## Empty mini-Area (EmA): EmR and EmL

Advanced methods and approaches for solving Sudoku puzzles

### Empty mini-Area (EmA): EmR and EmL

This is done to formally put this into a separate thread as it was started in a sperate discussion. It stems from our use of confusing terms specifically the term Empty rectangle to describe a technique when it is should be a term to describe a grouped strong link in a box. I will follow this with some observations from 9x9 variant solving as well.

I came to the conclusion that the term Empty rectangle must only be used to describe a Single grouped strong link in a box where the inference of the strong link is derived from a mini-column and mini-row within that box.

We shouldn't therefore use "Empty rectangle" to describe a pattern of 2 strong links (one in a line and the other a grouped strong link in box). The term should be used to simplify the description of that Grouped strong link when used within a pattern (e.g. W-wing using an empty rectangle) (e.g. 2 strong links pattern using an empty rectangle)

the following is an attempt to categorize these Grouped strong links in location with a view to simplify description and reduce current confusion about terminology:

Code: Select all
`Empty mini-Area (EmA):======================1. Empty mini-Rectangle (EmR):------------------------------+---------+| .  X (X)|| X  /  / ||(X) /  / |+---------+a.k.a. Empty Rectangle(ER) *(see note)2. Empty mini-Line (EmL):-------------------------These have 4 varietiesa. in a Line (rows)+-------+------+-------+|X(X) . |X . . | / / / ||. .  . |. . . | . . . ||. .  . |. . . | . . . |+-------+------+-------+b. in a Line (columns)+-------+| X . . ||(X). . || . . . |+-------+| X . . || . . . || . . . |+-------+| / . . || / . . || / . . |+-------+c. in a box (mini-rows)+--------+| X(X)(X)|| X(X) . || / /  / |+--------+d. in a box (mini-columns)+-------+| X  X /||(X)(X)/||(X) . /|+-------+`

The exemplars above are what defines each category when all parenthesized candidates are present (regardless of how the dotted cells are filled). Any reduction in the number of candidates within parenthesis can lead to equivalence between some of these categories and may lead to a simple strong link in location (conjugate pair) when reduced to 2 candidate exemplar.

*: If there is an X candidate in the intersection cell of the EmR (also referred to as the heart cell) then this candidate is a potential cannibalistic elimination (only eliminated if it can see all fin cells or if the pattern is a ring/loop)

[EDIIT: added a note to explain the nature of candidate X in the intersection cell of an EmR (Thanks to rjamil)]
Last edited by tarek on Mon Mar 02, 2020 8:27 pm, edited 1 time in total.

tarek

Posts: 3748
Joined: 05 January 2006

### Re: Grouped strong links in 9x9 variants

With Regular shaped constraints varieties, the solver can observe the following:

Box or Skewed box constraints: In these constraints a box (or block) would be a region formed of 9 cells that are in the intersection of 3 parallel lines (e.g rows) and 3 other parallel lines (e.g.) columns.

Our vanilla sudoku is the main example of that with identical looking 3x3 boxes (blocks).

Windows (as in Windoku, Hypersudoku or SudokuW) there are 4 skewed boxes (blockes) that force the formation of 5 other regions (but still every region has 9 cells in 3x3 lines)

Disjoint groups (as in sudokuDG, SudokuP of Sudoku4D) is another example where each region has cells in a 3x3 line formation.

The above examples would have grouped strong links behaviour similar to the regular 3x3 boxes (blocks) that you would observe in vanilla sudoku. In particular, the Empty mini-Rectangle or what is described as "empty rectangle" applies to all of these regions. The EmL types c and d would also apply.

Line constraints: With these the 9-cell region intersects with 3 different regions.
The main example would be rows or columns in a vanilla sudoku where each line intersects with 3 different boxes (blocks).

A close similar example would be the X diagonal constraints in sudokuX.

Similar examples can be said on how row or columns would interact with Windows or disjoint groups. These lines would exhibit grouped strong link behaviour similar to the Empty mini-Line (EmL) types a and b
Last edited by tarek on Fri Dec 13, 2019 8:04 pm, edited 4 times in total.

tarek

Posts: 3748
Joined: 05 January 2006

### Re: Empty mini-Area (EmA): EmR and EmL

Examples with illustrations:

SudokuX: Grouped Strong link in the "Anti diagonal" weary linking through B7 with Grouped Strong link in a Row

SudokuX: Grouped Strong link in the "Main diagonal" weakly linked through a Row to an EmR (Empty rectangle) in B3

SudokuX: In this example there are Grouped Strong links in both diagonals weakly linked through B5

WindokuX: In this example the Windows Group (r678c159) has a grouped strong with a EmR (Empty rectangle) behaviour

WindokuX: Note that with this example the windows group in r159c159 has the grouped strong link having an EmR (empty rectangle) behaviour

SudokuDG (SudokuP): Observe DG2 which is Position 2 in each box (r147c258) in the light green cells has an EmR (empty rectangle) type grouped strong link that is weakly linked via C2 to the Grouped strong link in B7 (another empty rectangle) which results in r3c4<>1

SudokuDG (SudokuP): Observe DG1 which is position 1 in each box (r147c147) with has the EmR weakly linked via c1 to a strong link in r8

tarek

Posts: 3748
Joined: 05 January 2006