Why can't a digit repeat within a kakuro run?

It's the core rule of kakuro, the same way a sudoku row can't repeat a digit. A run of 2 cells summing to 10 can't be 5+5; it must be one of 1+9, 2+8, 3+7, or 4+6. That restriction is exactly what makes the puzzles logically solvable.

How do I use this to solve a kakuro puzzle?

Start with the runs whose sum has the fewest combinations (the unique ones first). Where two runs cross, the cell must hold a digit that appears in a legal combination of both runs — intersect the digit sets to narrow the cell. As digits get placed, use the include/exclude filters to update each run's remaining combinations.

What range of sums is legal for each run length?

A run of N cells can sum from 1+2+...+N up to 9+8+... down N digits. So 2 cells: 3-17, 3 cells: 6-24, 4 cells: 10-30, 5 cells: 15-35, up to 9 cells which must sum to exactly 45. The calculator rejects anything outside these ranges.

Does this tool solve a whole kakuro grid for me?

It's a combination calculator and technique reference — the tool every kakuro solver keeps beside the puzzle — rather than a full-grid autosolver. It gives you the exact digit sets for any run instantly, which is 90% of the work; the crossing logic stays your part of the fun.

Is the calculator free?

Yes, completely free with no signup. Everything is computed in your browser instantly.

Kakuro Combination Calculator & Solver Help

Every kakuro run is a little math puzzle: distinct digits 1–9 that add to the printed clue. Enter a run’s sum and length to see every legal combination instantly, mark digits you know are in or out, and use the reference tables below — including the unique “magic block” sums — to crack any cross-sums puzzle.

Your run

Clue sum (3–45)

Cells in the run

Digit filters — click once to require, twice to exclude

Legal ranges per run length

Cells	Min sum	Max sum
2	3	17
3	6	24
4	10	30
5	15	35
6	21	39
7	28	42
8	36	44
9	45	45

Combinations16 in 3 · 8 found

1+6+91+7+82+5+92+6+83+4+93+5+83+6+74+5+7

What Is Kakuro?

Kakuro (also called cross sums) is a crossword built from arithmetic. Each “word” is a run of white cells with a clue giving the run’s total; you fill the run with digits 1–9, never repeating a digit within a run, so that every clue is satisfied where the across and down runs cross. Because digits can’t repeat, each clue has a limited menu of digit sets — and knowing that menu is the whole game. That’s what this calculator gives you instantly.

Unique Combinations: the Solving Key

Some clue sums have exactly onelegal digit set. Spot one of these and you know the run’s digits with certainty before placing anything — only their order remains. Experienced solvers memorize this table and scan a fresh puzzle for these runs first:

Cells	Sum	Only combination
2	3	1+2
2	4	1+3
2	16	7+9
2	17	8+9
3	6	1+2+3
3	7	1+2+4
3	23	6+8+9
3	24	7+8+9
4	10	1+2+3+4
4	11	1+2+3+5
4	29	5+7+8+9
4	30	6+7+8+9
5	15	1+2+3+4+5
5	16	1+2+3+4+6
5	34	4+6+7+8+9
5	35	5+6+7+8+9
6	21	1+2+3+4+5+6
6	22	1+2+3+4+5+7
6	38	3+5+6+7+8+9
6	39	4+5+6+7+8+9
7	28	1+2+3+4+5+6+7
7	29	1+2+3+4+5+6+8
7	41	2+4+5+6+7+8+9
7	42	3+4+5+6+7+8+9
8	36	1+2+3+4+5+6+7+8
8	37	1+2+3+4+5+6+7+9
8	38	1+2+3+4+5+6+8+9
8	39	1+2+3+4+5+7+8+9
8	40	1+2+3+4+6+7+8+9
8	41	1+2+3+5+6+7+8+9
8	42	1+2+4+5+6+7+8+9
8	43	1+3+4+5+6+7+8+9
8	44	2+3+4+5+6+7+8+9
9	45	1+2+3+4+5+6+7+8+9

Full Combination Reference (Runs of 2–5 Cells)

Every legal combination for the run lengths you meet most often. Unique sums are the single-entry rows; the busiest sums (like 20-in-four with 12 combinations) are where crossing-run logic does the work instead.

2-cell runs

Sum	Combinations
3	1+2
4	1+3
5	1+4 · 2+3
6	1+5 · 2+4
7	1+6 · 2+5 · 3+4
8	1+7 · 2+6 · 3+5
9	1+8 · 2+7 · 3+6 · 4+5
10	1+9 · 2+8 · 3+7 · 4+6
11	2+9 · 3+8 · 4+7 · 5+6
12	3+9 · 4+8 · 5+7
13	4+9 · 5+8 · 6+7
14	5+9 · 6+8
15	6+9 · 7+8
16	7+9
17	8+9

3-cell runs

Sum	Combinations
6	1+2+3
7	1+2+4
8	1+2+5 · 1+3+4
9	1+2+6 · 1+3+5 · 2+3+4
10	1+2+7 · 1+3+6 · 1+4+5 · 2+3+5
11	1+2+8 · 1+3+7 · 1+4+6 · 2+3+6 · 2+4+5
12	1+2+9 · 1+3+8 · 1+4+7 · 1+5+6 · 2+3+7 · 2+4+6 · 3+4+5
13	1+3+9 · 1+4+8 · 1+5+7 · 2+3+8 · 2+4+7 · 2+5+6 · 3+4+6
14	1+4+9 · 1+5+8 · 1+6+7 · 2+3+9 · 2+4+8 · 2+5+7 · 3+4+7 · 3+5+6
15	1+5+9 · 1+6+8 · 2+4+9 · 2+5+8 · 2+6+7 · 3+4+8 · 3+5+7 · 4+5+6
16	1+6+9 · 1+7+8 · 2+5+9 · 2+6+8 · 3+4+9 · 3+5+8 · 3+6+7 · 4+5+7
17	1+7+9 · 2+6+9 · 2+7+8 · 3+5+9 · 3+6+8 · 4+5+8 · 4+6+7
18	1+8+9 · 2+7+9 · 3+6+9 · 3+7+8 · 4+5+9 · 4+6+8 · 5+6+7
19	2+8+9 · 3+7+9 · 4+6+9 · 4+7+8 · 5+6+8
20	3+8+9 · 4+7+9 · 5+6+9 · 5+7+8
21	4+8+9 · 5+7+9 · 6+7+8
22	5+8+9 · 6+7+9
23	6+8+9
24	7+8+9

4-cell runs

Sum	Combinations
10	1+2+3+4
11	1+2+3+5
12	1+2+3+6 · 1+2+4+5
13	1+2+3+7 · 1+2+4+6 · 1+3+4+5
14	1+2+3+8 · 1+2+4+7 · 1+2+5+6 · 1+3+4+6 · 2+3+4+5
15	1+2+3+9 · 1+2+4+8 · 1+2+5+7 · 1+3+4+7 · 1+3+5+6 · 2+3+4+6
16	1+2+4+9 · 1+2+5+8 · 1+2+6+7 · 1+3+4+8 · 1+3+5+7 · 1+4+5+6 · 2+3+4+7 · 2+3+5+6
17	1+2+5+9 · 1+2+6+8 · 1+3+4+9 · 1+3+5+8 · 1+3+6+7 · 1+4+5+7 · 2+3+4+8 · 2+3+5+7 · 2+4+5+6
18	1+2+6+9 · 1+2+7+8 · 1+3+5+9 · 1+3+6+8 · 1+4+5+8 · 1+4+6+7 · 2+3+4+9 · 2+3+5+8 · 2+3+6+7 · 2+4+5+7 · 3+4+5+6
19	1+2+7+9 · 1+3+6+9 · 1+3+7+8 · 1+4+5+9 · 1+4+6+8 · 1+5+6+7 · 2+3+5+9 · 2+3+6+8 · 2+4+5+8 · 2+4+6+7 · 3+4+5+7
20	1+2+8+9 · 1+3+7+9 · 1+4+6+9 · 1+4+7+8 · 1+5+6+8 · 2+3+6+9 · 2+3+7+8 · 2+4+5+9 · 2+4+6+8 · 2+5+6+7 · 3+4+5+8 · 3+4+6+7
21	1+3+8+9 · 1+4+7+9 · 1+5+6+9 · 1+5+7+8 · 2+3+7+9 · 2+4+6+9 · 2+4+7+8 · 2+5+6+8 · 3+4+5+9 · 3+4+6+8 · 3+5+6+7
22	1+4+8+9 · 1+5+7+9 · 1+6+7+8 · 2+3+8+9 · 2+4+7+9 · 2+5+6+9 · 2+5+7+8 · 3+4+6+9 · 3+4+7+8 · 3+5+6+8 · 4+5+6+7
23	1+5+8+9 · 1+6+7+9 · 2+4+8+9 · 2+5+7+9 · 2+6+7+8 · 3+4+7+9 · 3+5+6+9 · 3+5+7+8 · 4+5+6+8
24	1+6+8+9 · 2+5+8+9 · 2+6+7+9 · 3+4+8+9 · 3+5+7+9 · 3+6+7+8 · 4+5+6+9 · 4+5+7+8
25	1+7+8+9 · 2+6+8+9 · 3+5+8+9 · 3+6+7+9 · 4+5+7+9 · 4+6+7+8
26	2+7+8+9 · 3+6+8+9 · 4+5+8+9 · 4+6+7+9 · 5+6+7+8
27	3+7+8+9 · 4+6+8+9 · 5+6+7+9
28	4+7+8+9 · 5+6+8+9
29	5+7+8+9
30	6+7+8+9

5-cell runs

Sum	Combinations
15	1+2+3+4+5
16	1+2+3+4+6
17	1+2+3+4+7 · 1+2+3+5+6
18	1+2+3+4+8 · 1+2+3+5+7 · 1+2+4+5+6
19	1+2+3+4+9 · 1+2+3+5+8 · 1+2+3+6+7 · 1+2+4+5+7 · 1+3+4+5+6
20	1+2+3+5+9 · 1+2+3+6+8 · 1+2+4+5+8 · 1+2+4+6+7 · 1+3+4+5+7 · 2+3+4+5+6
21	1+2+3+6+9 · 1+2+3+7+8 · 1+2+4+5+9 · 1+2+4+6+8 · 1+2+5+6+7 · 1+3+4+5+8 · 1+3+4+6+7 · 2+3+4+5+7
22	1+2+3+7+9 · 1+2+4+6+9 · 1+2+4+7+8 · 1+2+5+6+8 · 1+3+4+5+9 · 1+3+4+6+8 · 1+3+5+6+7 · 2+3+4+5+8 · 2+3+4+6+7
23	1+2+3+8+9 · 1+2+4+7+9 · 1+2+5+6+9 · 1+2+5+7+8 · 1+3+4+6+9 · 1+3+4+7+8 · 1+3+5+6+8 · 1+4+5+6+7 · 2+3+4+5+9 · 2+3+4+6+8 · 2+3+5+6+7
24	1+2+4+8+9 · 1+2+5+7+9 · 1+2+6+7+8 · 1+3+4+7+9 · 1+3+5+6+9 · 1+3+5+7+8 · 1+4+5+6+8 · 2+3+4+6+9 · 2+3+4+7+8 · 2+3+5+6+8 · 2+4+5+6+7
25	1+2+5+8+9 · 1+2+6+7+9 · 1+3+4+8+9 · 1+3+5+7+9 · 1+3+6+7+8 · 1+4+5+6+9 · 1+4+5+7+8 · 2+3+4+7+9 · 2+3+5+6+9 · 2+3+5+7+8 · 2+4+5+6+8 · 3+4+5+6+7
26	1+2+6+8+9 · 1+3+5+8+9 · 1+3+6+7+9 · 1+4+5+7+9 · 1+4+6+7+8 · 2+3+4+8+9 · 2+3+5+7+9 · 2+3+6+7+8 · 2+4+5+6+9 · 2+4+5+7+8 · 3+4+5+6+8
27	1+2+7+8+9 · 1+3+6+8+9 · 1+4+5+8+9 · 1+4+6+7+9 · 1+5+6+7+8 · 2+3+5+8+9 · 2+3+6+7+9 · 2+4+5+7+9 · 2+4+6+7+8 · 3+4+5+6+9 · 3+4+5+7+8
28	1+3+7+8+9 · 1+4+6+8+9 · 1+5+6+7+9 · 2+3+6+8+9 · 2+4+5+8+9 · 2+4+6+7+9 · 2+5+6+7+8 · 3+4+5+7+9 · 3+4+6+7+8
29	1+4+7+8+9 · 1+5+6+8+9 · 2+3+7+8+9 · 2+4+6+8+9 · 2+5+6+7+9 · 3+4+5+8+9 · 3+4+6+7+9 · 3+5+6+7+8
30	1+5+7+8+9 · 2+4+7+8+9 · 2+5+6+8+9 · 3+4+6+8+9 · 3+5+6+7+9 · 4+5+6+7+8
31	1+6+7+8+9 · 2+5+7+8+9 · 3+4+7+8+9 · 3+5+6+8+9 · 4+5+6+7+9
32	2+6+7+8+9 · 3+5+7+8+9 · 4+5+6+8+9
33	3+6+7+8+9 · 4+5+7+8+9
34	4+6+7+8+9
35	5+6+7+8+9

How to Use Combinations to Solve

Start with unique runs. Scan the grid for magic-block sums (3, 4, 16, 17 in two cells; 6, 7, 23, 24 in three; and so on). Their digits are certain immediately.
Intersect crossing runs. A cell belongs to one across run and one down run, so its digit must appear in a legal combination of both. If 4-in-two ({1,3}) crosses 23-in-three ({6,8,9}), the shared cell has no legal digit — but crossing 24-in-three ({7,8,9}) with 17-in-two ({8,9}) narrows the cell to 8 or 9.
Update as you place digits. Each placed digit removes combinations from the rest of its run. Use the include/exclude filters above to see what remains, and watch the certain-digits callout.
Mind the no-repeat rule.A digit placed in a run excludes it from the rest of that run only — the same digit can appear again elsewhere in the puzzle.

Frequently Asked Questions

What does the kakuro combination calculator do?

It lists every set of distinct digits 1–9 that adds to your run’s clue. Mark digits as definitely-in or definitely-out and the list narrows accordingly — with certain digits called out.

Why can’t a digit repeat within a run?

It’s kakuro’s core rule, like a sudoku row. 10-in-two can never be 5+5 — only 1+9, 2+8, 3+7, or 4+6. The restriction is what makes the puzzle logical rather than arithmetic guesswork.

What are unique combinations or “magic blocks”?

Sums with exactly one legal digit set — like 3-in-two ({1,2}) or 24-in-three ({7,8,9}). They hand you certain digits for free, so start solving there. The full list is in the table above.

How do I solve where two runs cross?

The shared cell’s digit must appear in some legal combination of both runs. Intersect the two digit menus; if only one digit survives, place it.

What sums are legal for each run length?

From the smallest distinct digits (1+2+…) to the largest (9+8+…): 2 cells is 3–17, 3 cells is 6–24, 4 cells is 10–30, 5 cells is 15–35, and a 9-cell run is always exactly 45.

Does this solve the whole grid for me?

It’s the combination reference every solver keeps at hand rather than a full-grid autosolver — it does the arithmetic legwork and leaves the crossing deductions to you.

Is it free?

Yes — free, no signup, and everything is computed instantly in your browser.

Want to Make Kakuro Puzzles Instead?

This page helps you solve kakuro. To generate printable kakuro puzzles — for practice sheets or a KDP puzzle book — use the free kakuro generator with adjustable grid sizes and answer keys.

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