How to use the kakuro combinations table
A kakuro clue gives the sum of a run of white cells, and the digits in a run must all be different, from 1 to 9. This kakuro combinations chart lists every set of digits for every sum and every run length from 2 to 9 cells: 120 sums and 502 combinations in all. Combinations are written as digit sets in ascending order, so 124 means a 1, a 2 and a 4 in some order.
For one clue, type the sum and choose the number of cells. You get the combinations, the digits that appear in every one of them (they must be somewhere in the run) and the digits that never appear (cross them out of every cell of the run). When a crossing run or a given digit already tells you something, tap it under Must include or Must not include to narrow the list. The same filters work on the full table, for example all runs that can hold a 9 next to a 1.
The combinations worth learning by heart
Sums with only one combination are highlighted. There are 34 of them for runs of 2 to 9 cells, and they are the usual way into a kakuro puzzle:
- Lowest sums: 3 in 2 is always 1+2, 4 in 2 is 1+3, 6 in 3 is 1+2+3, 7 in 3 is 1+2+4, 10 in 4 is 1+2+3+4 and 11 in 4 is 1+2+3+5.
- Highest sums: 16 in 2 is 7+9, 17 in 2 is 8+9, 23 in 3 is 6+8+9, 24 in 3 is 7+8+9, 29 in 4 is 5+7+8+9 and 30 in 4 is 6+7+8+9.
- Five, six and seven cells: 15 and 16 in 5, 34 and 35 in 5, 21 and 22 in 6, 38 and 39 in 6, 28 and 29 in 7, 41 and 42 in 7.
- Eight cells: every sum from 36 to 44 has one combination, because 8 different digits leave out exactly one digit from 1 to 9 (whose total is 45). A run of 8 adding up to 40 leaves out the 5.
- Nine cells: the only sum is 45, all nine digits.
Worked example
A run of 3 cells with the clue 23 can only be 6, 8 and 9. If one of those cells is also in a down run of 2 cells with the clue 4 (always 1 and 3), the shared cell must hold a digit from both sets. The two sets have nothing in common, so the puzzle is wrong or the runs were read incorrectly. Another case: an across run of 2 adding up to 10 has four combinations (19, 28, 37, 46), but if a crossing clue rules out 1, 2 and 3, only 4+6 is left. Choose 10, 2 cells and exclude 1, 2 and 3 to see it.
How the table is made
The table is generated from all 511 non-empty sets of the digits 1 to 9, grouped by size and sum. Our test rebuilds it a second way, by trying every ordered list of digits for runs of up to 6 cells and every set of left-out digits for longer runs, and checks that both agree for every sum, that each combination adds up correctly, and that the printed table on this page matches. Each run length covers the sums from the smallest possible (1+2+...) to the largest (9+8+...); any clue outside that range is impossible, and the lookup says so.
Printing
Print the table gives the full chart on paper in two columns, without the filter, to keep next to a puzzle book. To solve a whole grid, enter it in the kakuro solver, which uses the same combinations to explain every step.
The eight ways to make 15 from three different digits are also the eight lines of the 3 x 3 magic square: three rows, three columns and two diagonals.
Practice with the chart open: play kakuro online or try the cage sums of killer sudoku.
Solver tested . Found a problem? Tell us.
Questions
How many kakuro combinations are there?
For runs of 2 to 9 cells there are 502 combinations spread over 120 possible sums. That is every set of two or more different digits from 1 to 9.
What does a unique combination mean in kakuro?
It is a sum that only one set of digits can make for that run length, such as 17 in two cells (8 and 9) or 7 in three cells (1, 2 and 4). You know which digits go in the run straight away, only not yet their order.
Can a digit repeat in a kakuro run?
No. Every run uses different digits from 1 to 9, which is why a run can be at most 9 cells long and why each run length has a smallest and a largest possible sum.
What is the highest sum for 4 cells?
30, which is 6+7+8+9. The lowest is 10 (1+2+3+4). In general the lowest sum of n cells adds the n smallest digits and the highest adds the n largest.
How do I use the combinations to solve a kakuro?
Look up each clue, cross out the digits that never appear, then compare crossing runs: the shared cell can only hold a digit allowed by both. The kakuro solver shows these steps on your own grid.