Kakuro Combinations: Complete Cheat Sheet and Reference Guide
Kakuro Combinations: Complete Cheat Sheet and Reference Guide
TL;DR
Kakuro “combinations” refer to the possible sets of digits that, when added, are equal to the clue number associated with that run. A “run” refers to the grouping of cells whose sum must equal the clue number.
Some combinations are unique, meaning that only one set of digits will complete the run. The digits included in such a combination serve as guaranteed cell answers for the rest of the board.
This guide lists every unique combination within the full reference table, so you can solve Kakuro faster without guessing.
What are Unique Combinations in Kakuro?
In Kakuro, some runs correspond to clue numbers that can be solved only by a single set of numbers, known as a unique combination.
Examples of unique combinations include:
- A 2-cell run summing to 3, only solvable by 1 and 2.
- A 2-cell run summing to 17, only solvable by 8 and 9.
Learning all of the unique combinations helps eliminate guesswork.
This guide serves as a complete reference for all Kakuro combinations, organized by run length and sum. Memorize these combinations until the patterns become second nature! Think of it as a Kakuro cheat sheet to help you as you play.
New to Kakuro?
Start with the How to Play Kakuro guide first!
Why are Kakuro Combinations Important?
Key concept: To solve a Kakuro run, players must fill the cells with different digits from 1 to 9 that add up to the run’s clue number.
For any given run length (number of cells) and clue number (sum of said cells), the number of valid digit combinations varies. Some runs will have many eligible combinations, providing the player with a lot of flexibility in terms of digit values and placement.
That said, some runs will only have one unique combination of digits whose sum equals its clue number.
Why unique combinations matter:
When only one unique combination suffices, you know which EXACT digits belong in that run. You don't know their order yet, but you've eliminated all the other digits from contention.
These forced answers are the most powerful shortcuts in Kakuro. They are also useful when solving the greater board, as unique combination digits are guaranteed to appear within the run in question.
Unique Combinations: Forced Answers
The following table includes all of the unique combinations possible in Kakuro.
Run Length | Clue Sum | Only Possible Digits |
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 | 2,3,4,5,6,7,9 |
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 | 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 |
How to Use Combinations While Solving
In Kakuro, players should work outward from certainty, not inward from ambiguity.
Step 1 - Identify Forced Answers:
Before writing anything, scan every run for unique combinations. These will give you guaranteed sets of digits. Upon finding a unique combination, immediately write those digits as candidates.
Step 2 - Use intersections to determine order:
Once you’ve identified your unique combination(s), use the intersecting run (the one that shares a cell with your original run) to determine the order of digits in your original run. If the intersecting cell can't contain a specific value, then that digit must be placed elsewhere in the run.
Step 3 - Eliminate values from non-unique runs:
Leverage the values from forced answer runs to help narrow your candidates for runs with multiple valid combinations.
If an intersecting cell is confirmed to contain the value 7, you can cross off 7 as a candidate for all of the other cells in that intersecting run.
Full Combinations Reference Table
2-cell runs
Clue Sum | Valid Digit 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
Clue Sum | Valid Digit 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
Clue Sum | Valid Digit 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 · 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 · 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 · 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 · 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 · 2,3,6,9 · 2,3,7,8 · 2,4,5,9 · 2,4,6,8 · 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
Clue Sum | Valid Digit 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 · 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 · 3,4,5,6,7 |
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 · 3,4,5,6,8 |
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 · 3,4,5,6,9 |
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 |
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 |
6-cell runs
Clue Sum | Valid Digit Combinations |
|---|---|
21 | 1,2,3,4,5,6 |
22 | 1,2,3,4,5,7 |
23 | 1,2,3,4,5,8 · 1,2,3,4,6,7 |
24 | 1,2,3,4,5,9 · 1,2,3,4,6,8 · 1,2,3,5,6,7 |
25 | 1,2,3,4,6,9 · 1,2,3,4,7,8 · 1,2,3,5,6,8 · 1,2,4,5,6,7 |
26 | 1,2,3,4,7,9 · 1,2,3,5,6,9 · 1,2,3,5,7,8 · 1,2,4,5,6,8 · 1,3,4,5,6,7 |
27 | 1,2,3,4,8,9 · 1,2,3,5,7,9 · 1,2,3,6,7,8 · 1,2,4,5,6,9 · 1,2,4,5,7,8 · 1,3,4,5,6,8 · 2,3,4,5,6,7 |
28 | 1,2,3,5,8,9 · 1,2,3,6,7,9 · 1,2,4,5,7,9 · 1,2,4,6,7,8 · 1,3,4,5,6,9 · 1,3,4,5,7,8 · 2,3,4,5,6,8 |
29 | 1,2,3,6,8,9 · 1,2,4,5,8,9 · 1,2,4,6,7,9 · 1,2,5,6,7,8 · 1,3,4,5,7,9 · 1,3,4,6,7,8 · 2,3,4,5,6,9 · 2,3,4,5,7,8 |
30 | 1,2,3,7,8,9 · 1,2,4,6,8,9 · 1,2,5,6,7,9 · 1,3,4,5,8,9 · 1,3,4,6,7,9 · 1,3,5,6,7,8 · 2,3,4,5,7,9 · 2,3,4,6,7,8 |
31 | 1,2,4,7,8,9 · 1,2,5,6,8,9 · 1,3,4,6,8,9 · 1,3,5,6,7,9 · 1,4,5,6,7,8 · 2,3,4,5,8,9 · 2,3,4,6,7,9 · 2,3,5,6,7,8 |
32 | 1,2,5,7,8,9 · 1,3,4,7,8,9 · 1,3,5,6,8,9 · 1,4,5,6,7,9 · 2,3,4,6,8,9 · 2,3,5,6,7,9 · 2,4,5,6,7,8 |
33 | 1,2,6,7,8,9 · 1,3,5,7,8,9 · 1,4,5,6,8,9 · 2,3,4,7,8,9 · 2,3,5,6,8,9 · 2,4,5,6,7,9 · 3,4,5,6,7,8 |
34 | 1,3,6,7,8,9 · 1,4,5,7,8,9 · 2,3,5,7,8,9 · 2,4,5,6,8,9 · 3,4,5,6,7,9 |
35 | 1,4,6,7,8,9 · 2,3,6,7,8,9 · 2,4,5,7,8,9 · 3,4,5,6,8,9 |
36 | 1,5,6,7,8,9 · 2,4,6,7,8,9 · 3,4,5,7,8,9 |
37 | 2,5,6,7,8,9 · 3,4,6,7,8,9 |
38 | 3,5,6,7,8,9 |
39 | 4,5,6,7,8,9 |
7-cell runs
Clue Sum | Valid Digit Combinations |
|---|---|
28 | 1,2,3,4,5,6,7 |
29 | 1,2,3,4,5,6,8 |
30 | 1,2,3,4,5,6,9 · 1,2,3,4,5,7,8 |
31 | 1,2,3,4,5,7,9 · 1,2,3,4,6,7,8 |
32 | 1,2,3,4,5,8,9 · 1,2,3,4,6,7,9 · 1,2,3,5,6,7,8 |
33 | 1,2,3,4,6,8,9 · 1,2,3,5,6,7,9 · 1,2,4,5,6,7,8 |
34 | 1,2,3,4,7,8,9 · 1,2,3,5,6,8,9 · 1,2,4,5,6,7,9 · 1,3,4,5,6,7,8 |
35 | 1,2,3,5,7,8,9 · 1,2,4,5,6,8,9 · 1,3,4,5,6,7,9 · 2,3,4,5,6,7,8 |
36 | 1,2,3,6,7,8,9 · 1,2,4,5,7,8,9 · 1,3,4,5,6,8,9 · 2,3,4,5,6,7,9 |
37 | 1,2,4,6,7,8,9 · 1,3,4,5,7,8,9 · 2,3,4,5,6,8,9 |
38 | 1,2,5,6,7,8,9 · 1,3,4,6,7,8,9 · 2,3,4,5,7,8,9 |
39 | 1,3,5,6,7,8,9 · 2,3,4,6,7,8,9 |
40 | 1,4,5,6,7,8,9 · 2,3,5,6,7,8,9 |
41 | 2,4,5,6,7,8,9 |
42 | 3,4,5,6,7,8,9 |
8-cell runs
Clue Sum | Valid Digit Combination |
|---|---|
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-cell runs
Clue Sum | Valid Digit Combination |
|---|---|
45 | 1,2,3,4,5,6,7,8,9 |
FAQ
What are Kakuro combinations?
Kakuro combinations are valid sets of different digits between 1 and 9 that simultaneously match the number of cells within a run and add up to the run’s corresponding clue number.
What are unique combinations in Kakuro?
Unique combinations in Kakuro refer to runs that can only be solved by a single set of digits. You can think of these combinations as Kakuro magic numbers to help reduce your solve times.
Where can I play Kakuro online?
You can play free daily Kakuro puzzles at kakuroconquest.com. The site offers beginner through expert difficulty levels, and does not require an account to play.