Sums with only one combination
The table lists, for each number of cells, the sums that can be made in exactly one way. The pattern is easy to remember: the smallest and largest possible sums for each size are always unique, and so are the sums just next to them.
Knowing the set does not tell you the order, but it removes every other digit from those cells — often enough to settle a crossing line.
| Cells | Sum = digits |
|---|---|
| 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 |
| 5 | 15 = 1 + 2 + 3 + 4 + 5 16 = 1 + 2 + 3 + 4 + 6 34 = 4 + 6 + 7 + 8 + 9 35 = 5 + 6 + 7 + 8 + 9 |
| 6 | 21 = 1 + 2 + 3 + 4 + 5 + 6 22 = 1 + 2 + 3 + 4 + 5 + 7 38 = 3 + 5 + 6 + 7 + 8 + 9 39 = 4 + 5 + 6 + 7 + 8 + 9 |
| 7 | 28 = 1 + 2 + 3 + 4 + 5 + 6 + 7 29 = 1 + 2 + 3 + 4 + 5 + 6 + 8 41 = 2 + 4 + 5 + 6 + 7 + 8 + 9 42 = 3 + 4 + 5 + 6 + 7 + 8 + 9 |
| 8 | 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 37 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 9 38 = 1 + 2 + 3 + 4 + 5 + 6 + 8 + 9 39 = 1 + 2 + 3 + 4 + 5 + 7 + 8 + 9 40 = 1 + 2 + 3 + 4 + 6 + 7 + 8 + 9 41 = 1 + 2 + 3 + 5 + 6 + 7 + 8 + 9 42 = 1 + 2 + 4 + 5 + 6 + 7 + 8 + 9 43 = 1 + 3 + 4 + 5 + 6 + 7 + 8 + 9 44 = 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 |
| 9 | 45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 |
The rule of 45
In killer sudoku every row, column and box adds up to 45. Add the cages that lie completely inside a box: if they cover all but one cell, that cell is 45 minus their total. If a cage sticks out of the box by one cell, that cell is the cage total minus what the box still needs. The same works across two or three rows at once (90, 135).
The missing digit
Long runs are easier through what they leave out. Eight distinct digits adding to s use every digit except 45 − s. Seven digits adding to s leave out two digits that add up to 45 − s. In kakuro, a run of 8 with clue 39 is missing 6, and a run of 9 always uses all nine digits.
Crossing runs
In kakuro, each white cell belongs to one run across and one run down. Its digit must be allowed by both. A two-cell 4 (1+3) crossing a two-cell 16 (7+9) would be impossible, so such clues never meet; but a 4 crossing a three-cell 6 (1+2+3) can only meet at 1 or 3. Write the shared candidates in the crossing cell and the run usually collapses.
Cages that cross a box
A killer cage can stretch across two boxes. Its digits still may not repeat inside the cage, but the box rule applies separately in each part. That is useful: if a cage of 3 cells adding to 6 crosses a box boundary, the digits 1, 2 and 3 are split between two boxes, and the split often settles which box needs which.