Kakuro

A simple cross-sum puzzle (kakuro).

Associated solution

Kakuro or Kakro ( Japanese カックロ kakkuro from 加算クロス kasan kurosu , German 'sum + cross' ) is a type of logic puzzle .

In the German-speaking world, besides Kakuro, there are also the names cross number puzzle, cross sum and number swede in use.

regulate

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

The numbers in the upper corner of a box indicate the total of the numbers in the row of free fields to the right of it, the numbers in the lower corner describe the vertical fields directly below in the same way.

The following rules apply:

Only numbers from 1 to 9 may appear
Each digit may only appear once in each sum
Only one number may be entered in each free field

Usually the given numbers clearly define the solution.

history

Kakuro is one of the oldest logic puzzles that is played on a grid. A Kakuro is already in the April / May issue of Dell Publishing Company from 1950 and is as popular as Sudoku in Japan .

Representations

Since the Japanese word Kakuro has the root kuro (= black), the total fields are often shown as black areas (with white letters). Except for the display of the total fields, the display of a kakuro is quite uniform.

Traditional form of a kakuro with black total fields
Kakuro with white sum fields
Kakuro with gray sum fields

Mathematical considerations

The free fields belonging to a number (to the right of or below it, as the case may be) are called “digits” of this number in the following. With all considerations of sums, however, one must not forget that they only say something about the numbers in the series , but generally nothing about their order . The exact sequence must then be deduced from further logical conclusions or combinations of sum considerations.

Uniqueness of sums

The simplest mathematical consideration that one makes use of when solving Kakuro is the following: For a given number of places (fields), certain sums are uniquely determined. For example, if you look at the number 7 at the bottom right in the above example, it should be formed with three digits. Only the number combination (1, 2, 4) in a previously unknown order comes into question for this. The number 6 horizontally at the bottom left can only be represented by a combination of the numbers (1, 2, 3) in three places.

The following numbers have only a single decomposition into two admissible summands: 3 (1, 2), 4 (1, 3), 16 (7, 9) and 17 (8, 9). Numbers can also be found that only permit a single decomposition into even more summands.

With 2 to 9 fields there are 502 sets of numbers from which the solutions can consist. With some field-sum combinations there is only one set of numbers, the maximum of 12 number sets can be found with the sum of 20 and 4 fields and with the sum of 25 and 5 fields. The permutation increases the number of possibilities in the latter case to 1440 (12 × 5!) . With 9 fields there is only one set of numbers - namely all digits from 1 to 9 together - but the permutation increases the number of possibilities to 9! = 362880. With some fields-sum combinations, some digits are not included at all. With 6 fields and a total of 37 z. For example, there are 2 sets of numbers, but none of them contain a 1. If a 2 or a 5 is secured, neither a 3 nor a 4 can occur. Conversely, neither a 2 nor a 5 can occur if a 3 or a 4 is secured. Removing these candidates further narrows the solution.

Minimum and maximum amounts

Since only the digits 1 to 9 can be entered in the fields of the Kakuro, their sum has a maximum value for a given number of digits. For example, a row of four fields has a maximum value of . Likewise, the same set of fields has minimal value . ${\ displaystyle 30 = 9 + 8 + 7 + 6}$ ${\ displaystyle 10 = 1 + 2 + 3 + 4}$

In terms of a formula, the maximum or minimum value of a sum can be expressed as follows, depending on the number of digits ( denotes the maximum, the minimum value): ${\ displaystyle n}$ ${\ displaystyle u (n)}$ ${\ displaystyle l (n)}$

${\ displaystyle u (n) = \ sum _ {k = 9-n + 1} ^ {9} k = {\ frac {19 \ cdot nn ^ {2}} {2}}}$

${\ displaystyle l (n) = \ sum _ {k = 1} ^ {n} k = {\ frac {n \ cdot (n + 1)} {2}}}$ ( Gaussian empirical formula )

Restrictions

Since the digits must not be repeated in a row, a row has a maximum of nine digits and thus a maximum of 45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9.

Solution methods

To solve a Kakuro one tries to enter more and more numbers in the corresponding fields, which then add up to ever larger areas.

Simple kakuros contain groups of just one square, in which consequently the sum corresponds to the number itself. Furthermore, the specified sum can often be used to narrow down the numbers that are still possible in the fields of this sum. As mentioned above, there are sometimes clear decompositions. For example, a sum with two fields and sum 3 can only contain the numbers 1 and 2, and one with sum 7 only contains the summands 1 and 6 or 2 and 5 or 3 and 4. All other sums and even larger Specify the corresponding combination options for areas. Sometimes complete tables of these sum breakdowns are used to solve a kakuro. In general, however, these can be found very quickly through logical considerations.

Often, individual numbers can be found by looking at the decomposition of sums of different numbers at the same time. In the example on the left, both the division of the sum 34 into 5 fields (4 + 6 + 7 + 8 + 9) and the sum 7 into 3 fields (1 + 2 + 4) are unique, so that only the Number 4 remains.

variants

Japanese sums : The positions of the empty fields are not known here.
Killer Sudoku : It combines Kakuro with Sudoku .

Individual evidence

↑ Kakuro history. Retrieved on March 13, 2015 (English, when accessing a web browser in the German language setting, another page with the German title "Kakuro Geschichte" is forwarded, but the text itself remains in English ).

Web links

Commons : Kakuro - collection of images, videos and audio files

Link catalog on the topic of Kakuro at curlie.org (formerly DMOZ )

Tables of possible sums. ( PDF ; 37.1 KB) In: Kakuro-Knacker . Retrieved March 13, 2015 .

Kakuro Total Tables. In: PuzzleBooks.net . Retrieved March 13, 2015 .

[1] Kakuro history. Retrieved on March 13, 2015 (English, when accessing a web browser in the German language setting, another page with the German title "Kakuro Geschichte" is forwarded, but the text itself remains in English ).