Kakuro Rules & Solving Strategies for Beginners

Read the clues before entering digits

A run is an uninterrupted group of white cells that ends at a dark cell or the grid boundary. Count its cells before thinking about its sum. A clue of 7 over two cells describes a different problem from 7 over three cells: the number of available positions changes every possible combination.

In the GooferThings grid, the number in the upper-right part of a dark clue cell totals the white cells to its right. The lower-left number totals the cells below it. A clue cell can show both numbers because it can start an across run and a down run at the same time.

Every white cell participates in an across run and a down run. Its digit must satisfy both. Unlike Sudoku, Kakuro does not require a whole row or column to contain 1–9. Two separate runs in the same row can reuse digits, provided neither individual run contains a duplicate.

Separate a combination from its order

Consider a two-cell run with a total of 4. Its distinct digits must be 1 and 3. The apparent alternative 2 + 2 is forbidden because a run cannot repeat a digit. Knowing the combination does not tell you whether the first cell is 1 or 3; the crossing clues settle that.

Write the remaining possibilities as small notes. For this run, both cells initially have candidates {1,3}. These are alternatives, not two entries to add into the same cell. Once one cell becomes 1, the other must become 3, and both crossing runs need to be checked again.

Work through a crossing example

Imagine a small fragment with two white cells, A and B, in an across run totaling 4. A also belongs to a two-cell down run totaling 3. B belongs to a different two-cell down run totaling 7. These clues are enough to resolve the fragment without guessing.

The across clue gives A and B the candidates {1,3}. A's down clue permits only {1,2}. A must belong to both sets, so its only candidate is 1. The across run then requires B = 3. A's down partner becomes 2; B's down partner becomes 4.

Check the result in both directions: 1 + 3 = 4, 1 + 2 = 3, and 3 + 4 = 7. Each run has distinct digits. This example illustrates the central move in Kakuro: intersect the possibilities from two clues, then carry a confirmed value back into each affected run.

1. Count the cells in each run touching your chosen cell.
2. List the distinct-digit combinations that produce each clue.
3. Keep only digits allowed by both crossing runs.
4. Enter a digit only when one candidate remains.
5. Subtract confirmed digits from their clues and update the remaining cells.

Use the remaining sum carefully

After some digits are filled, subtract their total from the clue. If a four-cell run totaling 18 already contains 2 and 7, its other two cells must total 9. They cannot use 2 or 7 again. The possible pairs are therefore {1,8}, {3,6}, and {4,5}.

A crossing clue can reduce this further. If one remaining cell permits only {3,4}, the pair {1,8} disappears, but you still cannot choose between {3,6} and {4,5}. Keep the uncertainty in your notes. A valid partial deduction is useful even when it does not immediately produce an entry.

Choose productive places to look

Short runs are often useful because they have fewer combinations, but a nearly completed long run can be even easier. When a clue has one blank cell, subtract the filled digits and check that the result is between 1 and 9 and does not duplicate a digit already present.

Very low and very high sums are restrictive. Three different positive digits total at least 6 because 1 + 2 + 3 = 6. Three different digits from 1–9 total at most 24 because 7 + 8 + 9 = 24. These bounds can expose a mistaken entry before it spreads.

1. Scan for a run with one blank cell before listing large combinations.
2. Mark forced combinations such as two cells totaling 3 or 17.
3. After every entry, revisit both crossing runs instead of continuing only across.
4. Erase a candidate only when you can identify the clue or used digit that excludes it.

Check completion rather than appearance

A filled grid is not necessarily a solved grid. Review every across and down run independently: count its cells, add its digits, compare the total with the clue, and look for duplicates. A wrong digit can leave one direction correct while breaking the other.

If you reach a contradiction, return to the earliest entry whose reason you cannot explain. Common causes are overlooking a crossing clue, allowing a repeated digit, or treating a known combination as a known order. Recalculate the relevant run before changing several unrelated cells.

Worked example: follow each deduction

Read rows from top to bottom and columns from left to right, starting at 1. Read each instruction, then compare the orange changes with the previous picture. Keep undecided cells or segments open until the explanation settles them.

? = undecided

1. Follow these changes in order; the picture below includes all of them.

   • Write 1 at row 2, column 2. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 2; row 3, column 2; row 4, column 2; row 5, column 2, the remaining arrangements are (1, 2, 3, 4) or (1, 2, 4, 3); they all give 1 here.

   • Write 4 at row 5, column 2. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 5, column 2; row 5, column 3; row 5, column 4; row 5, column 5, the remaining arrangements are (4, 1, 2, 3) or (4, 2, 1, 3); they all give 4 here.

   • Write 1 at row 3, column 5. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 5; row 3, column 5; row 4, column 5; row 5, column 5, the remaining arrangements are (4, 1, 2, 3); they all give 1 here.

2. Follow these changes in order; the picture below includes all of them.

   • Write 2 at row 4, column 5. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 5; row 3, column 5; row 4, column 5; row 5, column 5, the remaining arrangements are (4, 1, 2, 3); they all give 2 here.

   • Write 2 at row 2, column 3. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 2; row 2, column 3; row 2, column 4; row 2, column 5, the remaining arrangements are (1, 2, 3, 4); they all give 2 here.

   • Write 3 at row 4, column 2. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 2; row 3, column 2; row 4, column 2; row 5, column 2, the remaining arrangements are (1, 2, 3, 4); they all give 3 here.

3. Follow these changes in order; the picture below includes all of them.

   • Write 3 at row 3, column 3. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 3, column 2; row 3, column 3; row 3, column 4; row 3, column 5, the remaining arrangements are (2, 3, 4, 1); they all give 3 here.

   • Write 4 at row 3, column 4. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 3, column 2; row 3, column 3; row 3, column 4; row 3, column 5, the remaining arrangements are (2, 3, 4, 1); they all give 4 here.

   • Write 4 at row 4, column 3. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 3; row 3, column 3; row 4, column 3; row 5, column 3, the remaining arrangements are (2, 3, 4, 1); they all give 4 here.

4. Follow these changes in order; the picture below includes all of them.

   • Write 1 at row 5, column 3. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 3; row 3, column 3; row 4, column 3; row 5, column 3, the remaining arrangements are (2, 3, 4, 1); they all give 1 here.

   • Write 1 at row 4, column 4. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 4, column 2; row 4, column 3; row 4, column 4; row 4, column 5, the remaining arrangements are (3, 4, 1, 2); they all give 1 here.

   • Write 2 at row 5, column 4. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 4; row 3, column 4; row 4, column 4; row 5, column 4, the remaining arrangements are (3, 4, 1, 2); they all give 2 here.

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