Kakuro is combinatorics, not arithmetic

The clues give you sums, but solving never hinges on addition. The real question is always: for this sum in this many cells, how many digit combinations exist? Digits 1–9 cannot repeat within a run (they may repeat across runs — the key difference from sudoku), so "17 in two cells" is not any pair adding to 17: it is 8+9, full stop. Runs with a unique combination are where every puzzle cracks open.

The magic combination tables

These sums decompose exactly one way — meet one and the candidates lock instantly:

The pattern is easy to remember: for each run length, the two smallest and the two largest sums are unique — the minimum can only be built from the smallest digits, the maximum only from the largest.

Intersections: take the overlap of two sets

Every cell belongs to one across-run and one down-run, and its candidates are the intersection of the two combination sets. A typical case: "17 in two" across (candidates 8, 9) crossing "12 in three" down — of the combinations 1+2+9, 1+3+8, 1+4+7, 2+3+7, 2+4+6 and 3+4+5, only the first two contain an 8 or 9, so that cell narrows to whichever of 8 and 9 survives both runs. The more magic runs a puzzle has, the faster its intersections collapse. An empty intersection means an earlier entry is wrong — a useful alarm, not a disaster.

Short runs first, extremes as filters

Two-cell runs never exceed four combinations, so process them first. Long runs of five or more have too many combinations to enumerate early — constrain them with extremes instead: n cells sum to at least 1+2+…+n and at most the top n digits. Clues near those bounds squeeze hard: a four-cell 12 cannot contain anything above 6, because the other three cells already cost at least 1+2+3.

The classic mistakes

Importing sudoku's "no repeats in the whole row" is the classic error: kakuro's no-repeat rule lives inside a single run only, and a black cell starts a new run where digits may recur. The other is writing digits that merely "look plausible" — every kakuro has exactly one solution, so every entry should carry a combination-based reason. A digit you cannot justify will be erased later, with interest.

On this site

Our kakuro offers easy, medium and hard boards, with difficulty controlling grid size and run lengths. The check button verifies a finished grid, and completion times are tracked per difficulty. Keep the tables beside you for a few games — they stop being necessary surprisingly fast.

Apply it immediately: open Kakuro and circle every magic run before entering a digit.