UNIQUE-SUM RUNS — START HERE
Some clue + length combinations have exactly one possible digit set. These are your free moves — no cross-referencing needed.
2-cell runs:
3 → {1, 2}
4 → {1, 3}
16 → {7, 9}
17 → {8, 9}
3-cell runs:
6 → {1, 2, 3}
7 → {1, 2, 4}
23 → {6, 8, 9}
24 → {7, 8, 9}
4-cell runs:
10 → {1, 2, 3, 4}
11 → {1, 2, 3, 5}
29 → {5, 7, 8, 9}
30 → {6, 7, 8, 9}
Scan every run in the grid on your first pass and mark the candidates for each unique-sum run. This populates a large chunk of the pencil marks before you've made a single placement.
CROSS-REFERENCING — THE CORE TECHNIQUE
Every cell sits at the intersection of one across-run and one down-run. When you know the candidate sets for both runs, the valid digits for that cell are only those in both sets.
Example: across-run candidates {1, 2, 4}, down-run candidates {2, 8, 9}. The only digit in both sets is 2 — so that cell is placed.
Most Kakuro puzzles are solvable by running this intersection check on every cell until the whole grid fills. The larger the candidate sets, the less this helps — but unique-sum runs keep the sets small enough to make it work.
NO REPEATS — THE HIDDEN ELIMINATOR
No digit can repeat within a single run. This is worth using actively, not just as a rule to avoid.
If a run of {1, 2, 4} already has a 1 placed somewhere in it, the other cells in that run can't be 1. Cross-referencing with the down-run candidates for those cells immediately eliminates 1 from each, often forcing a placement.
As the board fills, the no-repeat rule does more and more work — each placed digit prunes every run it touches.
PENCIL MARKS — WHEN AND HOW
Write the candidate set for each run lightly in the cells from the start. When a cell has only one candidate remaining after cross-referencing, place it.
When you place a digit: (1) erase it from the rest of the run, (2) update any crossing run that now has one fewer candidate. The cascade usually unlocks several cells in a row.
Don't wait to pencil-mark until you're stuck — mark early, update frequently. The bookkeeping is the technique.
THE SOLVING LOOP
- Mark all unique-sum run candidates on the first pass.
- For every empty cell, intersect its across and down candidate sets.
- Place any cell with a single candidate after intersection.
- Update both runs that cell belongs to, erasing the placed digit.
- Repeat from step 2 until the board is solved.
This loop handles most beginner and intermediate Kakuro grids. If you're stuck after a full pass, look for a run where a digit can only go in one position — the same as a hidden single in Sudoku.
THE BEGINNER TRAP
Trying to place digits by arithmetic alone — picking up a clue, listing possible sums, choosing one. That works but it's slow and bypasses the cross-reference step that actually narrows the candidates.
Kakuro puzzles are always solvable without guessing. If you feel like you need to guess, you have an unapplied cross-reference or a no-repeat elimination somewhere. Find it.