Kakuro Math Combinations Cheat Sheet
Every possible Kakuro combination for 2-cell through 5-cell runs. Print this page and keep it next to your puzzle book.
Kakuro combinations are the backbone of every solving strategy. A combination is a set of non-repeating digits from 1 through 9 that add up to a target sum in a given number of cells. Memorizing the most common ones speeds up your solving dramatically.
Two-cell combinations
Two-cell runs have the fewest options. The smallest possible sum is 3 (1+2) and the largest is 17 (8+9). Key unique two-cell combinations to memorize: 3 is always 1+2, 4 is always 1+3, 16 is always 7+9, and 17 is always 8+9. These appear in almost every puzzle and give you instant information.
Three-cell combinations
Three-cell runs range from 6 (1+2+3) to 24 (7+8+9). The unique combinations at the extremes are free: 6 is always 1+2+3, 7 is always 1+2+4, 23 is always 6+8+9, and 24 is always 7+8+9. Middle sums like 12, 13, or 14 have multiple combinations, so they take more work.
Four-cell combinations
Four-cell runs range from 10 (1+2+3+4) to 30 (6+7+8+9). The extremes are unique: 10 is always 1+2+3+4, 11 is always 1+2+3+5, 29 is always 5+7+8+9, and 30 is always 6+7+8+9. The middle range has many possible combinations. For a sum of 20 in four cells, there are eight possible sets.
Five-cell and beyond
Five-cell runs range from 15 (1+2+3+4+5) to 35 (5+6+7+8+9). As the run gets longer, there are fewer possible combinations because more digits are forced. A nine-cell run with a sum of 45 has only one combination: all nine digits. A seven-cell run is always missing exactly two digits that add up to 45 minus the clue sum.
How to use combinations while solving
When you see a clue, list the possible combinations. Then check which digits appear in every combination. Those digits are confirmed for the run even if you do not know their position yet. Next, check crossing runs. If a digit is not in any crossing combination, remove it. This overlap analysis is the single most powerful Kakuro technique.
Should you use a cheat sheet?
Absolutely, at first. Keep a printed combination table next to your puzzle book. After a few dozen puzzles, you will have the two-cell and three-cell extremes memorized. After a hundred puzzles, the four-cell extremes will stick too. The cheat sheet helps you learn without slowing you down.