The New Sudoku Players' Forum

[Serg]

Hi, people!
blue paid our attention to the property of SudokuP valid puzzles to contain more than 4 empty boxes (ordinary sudoku valid puzzles may contain not more than 4 empty boxes). (See his post in the thread SudokuP - Min Clue Project.)
I investigated all possible configurations of empty boxes for SudokuP valid puzzles. Here are my results.
Configuration of empty boxes is denoted by 3 x 3 matrix, where each symbol denotes alone box - symbol "0" denotes empty box, symbol "X" denotes non-empty box.

Let's consider all possible configurations of empty boxes in SudokuP puzzles. There are 26 essentially different box configurations.

Code: Select all

                0 0 0
9 empty boxes   0 0 0
                0 0 0

                0 0 0
8 empty boxes   0 0 0
                0 0 X

                0 0 0   0 0 0
7 empty boxes   0 0 0   0 0 X
                0 X X   0 X 0
                              *****************
                0 0 0   0 0 0 * 0 0 0   0 0 X *
6 empty boxes   0 0 0   0 0 X * 0 0 X   0 X 0 *
                X X X   0 X X * X X 0   X 0 0 *
*******************************               **********
                0 0 0   0 0 0   0 0 0   0 0 X   0 0 X
5 empty boxes   0 0 X   0 X X   0 X X   0 0 X   0 X 0
                X X X   0 X X   X 0 X   X X 0   X 0 X

                0 0 0   0 0 X   0 0 X   0 0 X   0 0 X
4 empty boxes   0 X X   0 0 X   0 X 0   0 X X   X X 0
                X X X   X X X   X X X   X X 0   X X 0

                0 0 0   0 0 X   0 0 X   0 X X
3 empty boxes   X X X   0 X X   X X 0   X 0 X
                X X X   X X X   X X X   X X 0

                0 0 X   0 X X
2 empty boxes   X X X   X 0 X
                X X X   X X X

                0 X X
1 empty boxes   X X X
                X X X

                X X X
0 empty boxes   X X X
                X X X


It turns out, configurations placed upper asterisk line have no valid SudokuP puzzles, configurations placed lower asterisk line have valid SudokuP puzzles.
For example, configuration

Code: Select all

0 0 0
0 0 X
X X 0

(another view of this configuration)

Code: Select all

         P1
+-----+-----+-----+
|. . .|. . .|. . .|
|. . .|. . .|. . .|
|. . .|. . .|. . .|
+-----+-----+-----+
|. . .|. . .|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|. . .|
+-----+-----+-----+

has valid SudokuP puzzle:

Code: Select all

SudokuP valid puzzle:
         S1
+-----+-----+-----+
|. . .|. . .|. . .|
|. . .|. . .|. . .|
|. . .|. . .|. . .|
+-----+-----+-----+
|. . .|. . .|2 9 5|
|. . .|. . .|8 4 3|
|. . .|. . .|1 7 6|
+-----+-----+-----+
|3 4 8|9 7 2|. . .|
|2 6 5|1 8 4|. . .|
|9 1 7|5 6 3|. . .|
+-----+-----+-----+

The same in Line Form:

Code: Select all

.................................295......843......176348972...265184...917563...

You can see that SudokuP valid pattern may not have more than 6 empty boxes. The only possible configurations with 6 empty boxes are

Code: Select all

0 0 0     0 0 X
0 0 X     0 X 0
X X 0     X 0 0

Configurations P2 and P3 have no valid SudokuP puzzles.

Code: Select all

         P2                      P3
+-----+-----+-----+     +-----+-----+-----+
|. . .|. . .|. . .|     |. . .|. . .|. . .|
|. . .|. . .|. . .|     |. . .|. . .|. . .|
|. . .|. . .|. . .|     |. . .|. . .|. . .|
+-----+-----+-----+     +-----+-----+-----+
|. . .|. . .|. . .|     |. . .|. . .|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 x x|x x x|     |. . .|x x x|x x x|
+-----+-----+-----+     +-----+-----+-----+

P2 has no valid puzzles because 2 empty bands can always be permuted to produce additional solutions. I've done exhaustive search for P3 pattern and didn't find any valid SudokuP puzzles.

Serg

[Edit. Thanks to blue, he pointed errors in earlier version of this publication.]
[Edit2. I fixed a bug in my code, so I am publishing now correct result (I hope it is correct).]

[blue]

Hi Serg,

That was good timing.
I found these puzzles, just yesterday.
(Two of them will be of interest).

Cheers,
Blue.

[Serg]

Hi, blue!

That was good timing.
I found these puzzles, just yesterday.
(Two of them will be of interest).

Thanks you for crosschecking my results. Your "two puzzles of interest" are counterexamples for my results, pointing a bug in my searching code. A bug in my code is not good news, but it's much more better, that I have now possibility of fixing that bug.

The time of my publication was not good really. I couldn't see your last examples before my publication.
It's rather strange for me that idea to do this investigation came to my head several days ago only. Thoughts translation

Serg

[Mathimagics]

.
Hello Serg!

12-clue SudokuP's can have 5 empty boxes:

Code: Select all

  +-------+-------+-------+
  | . . 3 | . . . | . . . |
  | 4 . 6 | . . . | . . 2 |
  | . . 9 | . . . | . . 4 |
  +-------+-------+-------+
  | . . . | . . . | . . . |
  | . . . | . . . | . . . |
  | . . . | . . . | . . . |
  +-------+-------+-------+
  | . . . | . . . | 3 1 . |
  | 2 1 . | . . . | . . . |
  | 7 . . | . . . | 5 . . |
  +-------+-------+-------+


Here are all 7 found so far:

Code: Select all

..3......4.6.....2..9.....4.................................31.21.......7.....5..
1.....7........23.9.....5....................................18.42.......3......6
......789............213.........6.5281............3.............................
123.....................654647........5.............28...........................
..3...7..4.....2.69....................................7........8.....15..1....9.
1.....7..4.....2.69....................................7........8.....15.3.....9.
..3........72.9........1.......3........6........4....6........9........28.......


[Serg]

Hi, Mathimagics!

12-clue SudokuP's can have 5 empty boxes:

Code: Select all

  +-------+-------+-------+
  | . . 3 | . . . | . . . |
  | 4 . 6 | . . . | . . 2 |
  | . . 9 | . . . | . . 4 |
  +-------+-------+-------+
  | . . . | . . . | . . . |
  | . . . | . . . | . . . |
  | . . . | . . . | . . . |
  +-------+-------+-------+
  | . . . | . . . | 3 1 . |
  | 2 1 . | . . . | . . . |
  | 7 . . | . . . | 5 . . |
  +-------+-------+-------+


Here are all 7 found so far:

Code: Select all

..3......4.6.....2..9.....4.................................31.21.......7.....5..
1.....7........23.9.....5....................................18.42.......3......6
......789............213.........6.5281............3.............................
123.....................654647........5.............28...........................
..3...7..4.....2.69....................................7........8.....15..1....9.
1.....7..4.....2.69....................................7........8.....15.3.....9.
..3........72.9........1.......3........6........4....6........9........28.......


Thank you for additional data! It is well known now that SudokuP valid puzzle can contain 5 empty boxes, but examples of valid minimal SudokuP puzzles with small number of clues are very useful in valid/invalid patterns analysis. (So, collection of minimal 12-clue SudokuP puzzles is very useful too, but it is sufficient to have 1 puzzle only for given pattern during valid/invalid patterns analysis.) When I was investigating "fully symmetrical" patterns, about 95% of search space was investigated by example puzzles, posted by sudoku enthusiasts years ago ...

Serg

[Serg]

Hi, people!
I fixed a bug in my code and finished this investigation. (I correct the first post in this thread to show investigation results.)

Serg

[Mathimagics]

.
Nice work Serg!

Do we know how many 6EB (6 empty boxes) puzzles there are?

[Serg]

Hi, Mathimagics!

Nice work Serg!

Thanks!

Do we know how many 6EB (6 empty boxes) puzzles there are?

I don't know the way to compute 6 empty boxes puzzles.

Instead I'm planning to find all possible patterns (clue cells configurations) for 5 empty boxes case, when SudokuP puzzle has 1 empty band and 1 empty stack (you and blue published several dozens valid SudokuP puzzles of this type). More precisely speaking, I'm planning to find all minimal patterns of this type. Such patterns have interesting property - they produce minimal puzzles only.

Serg