Kakuro is solved mostly by knowing which sets of digits can make a clue. Because every run uses each digit from 1 to 9 at most once, some clues allow only one combination: a 2-cell run adding to 3 must be 1 and 2, and a 3-cell run adding to 24 must be 7, 8 and 9. These unique sums are where every kakuro starts. Below are the unique sums for every run length, the complete lists for 2- and 3-cell runs (the ones you will meet most), and the techniques that turn the chart into solved squares. Try them on a real grid in our free kakuro puzzles.
For every run length, the two smallest and two largest totals can be made only one way. Learn this table first; it unlocks most easy and medium puzzles.
The 8-cell row has a neat shortcut: all nine digits add to 45, so an 8-cell run totalling 39 is every digit except 45 − 39 = 6.
| Cells | Possible totals | Unique sums and their digits |
|---|---|---|
| 2 | 3 to 17 | 3 = 12, 4 = 13, 16 = 79, 17 = 89 |
| 3 | 6 to 24 | 6 = 123, 7 = 124, 23 = 689, 24 = 789 |
| 4 | 10 to 30 | 10 = 1234, 11 = 1235, 29 = 5789, 30 = 6789 |
| 5 | 15 to 35 | 15 = 12345, 16 = 12346, 34 = 46789, 35 = 56789 |
| 6 | 21 to 39 | 21 = 123456, 22 = 123457, 38 = 356789, 39 = 456789 |
| 7 | 28 to 42 | 28 = 1234567, 29 = 1234568, 41 = 2456789, 42 = 3456789 |
| 8 | 36 to 44 | Every total is unique: the missing digit is 45 minus the total |
| 9 | 45 only | 45 = 123456789 |
Notice that even totals never use a double: 6 cannot be 3 + 3 and 10 cannot be 5 + 5, because a run never repeats a digit. That alone removes a candidate from many cells.
| Total | Combinations |
|---|---|
| 3 | 12 |
| 4 | 13 |
| 5 | 14, 23 |
| 6 | 15, 24 |
| 7 | 16, 25, 34 |
| 8 | 17, 26, 35 |
| 9 | 18, 27, 36, 45 |
| 10 | 19, 28, 37, 46 |
| 11 | 29, 38, 47, 56 |
| 12 | 39, 48, 57 |
| 13 | 49, 58, 67 |
| 14 | 59, 68 |
| 15 | 69, 78 |
| 16 | 79 |
| 17 | 89 |
Middle totals such as 15 have many options, so leave those runs until crossing runs have narrowed them down.
| Total | Combinations |
|---|---|
| 6 | 123 |
| 7 | 124 |
| 8 | 125, 134 |
| 9 | 126, 135, 234 |
| 10 | 127, 136, 145, 235 |
| 11 | 128, 137, 146, 236, 245 |
| 12 | 129, 138, 147, 156, 237, 246, 345 |
| 13 | 139, 148, 157, 238, 247, 256, 346 |
| 14 | 149, 158, 167, 239, 248, 257, 347, 356 |
| 15 | 159, 168, 249, 258, 267, 348, 357, 456 |
| 16 | 169, 178, 259, 268, 349, 358, 367, 457 |
| 17 | 179, 269, 278, 359, 368, 458, 467 |
| 18 | 189, 279, 369, 378, 459, 468, 567 |
| 19 | 289, 379, 469, 478, 568 |
| 20 | 389, 479, 569, 578 |
| 21 | 489, 579, 678 |
| 22 | 589, 679 |
| 23 | 689 |
| 24 | 789 |
1. Intersect two unique runs
Where a horizontal 2-cell 4 (digits 1 and 3) crosses a vertical 2-cell 3 (digits 1 and 2), the shared square must be in both sets, so it is 1. The other squares follow immediately: 3 across and 2 down.
2. Rule out digits that cannot appear
A 3-cell 23 is 6, 8 and 9. If the crossing run is a 2-cell 4 (1 and 3), there is no overlap at all, which tells you the puzzle has been misread. If it crosses a 2-cell 14 (5 and 9 or 6 and 8), the shared square can only be 6, 8 or 9, removing 5 entirely.
3. Look for required digits
Some totals have a digit in every combination. Every 2-cell 16 contains 7 and 9; every 3-cell 22 contains 9; every 4-cell 11 contains 1, 2 and 3. When only one square in the run can hold that digit, place it.
4. Use the remaining sum
Once some squares are filled, subtract them from the clue and treat what is left as a smaller run. A 4-cell 20 with a 9 and a 1 placed leaves 10 across two squares, and 10 without 9 or 1 is 28, 37 or 46.
Scan for every unique sum before writing anything else.
Start in corners and short runs, where crossings are tight.
Pencil in candidates rather than guessing; a correct kakuro never needs a guess.
Re-check each run total as you fill it, because one wrong digit spreads quickly.
If you enjoy the arithmetic side, killer sudoku and calcudoku use the same sum logic.
Want the general approach to grid puzzles? Our how to solve logic puzzles guide covers elimination and marking, and logic puzzle tips lists ten techniques that carry across puzzle types.
Totals that can be made from only one set of distinct digits for a given run length, such as 3 in two cells (1 and 2) or 24 in three cells (7, 8 and 9).
No. Each run uses distinct digits from 1 to 9, which is why 2-cell runs never contain doubles like 4 + 4.
24, made only by 7, 8 and 9. The lowest is 6, made only by 1, 2 and 3.
Four: 1 and 9, 2 and 8, 3 and 7, 4 and 6. 5 and 5 is not allowed.
No. A well-made kakuro has one solution that can be reached by logic alone using combinations and crossings.
Learn the unique sums, cross them with each other, and look for digits that every combination must contain. With this chart beside you, most kakuro grids fall into place square by square. Practise on easy, medium and hard grids in our free kakuro game.