A crossword built from arithmetic, fill every run so its digits are all different and add up to exactly the clue.
Each across and down run has a clue: the total its cells must add to. Fill the white cells with digits 1 to 9 so every run hits its sum and no digit repeats within a single run. Logic alone, no guessing required, solves a well-formed grid.
The puzzle is North American by birth and Japanese by name. In 1966 a Canadian working at Dell Magazines, Jacob E. Funk, gave it the English title Cross Sums — a fair description of the thing, a crossword whose answers are sums rather than words.
The name that stuck came later and from elsewhere. Kakuro is a contraction of the Japanese kasan kurosu (加算クロス), literally "addition cross", and it displaced Cross Sums so thoroughly that the puzzles now appear under the Japanese name in most publications — including the English-language ones that named it first.
In Japan it became a fixture of Nikoli, the publisher that built a house style out of hand-made logic puzzles, where it stands second in popularity only to Sudoku.
Further reading: A Crossword of Sums
Kakuro is the classic cross-sums number puzzle, rebuilt as an original, solver-verified free browser game with multiple difficulties and a daily challenge.
Kakuro is a logic puzzle often called a mathematical transliteration of the crossword, where digits 1–9 must sum to clues in horizontal and vertical entries without ever repeating a digit in the same