Kakuro

Kakuro, sometimes called Cross Sums or Cross Addition, is a number-based logic puzzle in which solvers enter digits into a crossword-like grid in order to make the sums in each row and column match the numbers at the beginning of them. Despite the Japanese name, the puzzle idea originated in an American puzzle magazine in 1950.

Background

Kakuro puzzles were invented by Jacob E. Funk, a Canadian puzzle writer for Dell Magazines, in 1950. At the time, the puzzle was called Cross Sums, and functioned like a number-based crossword puzzle (not to be confused with crossnumber puzzles), but with clues baked into the grid itself. The puzzles took a good bit of time to grow in popularity, eventually being featured in every issue by the mid 1960s.

In 1980, the puzzle was brought to Japan by Maki Kaji for use in Nikoli publications, calling them Kasan Kurosu (a combination of the Japanese word for 'addition' and the pronunciation of the English word 'cross'). This was eventually shortened to KaKuro, which has since become the most common name. After its adoption, Kakuro quickly became the most popular Nikoli-published puzzle genre, maintaining that lead until 1992, when Sudoku overtook it in popularity.

Puzzle Application

Kakuro puzzles can be identified by the presence of diagonally-split clue cells with numbers in a grid where the rest of the cells are either black or white. The goal of a Kakuro puzzle is to fill in the white cells with the digits 1-9 so that the sum of a given row/column's numbers is equal to the 'clue' number at the beginning of the row or column. Additionally, much like Sudoku, no number appears more than once in the same row or column.

Strategy

Much of the puzzle can be approach combinatorially. As such, it's very useful to know sums close to minimal or maximal, as any clue that is either minimal/maximal or 1 away has a unique set. For example, 4 cells summing to 11 must have a 1, 2, 3, and 5 in some order. For larger sets, being a few more away may still force certain digits to appear. For example, 4 cells summing to 13 must contain a 1, as 2+3+4+5=14 is too large.

Below is a table indicating the minimum and maximum for the number of clues in a row or column.

Clues Minimum Maximum
2 3 17
3 6 24
4 10 30
5 15 35
6 21 39
7 28 42
8 36 44

Depending on the shape of the grid, it may be possible to divide up a grid to get the sum of some cells when summing rows, and the sum of a few more cells when summing columns (or vice versa). Then you can subtract these sums to get the sum of one or a few cells.

Notable Examples

Played Straight

• Express Yourself (MITMH 2005) (web) - A relatively straightforward Kakuro puzzle, albeit a very large and difficult one. As long as you're not intimidated by the size, it should go down without encountering any serious gimmicks.
• Cross Something-Or-Others (web) (MITMH 2009) - While this puzzle does involve several variations, they're all explained and well-documented. Sure, a big part of the puzzle is determining which of the unlabelled grids belongs to each variation, but at least the puzzles are relatively standard!

Notable Twists

• The Joy of Accountancy (MITMH 2007) (web) - Normally, Kakuro puzzles require summing each individual digit together. In this case, though, some of the clues are way too high for that to be viable. Instead, solvers have to treat some strings as multi-digit numbers for adding purposes. While 9+8+2 would equal 19, 98+2 could equal 100, or 9+82 could equal 91.
• Stackuro (MIMTH 2017) (web) - Another Kakuro puzzle with clues too high to be attainable just from normal addition. Here, it's because two separate puzzles have been stacked on top of one another, resulting in the products of the clues being shown (and forcing solvers to do some factoring to figure out what the clues really are).