KAKURO

'Kakuro' is a kind of logic puzzle that is often referred to as a mathematical transliteration of the crossword. Kakuro puzzles are regular features in most, if not all, math-and-logic puzzle publications in the United States. Dell Magazines came up with the original English name ''Cross Sums'' and other names such as ''Cross Addition'' have also been used, but the Japanese name ''Kakuro,'' abbreviation of Japanese ''kasan kurosu,'' (加算クロス, addition cross) seems to have gained general acceptance and the puzzles appear to be titled this way now in most publications. The popularity of Kakuro in Japan is immense, second only to Sudoku among Nikoli's famed logic-puzzle offerings.[1]

Contents

Standard play and terminology

Solving techniques

Mathematics of Kakuro

Variants

See also

External links

Standard play and terminology

The canonical Kakuro puzzle is played in a grid of filled and empty cells — "black" and "white", respectively — usually 16×16 in size, but these dimensions can vary widely. Apart from the top row and leftmost column — which are entirely black — the grid, just like a crossword, is divided into "entries" — orthogonal lines of white cells — by the black cells. The black cells themselves — possibly barring those in a cluster — are not entirely solid but rather contain a diagonal slash from upper-left to lower-right and a number in one or both halves, such that each horizontal entry has a number in the black half-cell to its immediate left and each vertical entry has a number in the black half-cell immediately above it. These numbers, continuing the borrowed crossword terminology, are commonly called "clues".
The object of the puzzle is to insert a digit from 1 to 9 inclusive into each white cell such that the sum of the numbers in each entry matches the clue associated with it and that no digit is duplicated in any entry. It is that lack of duplication that makes creating Kakuro puzzles with unique solutions possible, and which means solving a Kakuro puzzle involves investigating combinations more, compared to Sudoku in which the focus is on permutations.
Some publishers prefer to print their Kakuro grids exactly like crossword grids, with no labeling in the black cells and instead numbering the entries, providing a separate list of the clues akin to a list of crossword clues. (This eliminates the row and column that are entirely black.) This is purely an issue of image and does not affect solving (at least, not beyond the degree of needing to look outside the grid to solve).
In discussing Kakuro puzzles and tactics, the typical shorthand for referring to an entry is "(clue, in numerals)-in-(number of cells in entry, spelled out)", such as "16-in-two" and "25-in-five". The exception is what would otherwise be called the "45-in-nine" — simply "45" is used, since the "-in-nine" is mathematically implied (nine cells is the longest possible entry, and since it cannot duplicate a digit it must consist of all the digits from 1 to 9 once). Curiously, "3-in-two", "4-in-two", "5-in-two", "43-in-eight", and "44-in-eight" are still frequently called as such, despite the "-in-two" and "-in-eight" being equally implied.

Solving techniques

Although brute-force guessing is of course possible, a better weapon is the understanding of the various combinatorial forms that entries can take for various pairings of clues and entry lengths. Those entries with sufficiently large or small clues for their length will have fewer possible combinations to consider, and by comparing them with entries that cross them, the proper permutation — or part of it — can be derived. The simplest example is where a 3-in-two crosses a 4-in-two: the 3-in-two must consist of '1' and '2' in some order; the 4-in-two (since '2' cannot be duplicated) must consist of '1' and '3' in some order. Therefore, their intersection must be '1', the only digit they have in common.
A "box technique" can also be applied on occasion, when the geometry of the unfilled white cells at any given stage of solving lends itself to it: by summing the clues for a series of horizontal entries (subtracting out the values of any digits already added to those entries) and subtracting the clues for a mostly-overlapping series of vertical entries, the difference can reveal the value of a partial entry, often a single cell.
It is common practice to mark potential values for cells in the cell corners until all but one have been proven impossible; for particularly challenging puzzles, sometimes entire ranges of values for cells are noted by solvers in the hope of eventually finding sufficient constraints to those ranges from crossing entries to be able to narrow the ranges to single values. Because of space constraints, instead of digits some solvers use a positional notation, where a potential numerical value is represented by a mark in a particular part of the cell, which makes it easy to place several potential values into a single cell. This also makes it easier to distinguish potential values from solution values.
Some solvers also use graph paper to try various digit combinations before writing them into the puzzle grid.

Mathematics of Kakuro

Kakuro puzzles are NP-complete [2].
There are two kinds of mathematical symmetry readily identifiable in kakuro puzzles: minimum and maximum constraints are duals, as are missing and required values.
All sum combinations can be represented using a bitmapped representation. This representation is useful for determining missing and required values using bitwise logic operations.

Variants

A relatively common variant of Kakuro is its logical successor, ''Cross Products'' (or ''Cross Multiplication''), where the clues are the product of the digits in the entries rather than the sum. Dell Magazines has produced such puzzles, but also allowed repeating of digits aside from 1 due to space limitations in the number of digits in each product in a puzzle. On the other hand, puzzles by Games Magazines are more like crossword puzzles, allowing the implementation of the no-repeating digits rule.
Another variant is ''Arrow Numbers'', where the combinations for each clue value cannot be repeated within the grid. Still another variant is having a different range of values that are inserted in the cells, such as 1 to 12, instead of the standard 1 to 9.
The final puzzle of the 2004 United States qualifier for the World Puzzle Championship is titled ''Cross Number Sums Place'': it is a ''Cross Sums'' where every row and column of the grid (except the top row and leftmost column as usual) contains exactly nine white cells, none of which — even across multiple entries — are allowed to use the same digit twice, like a ''Number Place'' (''Sudoku''); in addition, small circles are printed on the borders between some white cells; numerically adjacent digits must be placed astride those circles, and may not appear orthogonally adjacent when not astride a circle.

See also

★ Killer Sudoku, a variant of Sudoku which is solved using similar techniques.

External links

★ Tutorial at Nikoli (Macromedia Flash required)

★ Cross Sums Number Combination Guide: Guide detailing the various number combinations that are possible for a certain number of blanks and a certain number of spaces.

★ Complexity and Completeness of Finding Another Solution and its Application to Puzzles: Mathematical reference proving NP-completeness.

★ Will's Kakuro Masterclass: An introduction to Kakuro for beginners

★ The New Grid on the Block: ''The Guardian'' newspaper's introduction to Kakuro

★ Table of combinations for use in Kakuro