How to solve Kakuro
Kakuro is a crossword of sums. Fill the white squares with digits from 1 to 9 so that each run adds up to the clue at its start, across clues on the left and down clues above, with no digit repeated in a run. Solve it by finding sums with only one set of digits and letting crossing runs fix the order.
The rules
- Each white square holds a digit from 1 to 9.
- The digits in a run add up to its clue.
- No digit appears twice in the same run.
There are no rows of nine and no boxes. A run can be anything from two to nine squares long.
Sums with only one answer
Most sums can be made in several ways, but some cannot. Three in two squares can only be 1 and 2. Seventeen in two squares can only be 8 and 9. Six in three squares can only be 1, 2 and 3. Every such sum, worked out by computer from every set of different digits from 1 to 9:
| Squares | Sums with only one answer (and the 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 smallest and largest sums for each length are always on this list, and so is every sum of a run of eight: eight squares use all the digits but one, and the sum tells you which one is missing.
A tiny puzzle, solved
- The top run adds up to 3 in two squares, so it holds 1 and 2.
- The bottom run adds up to 17 in two squares, so it holds 8 and 9.
- The left column adds up to 11. Its top square is 1 or 2 and its bottom square is 8 or 9, and the only pair that makes 11 is 2 and 9.
- That leaves 1 above 8 in the right column, which adds up to 9 as its clue says.
Tips
- Start where a short run with a unique sum crosses another one; the crossing square often has only one digit in common.
- Write the possible digit sets beside long runs and cross them off as squares fill.
- Use the largest digit a run can hold: in a two-square run of 4, neither square can be more than 3.
- Count twice. A run of 3 squares that adds up to 23 must be 6, 8 and 9, never 7, 7 and 9, because digits do not repeat.
The same habit, in light
Kakuro is about the one combination that fits. Light Up is about the one square that fits: a courtyard with one place left, a dark square only one spot can light. Nightwatch has a new Light Up board every night, free in your browser.
Nightwatch is on Android. On iPhone, play tonight's board here.
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Questions
How do you solve Kakuro?
Start with runs whose sums can be made in only one way, such as 3 or 4 in two squares and 23 or 24 in three. Where two such runs cross, the shared square must hold a digit both sets contain. Fill those squares, then use the remaining sums to narrow down the rest.
What is the hardest part of Kakuro?
Long runs with middle sums, such as 20 in four squares, which can be made in many ways. Leave them for later and let the short runs that cross them decide their digits.
Can numbers repeat in Kakuro?
Not within a run. The same digit can appear in different runs, and a square belongs to one across run and one down run, so it must fit both.