How to solve Kakuro: sums, combinations and crossings
6 min read ·
Kakuro is a crossword made of numbers. Every run of white cells, across or down, has a sum written in the black cell before it, and its digits must add up to that sum without repeating. There are no starting digits at all: the sums are the only clues.
That makes Kakuro a puzzle about combinations. Once you know which digit sets can make a sum, and where an across run and a down run cross, the grid fills itself in faster than you'd expect.
The rules
- Fill every white cell with a digit from 1 to 9.
- Each run of white cells, across or down, adds up to the clue in the black cell before it: across sums sit above the diagonal, down sums below it.
- No digit repeats within a run.
Sums that can only be made one way
Because digits can't repeat, the smallest and largest sums for each run length have only one combination. Learn these and you'll always have a place to start:
- 2 cells: 3 = 1 + 2 · 4 = 1 + 3 · 16 = 7 + 9 · 17 = 8 + 9
- 3 cells: 6 = 1 + 2 + 3 · 7 = 1 + 2 + 4 · 23 = 6 + 8 + 9 · 24 = 7 + 8 + 9
- 4 cells: 10 = 1 + 2 + 3 + 4 · 11 = 1 + 2 + 3 + 5 · 29 = 5 + 7 + 8 + 9 · 30 = 6 + 7 + 8 + 9
- 5 cells: 15 = 1 + 2 + 3 + 4 + 5 · 16 = 1 + 2 + 3 + 4 + 6 · 34 = 4 + 6 + 7 + 8 + 9 · 35 = 5 + 6 + 7 + 8 + 9
Where runs cross
Every white cell belongs to one across run and one down run. Its digit must be possible in both, so intersect the two sets of candidates.
- Where a two-cell 4 (1 and 3) crosses a three-cell 6 (1, 2 and 3), the shared cell must be 1 or 3.
- Where a two-cell 3 (1 and 2) crosses a two-cell 4 (1 and 3), the shared cell can only be 1.
- Where the across set and the down set share only one digit, that cell is solved.
- Short runs near the edges have the smallest sets; start your crossings there.
Digits a run must contain
When a sum has several combinations, look at what they have in common. A 3-cell run of 8 can be 1 + 2 + 5 or 1 + 3 + 4: both contain a 1, so the run definitely holds a 1. If only one of its cells can take a 1, that cell is solved.
Likewise, a digit that appears in none of a run's combinations can be crossed out of all its cells. A 2-cell run of 12 is 3 + 9, 4 + 8 or 5 + 7, so it can never contain 1, 2 or 6.
A solving routine
- Mark every one-way sum and write its digits as notes.
- Intersect across and down candidates at every crossing.
- For the other runs, note the digits all their combinations share, and the digits none of them use.
- When a run has one cell left, its digit is the sum minus the others.
- Repeat: every placed digit shrinks the combinations of both its runs.
Practise
Puzzimo's Kakuro comes in 6×6, 8×8 and 10×10, counting the clue row and column along the top and left; the daily challenge is 6×6. Every puzzle is checked to have exactly one solution. Notes, undo and hints are built in (a hint adds 30 seconds to a ranked time), and a wrong digit counts as a mistake; the third ends the round.
Frequently asked questions
What's the best way to start a Kakuro?
Find the runs with one-way sums, like 3 or 4 in two cells and 23 or 24 in three, and write their digits as notes. Then look where those runs cross others.
How is Kakuro related to Killer Sudoku?
Both are about adding up distinct digits. Kakuro has only sums and no rows or columns of 1 to 9, while Killer Sudoku keeps the full Sudoku rules and adds cages on top.
Do I need to be good at maths for Kakuro?
No. You only add small numbers, and after a few puzzles the common combinations become automatic. The real skill is logic: which digits fit both runs at a crossing.