Kakuro

Kakuro or Kakro (Japaneseカックロkakkuro of加算 クロスkazan Kurosu, GV sum Cross ) is a number puzzle.

In German-speaking countries are also the names of cross number puzzles, cross sum and number Swede in use except Kakuro.

Regulate

The structure is similar to a crossword puzzle, but with numbers instead of letters and sums instead of word definitions.

The numbers in the top corner of a box indicate which sum the numbers in the set of free fields on the right to have it, the numbers in the bottom corner describe in the same way the vertical fields directly below.

The following rules apply:

• Only the digits 1 through 9
• In each sum each digit can appear only once
• Only one digit can be entered in any free field

Normally, the solution through the given numbers is clearly defined.

History

Kakuro is one of the oldest logic puzzle that is played on a grid. A Kakuro is already in the April / May issue of the Dell Publishing Company in 1950 and has been in Japan a similar popularity as Sudoku.

Representations

Since the Japanese word Kakuro, the root word kuro ( = black) plugged in, the total fields are often depicted as black areas ( with white letters ). Except for the representation of the sum fields the representation of a Kakuro is quite uniform.

Kakuro with white total fields

Kakuro with gray total fields

Mathematical considerations

Belonging to a number of free fields ( to the right of or below it, depending ) are called in the following " points" that number. And we are all reflections sum not forget that they testify only about the figures contained in the series, in general, but nothing about their order. The exact sequence must then be inferred from total reflections by more logical conclusions or combinations thereof.

Uniqueness of sums

The simplest mathematical reasoning, which one makes use when solving Kakuro is the following: for a given number of points ( fields) are specified sums uniquely determined. For example, considering the number 7 at the bottom right in the above example. It is to be formed with three points. For this purpose, only there is the combination of numbers (1, 2, 4) in an order not yet known in question. The number 6 can be left down only by a combination of numbers (1, 2, 3) represent in three places.

The following numbers have only one decomposition into two allowable summand: 3 (1 2 ), 4 ( 1 3 ), 16 ( 7 9) and 17 ( 8 9). Also can find numbers that allow only a single decomposition into even more summands.

At 2-9 fields there are 502 number sets (see discussion). In some fields / total combinations there is only one set of numbers. The maximum number of 12 sets in the sum of 20 and 4 fields, as well as in the sum of 25 and 5 fields. By the number of permutation possibilities increases in the latter case to 1440 (12 * 5 ). Although 9 fields have only one set of numbers, but by the permutation, the number of possibilities increases to 362,880th Some fields / total combinations do not have some digits. At 6 fields and sum 37, for example, there are two number sets, but none has a 1 if a 2 or a 5 is secured, neither a 3 nor a 4 occur. Conversely, no 2, and no 5 times when a is 3 or 4 secured. Strikethrough these candidates, the solution is further limited.

Minimum amount and maximum amount

Since in the fields of Kakuro only the digits may be entered 1 to 9, has a given number of digits whose sum is a maximum value. A series of four fields, for example, has a maximum value. As the same set of fields has a minimum value.

A formula, can the maximum or minimum value of a sum depending on the number of digits as express ( called the maximum, the minimum value ) follows:

( Gaussian sum formula )

Restrictions

Since the digits are not allowed to repeat in a row, a number has a maximum of nine digits and a maximum value of 45 = 1 2 3 4 5 6 7 8 9.

Solution methods

To solve Kakuro trying gradually to scheduling under more and more numbers in the appropriate fields, which then combine to form larger and larger areas.

Simple Kakuro contain groups of only one square, which require coincides with the sum of the number itself. Furthermore, often can be described by the sum of the specified at all possible numbers in the fields of this sum Narrow strong. As mentioned above sometimes arise unique decompositions. For example, a sum total of two fields and 3 contain only the numbers 1 and 2 and contain a total of seven with only one of 1 and 6 or 2 and 5 or 3 and 4. Also can provide the appropriate combination of options for all further sums and for larger areas. Sometimes complete tables of this sum decompositions to solve a Kakuro be used. In general, these can, however, find very quickly by logical considerations.

Often, individual numbers can be found that one considers the sum decompositions of different numbers at the same time. Thus, both the breakdown of the sum of 34 to 5 fields ( 4 6 7 8 9 ) as well as the sum of 7 to 3 fields (1 2 4) is left clear in the example, so that the common field for only the number remains 4 left.

Variants

• Japanese sums: Here are the positions of the blank fields not known.
• Killer Sudoku: Sudoku with Kakuro combines.