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Kakuro rules

Kakuro looks like a crossword with numbers: dark cells hold the clues, white cells take digits, and every run of white cells must add up to its clue.

  1. Write a digit from 1 to 9 in every white cell.
  2. The digits of each across run add up to the clue on its left; the digits of each down run add up to the clue above it.
  3. No digit appears twice in the same run.

Kakuro is also known as cross sums; one grid is a cross sum.

Try it on small grids

Three small Kakuro puzzles to get the feel. Hints work here too and explain each step.

Practice 1 · 5x5 · Easy

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Practice 2 · 5x5 · Easy

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Practice 3 · 6x6 · Medium

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Your first moves

Look for the runs with unique sums: two cells adding to 3, 4, 16 or 17; three cells adding to 6, 7, 23 or 24. Write their digits as notes. Where two such runs cross, the shared cell must hold a digit both allow; often only one does. Then fill the last cell of any run whose other cells are known: it is the clue minus the rest.

A worked example, step by step

Our solver finished this Kakuro (6x6) in 46 steps. Every step names its technique; step through it on the board, or read the whole solve below.

16251510169201822372254123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789
The start
162515101692018223722549712278712353512323898951212345689345678941234561212341313
After step 23
16251510169201822372254972772513985237461431
Solved, after step 46
All 46 steps
  1. Unique sums. 16 in 2 cells can only be made as 7+9, so B2, C2 use just those digits.
  2. Unique sums. 16 in 2 cells can only be made as 7+9, so B2, B3 use just those digits.
  3. Unique sums. 15 in 5 cells can only be made as 1+2+3+4+5, so E2, E3, E4, E5, E6 use just those digits.
  4. Unique sums. 3 in 2 cells can only be made as 1+2, so B5, B6 use just those digits.
  5. Unique sums. 4 in 2 cells can only be made as 1+3, so E6, F6 use just those digits.
  6. Combination intersection. The across 9 at E2 in 2 cells can still be 1+8, 2+7, 3+6, 4+5; F2 loses the digits none of them uses.
  7. Combination intersection. The down 10 at F2 in 2 cells can still be 1+9, 2+8, 3+7, 4+6; F2, F3 lose the digits none of them uses.
  8. Combination intersection. The across 18 at B3 in 5 cells can still be 1+2+3+4+8, 1+2+3+5+7, 1+2+4+5+6; B3, C3, D3, F3 lose the digits none of them uses.
  9. Naked single. B3 can only be 7, so neither of its runs uses 7 again.
  10. Naked single. B2 can only be 9, so neither of its runs uses 9 again.
  11. Naked single. C2 can only be 7, so neither of its runs uses 7 again.
  12. Combination intersection. The down 10 at F2 in 2 cells can still be 2+8, 3+7, 4+6; F2, F3 lose the digits none of them uses.
  13. Combination intersection. The across 18 at B3 in 5 cells can still be 1+2+3+5+7; C3, D3, E3, F3 lose the digits none of them uses.
  14. Combination intersection. The down 20 at D3 in 3 cells can still be 3+8+9, 4+7+9, 5+6+9, 5+7+8; D3, D4, D5 lose the digits none of them uses.
  15. Combination intersection. The across 22 at C4 in 3 cells can still be 5+8+9, 6+7+9; C4, D4, E4 lose the digits none of them uses.
  16. Naked single. E4 can only be 5, so neither of its runs uses 5 again.
  17. Combination intersection. The across 9 at E2 in 2 cells can still be 1+8, 2+7, 3+6; E2, F2 lose the digits none of them uses.
  18. Combination intersection. The down 10 at F2 in 2 cells can still be 2+8, 3+7; F2 loses the digits none of them uses.
  19. Combination intersection. The across 9 at E2 in 2 cells can still be 1+8, 2+7; E2, F2 lose the digits none of them uses.
  20. Combination intersection. The across 22 at C4 in 3 cells can still be 5+8+9; C4, D4 lose the digits none of them uses.
  21. Combination intersection. The down 7 at F5 in 2 cells can still be 1+6, 2+5, 3+4; F5 loses the digits none of them uses.
  22. Combination intersection. The across 5 at B6 in 2 cells can still be 1+4, 2+3; C6 loses the digits none of them uses.
  23. One place left. Every combination left for the down 15 at E2 (1+2+3+4+5) needs a 4, and only E5 can take it.
  24. Naked single. E5 can only be 4, so neither of its runs uses 4 again.
  25. Combination intersection. The down 7 at F5 in 2 cells can still be 1+6, 2+5; F5, F6 lose the digits none of them uses.
  26. Naked single. F6 can only be 1, so neither of its runs uses 1 again.
  27. Naked single. E6 can only be 3, so neither of its runs uses 3 again.
  28. Last cell of a run. The other cells of the down 7 at F5 add up to 1, so F5 is 7 - 1 = 6.
  29. Naked single. F5 can only be 6, so neither of its runs uses 6 again.
  30. Combination intersection. The across 22 at B5 in 5 cells can still be 1+2+4+6+9, 1+3+4+6+8, 2+3+4+6+7; C5, D5 lose the digits none of them uses.
  31. Full run analysis. Placing the digits of the down 25 at C2 cell by cell, with the candidates each cell has left, only 2+3+5+7+8, 1+4+5+7+8, 2+3+4+7+9, 1+3+5+7+9 can be completed; C3, C5 lose the digits that never fit.
  32. One place left. Every combination left for the across 18 at B3 (1+2+3+5+7) needs a 1, and only E3 can take it.
  33. Naked single. E3 can only be 1, so neither of its runs uses 1 again.
  34. Naked single. E2 can only be 2, so neither of its runs uses 2 again.
  35. Last cell of a run. The other cells of the across 9 at E2 add up to 2, so F2 is 9 - 2 = 7.
  36. Last cell of a run. The other cells of the down 10 at F2 add up to 7, so F3 is 10 - 7 = 3.
  37. Naked single. F3 can only be 3, so neither of its runs uses 3 again.
  38. Naked single. D3 can only be 5, so neither of its runs uses 5 again.
  39. Naked single. C3 can only be 2, so neither of its runs uses 2 again.
  40. Combination intersection. The down 25 at C2 in 5 cells can still be 2+3+4+7+9; C4, C5, C6 lose the digits none of them uses.
  41. Naked single. C4 can only be 9, so neither of its runs uses 9 again.
  42. Naked single. D4 can only be 8, so neither of its runs uses 8 again.
  43. Naked single. C5 can only be 3, so neither of its runs uses 3 again.
  44. Last cell of a run. The other cells of the down 20 at D3 add up to 13, so D5 is 20 - 13 = 7.
  45. Last cell of a run. The other cells of the across 22 at B5 add up to 20, so B5 is 22 - 20 = 2.
  46. Naked single. B5 can only be 2, so neither of its runs uses 2 again.

Common mistakes

  • Repeating a digit inside a run because the sum still works: 4 in two cells is 1+3, never 2+2.
  • Reading the across clue as the down clue: across is above the diagonal, down is below it.
  • Using 0: Kakuro digits run from 1 to 9.

Where Kakuro comes from

Cross-sum puzzles appeared in American puzzle magazines in the middle of the 20th century. The Japanese publisher Nikoli gave them the name Kakuro, and the puzzle came back to Western newspapers in the 2000s next to sudoku.

Play today's KakuroSolving techniques