Cross number puzzle

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A cross number puzzle is a puzzle that has a scheme of boxes like a crossword puzzle , but in which numbers have to be entered instead of words. Similar to the crossword puzzle, conditions are specified for these numbers, usually arithmetic ; for example, the condition for a 36 to be found could be “is a square number”. More often than in crossword puzzles, the condition on a number also formulates a connection with one or more of the other numbers in the scheme, e.g. B: "is the sum of A horizontally and B vertically".

Cross number puzzles are difficult because the numbers you are looking for cannot usually be entered in one go ; Rather, by using what is already known and information about other, transverse or otherwise incoming numbers, one has to accumulate piece by piece knowledge about individual digits and use it until the contents of a box are fixed somewhere and can be entered.

In the example of the sought-after square number 36 from above, one could already know from other information or considerations that the first digit is 1, 2 or 3; the last one is then restricted to 5 or 6, since only 16, 25 or 36 are possible as square numbers. If only 1 or 3 is possible for the first, the value of 6 is already definitely fixed and can be entered, although not all digits of the square number sought here have been determined.

It is often advantageous to keep an eye out for the most restrictive conditions possible. In the case of two numbers of the same digit length to be entered, for example, one with the condition “is biquad” (ie the square of a square) is more restricted without prior knowledge than one with the condition “is a square number”; because there are 100 squares with four digits, namely 0 × 0 = 0000, 1 × 1 = 0001, ... 99 × 99 = 9801, but only 10 biquadrates with four digits, namely 0 × 0 × 0 × 0 = 0000, 1 × 1 × 1 × 1 = 0001,… 9 × 9 × 9 × 9 = 6561, and with the biquadrat condition the possibilities are more restricted at the individual places: In the first place with the square every digit can be, with the biquadrat only 0, 1, 2, 4, 6. Fewer cases for a single number tend to result in fewer cases for a single box.

Solving hard cross number puzzles is often hardly possible without the solver doing additional calculations on paper or even creating auxiliary tables for possible entries or entry combinations to aid memory and updating them later if necessary. Without notation, it is difficult and prone to errors to follow up hypotheses over longer chains, with the refutation of which one can exclude part of the possible digits in a certain box. Because of the effort, it is particularly important here to restrict oneself closely to the promising and not to get involved in exhaustion, where other ways deliver partial results effortlessly.

Skillful solving is mainly based on recognizing which of the information provides a return that can be used quickly, but in particular which, using the partial knowledge that has already been acquired, may now be "ripe" to definitely define another number. This allows the solver to avoid laborious case distinctions and written auxiliary tables, which would become unavoidable in a more awkward way. In this way, cross number puzzles can also be solved automatically by a computer.

Typical conditions in the definitions are for example:

The requirement that a number must be a prime number, for example, only allows the final digits 1, 2, 3, 5, 7, 9; in the case of two or more digit prime numbers and prohibition of the 0 even only the last digits 1,3,7,9. Every square has a final digit of 0, 1, 4, 5, 6 or 9.If it is stated that a four-digit number is a square of a two-digit one, and you also know that the four-digit number does not have a leading 0, then the beginning digit of the two-digit number cannot must be less than 3 because 1024 = 32 × 32 the first square is greater than 0999. Etc.

There is exactly one solution to well-put cross number puzzles that goes together with all the requirements. The zero usually does not appear in such puzzles. Often this is tacitly assumed, which the solver may only become clear when he discovers that without this assumption he would find two or more solutions. Often there are specifications that would not logically be needed for a solution, but can make the way there a lot easier. Some puzzlers ensure that their task can be solved equally clearly with and without the prohibition of zero, or they expressly mark certain details as being dispensable for solving; the puzzle solver then has the choice between different levels of difficulty by using them or not.

In addition to the frequent cross number puzzles, for the solution of which one only needs arithmetic or at best little more mathematical knowledge, other external educational knowledge can be included in the number conditions, for example through a condition such as "is the year of Mozart's birth".

Cross number puzzles are now also used in schools to make math lessons more varied. Various publishers offer various teaching materials.

See also

literature

credentials

  1. ↑ M. Groß, D. Plümpe (today Schmidt), M. Schmidt: Cross number puzzles: Sudokus were yesterday. In: Springer Verlag (Hrsg.): Informatik-Spektrum. 32, No. 6, 2009, ISSN 0170-6012, pp. 538-545.
  2. ↑ Table of contents RAAbits Mathematik (several directory numbers pertaining to cross number puzzles), published by the Vechta University Library .
  3. ^ Expedition Mathematics (Westermann Verlag)
  4. ↑ Cross number puzzles for elementary school
  5. ↑ A. Paulitsch: cross-figure mathematics