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## Generalization : Multidimensional groups

If you have carefully read the descriptions of the solution procedures, you may have noticed that some of the methods are grouped together in families. It is Therefore, useful to describe their rules in an abstract, generalized form, so that they are universally applicable.

First, some definitions:
Dimension - the size of the puzzle. The dimension must always be a square number.
Character set - the symbol set used in the puzzle. It always contains a number of symbols equal to the dimension.
BSize - the size of each block. This refers to the length of the sides of the inside squares. It is always the root of the dimension.
Logical Unit - the general name for rows, columns and blocks. In other words, those logical units that must contain each symbol of the character set exactly once. Occasionally there are puzzles in which the diagonals are also considered logical units.
Order - the ordinal number of a group. 2nd order would be a Twin, 3rd order would be a Triplet, etc. In this context, the order is the number of rows or columns that are present in the pattern.

The general formulation of the criteria for multidimensional groups, as seen from a row:
We're looking for order rows that contain the candidate x exactly twice. Here, the occurences of x are distributed over a total of exactly order columns. If these criteria are met, x can be eliminated from all of the fields in the columns that do not belong to the rows.
(Note: The regular distribution of occurences over the columns, where each column in turn has exactly two occurences for x, is not necessary for the pattern, but is usually the precondition for using of the pattern. An irregular distribution implies simpler solutions.)

If we have already made the effort to abstract and generalize the methodology, it also seems necessary to briefly discuss how using them makes sense.
Technical analyses suggest the assessment that the patterns in this family rarely occur per se. That seems logical due to their complexity. In addition, they quickly become even rarer the larger the dimension of the puzzle. A natural border seems to occur at the threshold value (½ * dimension), because the multidimensional groups behave in complementary fashion, just like the Hidden/Naked groups.

In the discussion of their usefulness, it quickly becomes tempting to think about whether the pattern could possibly be "extended". The candidat x, for example, would be expandable. One could also look for a group of candidates. The basis of the method would also be expandable. Why be limited to two occurences per row, as the rule requires? Perhaps it would be interesting to operate with three or more sites.

Sounds interesting at first, but in the end it is likely to be a naive thought. The candidate group of the extended x can be reduced to smaller groups with less complex patterns until you finally arrive back at single candidates.
The same applies for the pattern with the higher base. Imagine as the smallest conceivable variant - equivalent to an X-wing - a pattern of 3 rows and 3 columns, which has nine evenly distributed occurences for x. Now imagine making a choice for the first column. Then a pattern with two occurences remains for the remaining 2 rows and columns - the X-Wing, which was embedded in the larger pattern from the outset.
The higher basis can thus be reduced to less complex patterns with a lower basis.
Do you remember proofs in mathematics, or mathematical induction? This is exactly the phenomenon that we're talking about.
Although it is in a sense the exact opposite. From a methodological point of view, the enclosed, less complex patterns torpedo the existence of a higher basis from the outset.

Special cases of the multidimensional group:

• The X-Wing is a special case , because all occurrences per row cover all columns of the pattern.
• The Swordfish is a special case , because the distribution of occurences is automatically regular over the columns.
• The Jellyfish is the first "normal" member of the family (as are all higher orders). This means that the occurences may be irregularly distributed over the columns. It is valid in this irregular form, but unnecessary.

(Note: You may be asking yourself, why not go on and put the regularity of the distribution into the rules, if irregular Jellyfish are unnecessary? We could, but why? The rules are only meant to describe the patterns clearly, not to judge whether there may be easier ways to solve the puzzle.)