 5+7+8+9, 30 = 6+7+8+9 |
A two-cell cage of 17 is an 8 and a 9 in some order. A two-cell cage of 13 could be 4 and 9, 5 and 8, or 6 and 7, and that is where the rest of the grid has to help.
The same subtraction works on a bigger cage. This five-cell cage of 30 runs across two boxes. Its three givens make 20, so the two empty cells share 10, and Sudoku has already narrowed both of them.
When a cage leaves two options, look along the row
A cage that still allows two splits is not stuck. It is waiting for one digit from somewhere else.
The two-cell cage of 13 below could be 4 and 9 or 8 and 5. The 8 from the last example sits in the same row as its top cell, so that cell is 4. Sudoku leaves the bottom cell with 5 or 9, and only 9 brings the cage to 13.
Use 45 on harder boards
Every row, column and box adds up to 45. When a group of cages almost exactly covers a box, add up their sums and compare the total with 45. The difference is the digit in the cell that sticks out, or the cell that is missing. On easier boards you won’t need it, but on harder boards it is often the first way in.
Try it
The practice board has four cages. Start with the cage of 15 at the top and the cage of 30 on the left, where subtraction does most of the work.
Where to go next
Arrow Sudoku also works with sums, but along a path that ends in a circle. Sudoku Fog has two mixed variants with Killer Cage: one with German Whispers and one with Renban.
If you want the version where most of the grid is hidden while you solve, fog of war Sudoku is that. Everything published so far is on the Learn index.
Sudoku Fog is on Android — What Sudoku Fog is, Get it on Google Play.