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Kakuro Combinations: Complete Cheat Sheet and Reference Guide

Kakuro puzzle grid showing clue numbers and valid digit combinations in geometric illustration style

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What are Unique Combinations in Kakuro?

Think of it as a Kakuro cheat sheet for solving Kakuro puzzles faster.

Kakuro puzzles are a logic puzzle built around one core constraint: each run can only be filled with non-repeating digits that sum to the clue.

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! 

New to Kakuro?

Start with the How to Play Kakuro guide first!

Why are Kakuro Combinations Important?

Kakuro game displayed on a mobile phone using a 9x17 grid size board

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.

In Kakuro puzzles, for any given run length and clue number, the number of possible combinations varies — some runs have many, others have exactly one. Some runs will have many eligible sum combinations, providing plenty of flexibility in 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 any logic puzzle — and Kakuro has more of them than most. 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

Kakuro run lengths from 2-cell to 9-cell shown as ascending columns representing increasing complexity

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

When solving Kakuro puzzles, 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:

In Kakuro puzzles, leverage forced answer values to narrow the possible combinations for every other run on the board.

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.

The more sum combinations you've memorized, the faster this elimination process becomes.

Full Combinations Reference Table

These tables cover every valid digit set in Kakuro puzzles. Bookmark this page — it's the logic puzzle equivalent of a multiplication table.

The possible combinations grow quickly as run length increases, which is why even experienced players keep this reference handy.

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.

How do I use a Kakuro combinations table?

Find the section matching your run length (e.g., "3-cell runs"), then locate your clue sum in the left column. Every digit set listed in the right column is a valid combination for that run. If only one combination is listed, it's a unique combination — those digits are guaranteed.

What is the most useful Kakuro combination to memorize?

Start with the 2-cell unique combinations: clue 3 = {1,2} and clue 17 = {8,9}. These appear frequently and give you instant forced answers. The 3-cell extremes (6 = {1,2,3} and 24 = {7,8,9}) are the next most useful.

How many combinations are possible in Kakuro?

It depends on run length and clue number. A 2-cell run has between 1 and 4 valid combinations, while longer runs have many more. The unique combinations table above lists every run where only one valid digit set exists.

Are Kakuro puzzles a type of logic puzzle?

Yes. Kakuro puzzles are a logic puzzle at their core — no guessing required. Every solution can be reached through deduction using run constraints and combination elimination alone.

Understanding valid digit sets is also essential for making Kakuro puzzles — puzzle designers use these combination tables to ensure every grid has exactly one solution.

What makes creating Kakuro puzzles difficult?

What makes creating Kakuro puzzles challenging is ensuring each grid has a unique solution. Every run length and clue combination must be carefully chosen so that only one valid digit set solves the puzzle.

What are the black cells in Kakuro called?

The black cells are called filled and barred cells. They display the clue sums and separate the white entry squares into runs.

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.

New to Kakuro entirely? Read why Kakuro is worth your time before diving into strategy.