Solving killer cages
A cage total rarely allows every digit. Some totals have exactly one set of digits. And because every row, column and box adds up to 45, the cages around a gap tell you what the gap holds.
Why it works
On the board above:
The cage over r1c4 and r1c5 adds up to 17. The only two different digits that make 17 are 8 and 9.
So the cage holds an 8 and a 9. We just don’t know the order yet.
Column 4 already has a 9, in r6c4. So r1c4 can’t be 9.
That makes r1c4 an 8, and the 9 goes in r1c5.
More moves
The 45 rule
Every box, row and column adds up to 45. When cages cover all of a box but one cell, that cell is 45 minus the cages.
Box 1 adds up to 45, like every box.
Its four cages cover every cell except r3c3. Together they add up to 11 + 7 + 10 + 9 = 37.
So r3c3 is 45 − 37, which makes r3c3 an 8.
Outies
When cages cover a whole box and stick out by one cell, the cages add up to 45 plus that cell.
The five cages cover all of box 5, plus one cell outside it: r4c7.
They add up to 13 + 5 + 17 + 8 + 6 = 49. Box 5 accounts for 45 of that.
So r4c7 is 49 − 45, which makes r4c7 a 4.
Cages lock digits out
A cage whose digits you know, sitting inside one box or line, uses those digits up there. Clear them from the rest of that house.
The cage over r7c1 and r8c1 adds up to 3, so it holds a 1 and a 2.
Both cells are in box 7. That box has only one 1 and one 2, and the cage takes them.
So r9c2 can’t be 1 or 2, and neither can any other cell in box 7 or column 1.
How to use it
Read the small totals first
Very small and very large totals have the fewest ways to be made. Start there.
Check the combinations
List the sets of different digits that reach the total. One set means you know the digits, just not the order.
Look for a house almost covered by cages
If cages fill every cell of a box, row or column but one, the 45 rule gives that cell.
Let cages clear their houses
A cage that sits inside one box or line uses up its digits there. The other cells can’t have them.
Totals with one answer
Practice
8 boards. Each has one right answer, checked by our solver.
Loading the boards…
Try it in a real puzzle
Each of these starts with cage logic.
Tips
Learn the short list
3 and 4 in two cells, 16 and 17 in two cells, 6, 7, 23 and 24 in three cells. These come up in almost every killer.
Outies work too
If cages cover a whole box and poke out by one cell, that cell is the cages’ total minus 45.
Rows next to each other add up
Two rows add to 90 and three add to 135. The 45 rule works on any group of whole houses.
Pencil the combos, not the digits
Writing 89 in a cage corner is faster than filling every cell with marks.