Largest number of empty groups?

Everything about Sudoku that doesn't fit in one of the other sections

Re: Largest number of empty groups?

Postby ghfick » Wed May 11, 2022 6:23 pm

I came across this thread a while back. It is noted in this thread that 9 empty groups might be the maximum and 4 empty blocks might be the maximum. Has this matter been explored more recently on this forum? Any definitive arguments in any case?
Gordon
ghfick
 
Posts: 162
Joined: 06 April 2016

Re: Largest number of empty groups?

Postby JPF » Fri May 13, 2022 4:59 pm

A "group" may be a box, a row or a column.
It's impossible to have more than 3 empty rows.
same thing for columns.
Then, to get 10 empty groups, one need at least 4 empty boxes.
More than 4 empty boxes is not possible for valid* pattern; see here
Finally,10 empty groups would require 4 empty boxes, 3 empty rows, 3 empty columns.
From here the only pattern that could be valid is this one or an equivalent:

Code: Select all
+---+---+---+
|...|...|...|
|...|...|.xx|
|...|...|.xx|
+---+---+---+
|...|...|...|
|.xx|.xx|...|
|.xx|.xx|...|
+---+---+---+
|...|...|...|
|.xx|.xx|...|
|.xx|.xx|...|
+---+---+---+

I leave it to Serg to remind us why that pattern is not valid!

*valid pattern : pattern containing at least a valid puzzle

JPF
JPF
2017 Supporter
 
Posts: 6017
Joined: 06 December 2005
Location: Paris, France

Re: Largest number of empty groups?

Postby Serg » Sat May 14, 2022 12:32 pm

Hi, JPF!
JPF wrote:
Code: Select all
+---+---+---+
|...|...|...|
|...|...|.xx|
|...|...|.xx|
+---+---+---+
|...|...|...|
|.xx|.xx|...|
|.xx|.xx|...|
+---+---+---+
|...|...|...|
|.xx|.xx|...|
|.xx|.xx|...|
+---+---+---+

I leave it to Serg to remind us why that pattern is not valid!

This pattern is not valid because it is subset of maximal invalid pattern P141
Code: Select all
        P141
+-----+-----+-----+
|. . .|. . x|. . .|
|. x x|. . x|. . .|
|. x x|. . x|. . .|
+-----+-----+-----+
|. . .|x x x|x x x|
|. . .|x x x|x x x|
|. . .|x x x|x x x|
+-----+-----+-----+
|. . .|x x x|x x x|
|. . .|x x x|x x x|
|. . .|x x x|x x x|
+-----+-----+-----+

from 40 maximal patterns list.

Serg
Serg
2018 Supporter
 
Posts: 801
Joined: 01 June 2010
Location: Russia

Re: Largest number of empty groups?

Postby JPF » Sat May 14, 2022 3:31 pm

To close the topic:

9 empty groups
    * with 4 empty boxes, here are all the possible ed patterns:
    Code: Select all
          A               B               C               D
    +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
    |...|...|...|   |...|...|...|   |...|...|.xx|   |...|...|...|
    |...|...|xxx|   |...|...|.xx|   |...|...|.xx|   |...|...|.xx|
    |...|...|xxx|   |...|...|.xx|   |...|...|.xx|   |...|...|.xx|
    +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
    |...|...|...|   |...|...|...|   |...|...|...|   |...|...|...|
    |.xx|.xx|...|   |.xx|xxx|...|   |.xx|.xx|...|   |.xx|.xx|...|
    |.xx|.xx|...|   |.xx|xxx|...|   |.xx|.xx|...|   |.xx|.xx|...|
    +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
    |...|...|...|   |...|...|...|   |...|...|...|   |.xx|.xx|...|
    |.xx|.xx|...|   |.xx|xxx|...|   |.xx|.xx|...|   |.xx|.xx|...|
    |.xx|.xx|...|   |.xx|xxx|...|   |.xx|.xx|...|   |.xx|.xx|...|
    +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+

    I'm sure Serg will find good reasons to prove that these patterns are invalid.

    * with 3 empty boxes, there are only two ed patterns:
    Code: Select all
          E               F
    +---+---+---+   +---+---+---+
    |...|...|...|   |...|...|...|
    |...|.12|.34|   |...|...|.xx|
    |...|.56|.78|   |...|...|.xx|
    +---+---+---+   +---+---+---+
    |...|...|...|   |...|...|...|
    |.74|...|.21|   |.xx|.xx|...|
    |.83|...|.47|   |.xx|.xx|...|
    +---+---+---+   +---+---+---+
    |...|...|...|   |...|...|...|
    |.68|.21|...|   |.xx|.xx|.xx|
    |.52|.64|...|   |.xx|.xx|.xx|
    +---+---+---+   +---+---+---+
    The first one E, valid, has been found by tso.
    I doubt that F is valid...
JPF
JPF
2017 Supporter
 
Posts: 6017
Joined: 06 December 2005
Location: Paris, France

Re: Largest number of empty groups?

Postby ghfick » Sat May 14, 2022 7:32 pm

Hi JPF,
This is very interesting. Can you elaborate on why there can be no more than three empty rows or three empty columns?
Gordon
ghfick
 
Posts: 162
Joined: 06 April 2016

Re: Largest number of empty groups?

Postby mith » Sat May 14, 2022 8:31 pm

More than three empty rows implies two empty rows in the same band, and any solution would have a second where those two rows are swapped. (Likewise for columns.)
mith
 
Posts: 889
Joined: 14 July 2020

Re: Largest number of empty groups?

Postby JPF » Sat May 14, 2022 10:36 pm

About his puzzle,
tso wrote:The puzzle has a unique solution and is not difficult.

Here is the highest rating I could get for the same equivalent pattern:
Code: Select all
+---+---+---+
|...|...|...|
|...|.12|.34|
|...|.53|.16|
+---+---+---+
|...|...|...|
|.45|...|.67|
|.87|...|.23|
+---+---+---+
|...|...|...|
|.32|.65|...|
|.58|.47|...|
+---+---+---+    SER=8.9

JPF
JPF
2017 Supporter
 
Posts: 6017
Joined: 06 December 2005
Location: Paris, France

Re: Largest number of empty groups?

Postby ghfick » Sat May 14, 2022 11:33 pm

A beautiful puzzle and very tough too!
Fascinating properties.
ghfick
 
Posts: 162
Joined: 06 April 2016

Previous

Return to General