How to Solve Kakuro: Cross-Sums Made Simple

Updated June 2026

Kakuro is a crossword built from sums: fill each run of white cells with digits that add up to the clue, using no digit twice in a run. It looks like arithmetic, but the real solving is combinatorics — knowing which sets of digits can possibly make a clue, then letting the crossings pin down the order.

The two rules everything rests on

A run is a straight line of white cells ending at a clue. Two rules govern it: the digits (1–9) must add up to the clue, and no digit repeats within the same run. The no-repeat rule is what makes Kakuro solvable by logic instead of trial and error, because it sharply limits which combinations are even possible for a given length and total.

Learn the magic combinations

Short runs with extreme totals have only one possible set of digits — these "magic combinations" are free information:

• Two cells totalling 3 → {1,2}; totalling 4 → {1,3}; 16 → {7,9}; 17 → {8,9}.
• Three cells totalling 6 → {1,2,3}; 7 → {1,2,4}; 23 → {6,8,9}; 24 → {7,8,9}.

When a clue has a unique combination, you know exactly which digits fill that run — only their arrangement is left, and the crossing clues decide that. Memorising a handful of these is the biggest single speed-up in Kakuro.

Solve at the intersections

Every white cell belongs to one horizontal run and one vertical run, and its digit must satisfy both. So the cell's value is the overlap of the two sets. If the across run can only be {7,9} and the down run can only be {7,8}, the shared cell must be 7 — the only digit both allow. Hunting for cells where two restrictive clues cross is how you find your first certain entries.

Eliminate, don't just place

Treat each cell as a small list of candidate digits and cross off what cannot fit. A digit already forced elsewhere in the same run is gone; a digit absent from one of the cell's two possible combination sets is gone. Often you cannot place a value directly, but you can reduce a cell to two candidates, and then a crossing run removes one of them. Steady elimination beats staring for the answer.

The 45 trick for blocks

A full row or column of a 9-cell band sums to 45 (1 through 9 once each). When a cluster of runs nearly fills such a band, you can total the clues that cover it and compare with 45 to deduce a single leftover cell — the "magic 45" technique. Even on smaller grids, summing the across clues of a region and the down clues of the same region must agree, which catches errors and sometimes forces a lone cell.

Build outward from certainty

Start where constraints are tightest — the shortest runs and the most extreme totals — and let each placed digit tighten its crossing run. One forced cell typically cascades: it removes a candidate from a crossing run, which makes that run unique, which forces another cell. When the cascade stalls, look for the next unique combination or the next two-restrictive-clue intersection and start a new chain.

A practical plan

1. Mark the runs whose clue has a unique combination and fill those digit sets.
2. Solve cells at the intersection of two restrictive runs first.
3. Keep candidate lists per cell and eliminate using the no-repeat rule.
4. Use the 45 rule on nearly-complete rows, columns, and blocks.
5. Follow each forced digit's cascade through its crossings before searching anew.

Kakuro stops being intimidating once you see it as sets meeting at crossings rather than sums to compute. Learn the magic combinations, work the intersections, and the grid fills itself one forced digit at a time.