Kakuro Rules: How to Play, Combinations and Strategies

What is Kakuro?

Kakuro is a number puzzle laid out like a crossword. Instead of words, you write single digits from 1 to 9 into the white cells. Each horizontal or vertical stretch of white cells, called a run, has a clue, and the digits in the run must add up to that clue without repeating.

The clues live in black cells split by a diagonal line. The number above the diagonal is the target for the run going across to its right. The number below the diagonal is the target for the run going down beneath it. A clue cell can hold one number or two, depending on whether runs start next to it in one direction or both.

That is the whole puzzle: addition and a no-repeat rule. The depth comes from how the runs interlock. Every white cell sits in one across run and one down run, so each digit has to satisfy two sums at once. Fans of Sudoku usually find Kakuro the natural next step, since it adds light arithmetic to the same kind of elimination.

A short history

The puzzle began in the United States under the name Cross Sums, and it appeared for decades in Dell's puzzle magazines. The English name is usually credited to Jacob E. Funk, who worked for Dell.

The Japanese puzzle publisher Nikoli took up the format and called it kasan kurosu, from the Japanese word for addition and the English word "cross". That was shortened to kakkuro, and the Western name Kakuro comes from there. Nikoli is also the publisher that popularized Sudoku, and the two puzzles have traveled together ever since.

When the Sudoku craze swept British newspapers in the mid-2000s, editors looked for the next number puzzle, and Kakuro began appearing daily alongside it. Today it is a fixture of puzzle magazines, newspaper puzzle pages and apps. For more background, see Wikipedia's Kakuro article.

The core rules

Kakuro has four rules. Everything else in this guide is technique for applying them.

1. 1Write one digit from 1 to 9 in every white cellZero is never used. Black cells stay empty, and clue cells only hold the printed sums.
2. 2Each run adds up to its clueThe across clue sits to the left of its run and the down clue sits above it. A clue of 17 over two cells means the two digits total 17.
3. 3No digit repeats within a runA two-cell run clued 4 must be 1 and 3, never 2 and 2. Because of this rule, a run can be at most nine cells long.
4. 4Digits may repeat in different runsThere are no row, column or box rules as in Sudoku. The same digit can appear many times in a grid, as long as it never appears twice in one run.
5. 5There is exactly one solutionA properly constructed Kakuro can be solved by logic alone, with no guessing.

The no-repeat rule is what makes Kakuro a logic puzzle rather than a sum. Without it, a two-cell run clued 4 could be 2 + 2. With it, the only answer is 1 and 3, in some order.

A worked example

This small Kakuro is fully solved. Across clues are in gold, above each diagonal. Down clues are in blue, below it.

The highlighted cell shows how a solve starts. It belongs to the across run clued 17 over two cells, which can only be 8 + 9, and to the down run clued 16 over two cells, which can only be 7 + 9. The only digit in both sets is 9, so the cell must be 9. That immediately makes its neighbors 8 (to finish 17) and 7 (to finish 16), and the rest of the grid follows the same way: the three-cell runs clued 23 and 24 can only be {6, 8, 9} and {7, 8, 9}.

How to play: a step-by-step method

Here is a reliable routine for any Kakuro, from a beginner grid to a giant one:

1. Find the runs with only one combination. Scan every clue and note the ones that can be made just one way, such as 3 or 17 in two cells. These are your anchors.
2. Pencil in candidates. For each white cell, write the small digits that are allowed by both of its runs. Kakuro is much harder to hold in your head than Sudoku, so pencil marks pay off quickly.
3. Intersect the two runs. A cell's value must appear in the combinations for its across run and its down run. Cross out anything that appears in only one.
4. Place forced digits. When a cell is down to one candidate, write it in, then subtract it from both clues and remove it from the rest of each run.
5. Finish short runs. When all but one cell of a run is filled, the last digit is simply the clue minus the others.
6. Repeat across the grid. Each placed digit tightens the runs it touches. Keep sweeping until every cell is filled, then check each sum once.

Combinations worth memorizing

Experienced solvers know by heart which sums can be made only one way. These are the most useful, and they cover the smallest and largest total for each run length:

• Two cells. 3 = 1 + 2, 4 = 1 + 3, 16 = 7 + 9, 17 = 8 + 9.
• Three cells. 6 = 1 + 2 + 3, 7 = 1 + 2 + 4, 23 = 6 + 8 + 9, 24 = 7 + 8 + 9.
• Four cells. 10 = 1 + 2 + 3 + 4, 11 = 1 + 2 + 3 + 5, 29 = 5 + 7 + 8 + 9, 30 = 6 + 7 + 8 + 9.
• Five cells. 15 = 1 to 5, 16 = 1, 2, 3, 4, 6, 34 = 4, 6, 7, 8, 9, 35 = 5 to 9.
• Long runs. All nine digits always total 45, so an eight-cell run is missing exactly one digit: 45 minus the clue. A clue of 40 over eight cells means every digit except 5.

Strategies for faster solving

Once the basic routine is comfortable, these ideas crack the harder grids.

Related puzzles and variations

Kakuro's sum-and-no-repeat idea shows up in several close relatives:

• Cross Sums. The original American name for exactly the same puzzle, still used by some publishers.
• Mini and giant grids. Beginner Kakuro can be as small as a few runs. Magazine specials run to huge grids where the challenge is stamina and neat pencil marks.
• Killer Sudoku. A Sudoku grid overlaid with dotted cages, each with a sum and no repeated digits. It borrows Kakuro's arithmetic and adds the full Sudoku rules. See our Killer Sudoku guide.
• KenKen. A Latin-square puzzle whose cages can use addition, subtraction, multiplication or division. Read the KenKen rules for the details.

Common mistakes

The most common error is reading the wrong clue. Remember the geometry: the number above the diagonal belongs to the run on its right, and the number below the diagonal belongs to the run beneath it. Mixing them up leads to a contradiction many moves later.

Second, beginners often forget the no-repeat rule when a sum looks easy. An even clue over two cells, such as 8, is never split down the middle as 4 + 4. Third, solvers tend to skip pencil marks on medium grids and then lose track of which digits are still possible. Finally, a digit that is legal in its across run may still break its down run, so always check both before writing it in.

Kakuro vs Sudoku

Both puzzles use the digits 1 to 9 and both are solved by elimination, but they differ in important ways. Sudoku always uses a 9×9 grid with row, column and box rules, and its clues are given digits. Kakuro grids vary in size and shape, its clues are sums, and its only repeat rule applies inside each run.

In practice, Kakuro asks for more bookkeeping and a little mental arithmetic, while Sudoku asks for more pattern spotting. Many solvers enjoy switching between them. If the arithmetic side appeals to you, KenKen takes it further with four operations, and Futoshiki swaps sums for greater-than and less-than signs.

Frequently asked questions