Empty Boxes - I

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

Empty Boxes - I

Postby JPF » Mon Jun 26, 2006 9:55 pm

[Edit] Aug 29, 2006

I should have started this thread with some important results found in the past threads dealing with this subject :The following notation will be used :

0 if the Box is empty
X if the Box is not empty (at least 1 clue)
F if the Box has at least 8 clues( which means 9 clues in fact)

Here are the 25 non isomorphic cases :

Code: Select all
 0XX                                                                           
 XXX                                                                           
 XXX                                                                           
                                                                               
 00X  0XX                                                                       
 XXX  X0X                                                                       
 XXX  XXX                                                                       
                                                                               
 000  00X  00X  0XX                                                             
 XXX  0XX  XX0  X0X                                                             
 XXX  XXX  XXX  XX0                                                             
                                                                               
 000  00X  00X  00X  00X                                                       
 0XX  00X  0X0  XX0  XX0                                                       
 XXX  XXX  XXX  XX0  X0X                                                       
                                                                               
 000  000  000  00X  00X                                                       
 00X  0XX  0XX  00X  0X0                                                       
 XXX  0XX  X0X  XX0  X0X                                                       
                                                                               
 000  000  000  00X                                                             
 000  00X  00X  0X0                                                             
 XXX  0XX  XX0  X00                                                             
                                                                               
 000  000                                                                       
 000  00X                                                                       
 0XX  0X0                                                                       
                                                                               
 000                                                                           
 000                                                                           
 00X                                                                           
                                                                               
 000                                                                           
 000                                                                           
 000                                                                           
                                                                               

but only 6 shapes can have a valid puzzle.
Here is the Ocean’s post which gives the permitted shapes :

Ocean wrote:To sum it up, what you have established is that all Sudokus with emtpy boxes belong to (or can be transformed to) one of these six (nonisomorphic) cases:
Code: Select all
0XX  00X  0XX  00X  0XX  00X
XXX  XXX  X0X  XX0  X0X  XX0
XXX  XXX  XXX  XXX  XX0  XX0

No 5 (or more) empty boxes
Only one 4 empty boxes is possible.
It's worth noting that :
Code: Select all
00X   X0X
XX0   0X0
XX0   X0X

are isomorphic

[/edit]

Here are some variations on empty boxes.
Easy puzzles ; more patterns than puzzles.

3 empty boxes, 18 clues
Code: Select all
 . . . | 2 1 . | . . .
 . . 3 | 6 . . | . . .
 . 8 4 | . . . | . . .
-------+-------+-------
 2 1 . | . . . | . . .
 6 . . | . . . | . . 4
 . . . | . . . | . 8 3
-------+-------+-------
 . . . | . . . | 5 2 .
 . . . | . . 8 | 1 . .
 . . . | . 3 7 | . . .

4 empty boxes :
Code: Select all

 . . . | . . . | 1 2 3
 . . . | . . . | . 4 5
 . . . | . . . | . . 7
-------+-------+-------
 8 4 2 | 6 . . | . . .
 9 5 . | . . . | . . .
 1 . . | . . 7 | . . .
-------+-------+-------
 4 . . | . . 3 | . . .
 5 . . | . 7 1 | . . .
 6 3 1 | 4 8 2 | . . .

4 empty boxes :
Code: Select all
 . . . | . . . | 7 8 2
 . . . | . . . | . 6 .
 . . . | . . . | 9 3 5
-------+-------+-------
 . 3 . | . 8 . | . . .
 6 8 4 | 5 1 2 | . . .
 . 5 . | . 7 . | . . .
-------+-------+-------
 . 6 . | . 2 . | . . .
 3 4 7 | 6 9 8 | . . .
 . 1 . | . 4 . | . . .


4 empty boxes + 1 empty column :
Code: Select all

 . 2 . | . . . | . 5 .
 3 . 5 | . . . | 9 . 7
 . 8 . | . . . | . 6 .
-------+-------+-------
 . . . | 1 . 7 | . . .
 . . . | 6 . 5 | . . .
 . . . | 2 . 8 | . . .
-------+-------+-------
 . 1 4 | . . . | 8 7 .
 7 . 2 | . . . | 4 . 5
 . 6 . | . . . | . 3 .

4 empty boxes + 4 empty columns :
Code: Select all

 . . . | . . . | 9 . 6
 . . . | . . . | 3 . 5
 . . . | . . . | 8 . 4
-------+-------+-------
 5 . 4 | . 7 2 | . . .
 3 . 6 | . 5 9 | . . .
 9 . 7 | . 8 1 | . . .
-------+-------+-------
 . . . | . . . | . . .
 6 . 9 | . 4 3 | . . .
 2 . 3 | . 1 6 | . . .

4 empty boxes + 2 full rectangles :
Code: Select all

 . . . | . . . | 2 3 9
 . . . | . . . | 4 8 6
 . . . | . . . | 7 5 1
-------+-------+-------
 . . . | . . . | . . .
 5 4 1 | 8 7 . | . . .
 7 6 2 | 3 9 . | . . .
-------+-------+-------
 1 8 4 | 9 2 . | . . .
 3 9 7 | 4 8 . | . . .
 . . . | . . . | . . .

Any idea ?
JPF
Last edited by JPF on Tue Aug 29, 2006 2:06 pm, edited 1 time in total.
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Postby Ocean » Mon Jun 26, 2006 11:39 pm

Nice puzzles - pretty patterns!

This one is not new (a reprint from the superior thread):

Code: Select all
  8 . 5|. . .|4 . 6
  . 9 .|. . .|. 7 .
  6 . 1|. . .|8 . 9 
 ------+-----+-----
  . . .|8 1 5|. . .
  . . .|. 6 .|. . .
  . . .|2 4 3|. . .
 ------+-----+-----
  9 . 4|. . .|1 . 8 
  . 3 .|. . .|. 5 .
  5 . 8|. . .|3 . 7

And also a new pattern (hybrid symmetry; easy puzzle):
Code: Select all
  . . 8|. . .|4 1 6
  . . 9|. . .|. . 7
  6 2 5|. . .|. . 8
 ------+-----+-----
  . . .|1 3 9|. . .
  . . .|. . .|. . .
  . . .|4 5 8|. . .
 ------+-----+-----
  9 . .|. . .|5 2 4
  3 . .|. . .|7 . .
  1 5 6|. . .|9 . .
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Postby Moschopulus » Tue Jun 27, 2006 8:08 am

Any with this pattern?
Code: Select all
  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

Puzzles with this pattern should be called Fractal Sudokus.
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Postby Pi » Tue Jun 27, 2006 9:15 am

I have more recently been working on Latin Squares rather than sudoku and have found quite a few with this pattern

Code: Select all
+---------+
|XXXXXXXXX|
|XX-----XX|
|X-XXXXX-X|
|X-X---X-X|
|X-X---X-X|
|X-X---X-X|
|X-XXXXX-X|
|XX-----XX|
|XXXXXXXXX|
+---------+
[/code]
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Postby JPF » Tue Jun 27, 2006 9:58 am

Moschopulus wrote:Any with this pattern?
Code: Select all
  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

Puzzles with this pattern should be called Fractal Sudokus.

The idea of fractal sudokus seems good to me.
Here’s one :
Code: Select all
 . . . | . 8 9 | . 5 3
 . . . | 5 6 . | 2 8 .
 . . . | 7 3 . | 1 9 .
-------+-------+-------
 . 4 2 | . 9 7 | . . .
 5 1 . | 3 2 . | . . .
 3 8 . | 4 1 . | . . .
-------+-------+-------
 . 2 5 | . 4 3 | . . .
 8 9 . | 1 5 . | . . .
 4 3 . | 9 7 . | . . .

There are at the most 2^9= 512 possible patterns.
How many are valid and essentially different ?

Pi wrote:
Code: Select all
+---------+
|XXXXXXXXX|
|XX-----XX|
|X-XXXXX-X|
|X-X---X-X|
|X-X---X-X|
|X-X---X-X|
|X-XXXXX-X|
|XX-----XX|
|XXXXXXXXX|
+---------+

This pattern with diagonal reflection, has 9 empty groups (3 boxes + 3 rows + 3 columns) :
Code: Select all
 . . . | 9 . 7 | 1 . 6
 . . . | . . . | . . 2
 . . . | 6 . 8 | . . 3
-------+-------+-------
 8 . 3 | . . . | 2 . 7
 . . . | . . . | . . .
 2 . 5 | . . . | 8 . 9
-------+-------+-------
 5 . . | 1 . 2 | . . .
 . . . | . . . | . . .
 3 7 2 | 5 . 6 | . . .


and is included in yours.

JPF
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Postby tso » Wed Jun 28, 2006 4:32 pm

That last puzzle has only two emtpy rows. I think you meant something like this (a transposition of the TSO I pattern), which has 180 degree rotational and both diagonal mirror symmetry:


Code: Select all
+-------+-------+-------+
| . . . | 1 . 2 | 3 . 4 |
| . . . | . . . | . . . |
| . . . | 4 . 5 | 6 . 7 |
+-------+-------+-------+
| 1 . 5 | . . . | 7 . 2 |
| . . . | . . . | . . . |
| 7 . 6 | . . . | 8 . 3 |
+-------+-------+-------+
| 3 . 7 | 5 . 8 | . . . |
| . . . | . . . | . . . |
| 5 . 8 | 6 . 1 | . . . |
+-------+-------+-------+
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Postby JPF » Wed Jun 28, 2006 4:59 pm

tso wrote:That last puzzle has only two emtpy rows.
and only two empty columns !
"temporary blindness" ?

Your puzzle is just perfect.

JPF
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Postby udosuk » Thu Jun 29, 2006 11:51 am

JPF wrote:Your puzzle is just perfect.

Perfectly diabolical that is... I needed forcing chains to solve that one...
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Postby daj95376 » Thu Jun 29, 2006 6:46 pm

I don't know an X-Cycle from a Y-Cycle from a bicycle. However, here's what I found in an online dictionary.

Forcing Chains are Y-Cycles and Colors (and Multi-Colors) are X-Cycles.

Now, I don't know if it makes any difference in rating the difficulty of tso's puzzle, but it can be solved with: basic techniques, Colors*3, and Multi-Colors. (Two of the Colors could be viewed as one operation.)

[Edited] Demonstration ... as requested. My solver produced the following.

Code: Select all
r1c3    =  9     Naked  Single
r9c9    =  9     Naked  Single
r9      -  24    Naked  Pair
r6      -  249   Naked  Triple
    b9  -  1246  Naked  Quad
r8c7    =  5     Naked  Single
r8c9    =  8     Naked  Single
    b9  -  1     Locked Candidate (1)
    b9  -  6     Locked Candidate (1)
  c2    -  249   Naked  Triple
  c2b4  -  38    Naked  Pair   (column and box)
r3c2    =  1     Naked  Single
r8c2    =  6     Naked  Single
r8c3    =  1     Hidden Single
  c5    -  15    Hidden Pair
    b5  -  2     Locked Candidate (1)
r5      -  378   Hidden Triple
r6c4    =  2     Hidden Single
    b5  -  6     Locked Candidate (1)
  c2    -  2     Locked Candidate (2)
r8c5    =  2     Hidden Single
r5c8    <> 15    Unique Rectangle Type 1

 *--------------------------------------------------------------------*
 | 68     57     9      | 1      678    2      | 3      58     4      |
 | 2468   57     234    | 3789   36789  3679   | 129    12589  15     |
 | 28     1      23     | 4      389    5      | 6      289    7      |
 |----------------------+----------------------+----------------------|
 | 1      38     5      | 389    34689  3469   | 7      49     2      |
 | 249    38     24     | 378    15     37     | 149    469    156    |
 | 7      49     6      | 2      15     49     | 8      15     3      |
 |----------------------+----------------------+----------------------|
 | 3      249    7      | 5      49     8      | 124    1246   16     |
 | 49     6      1      | 379    2      3479   | 5      37     8      |
 | 5      24     8      | 6      37     1      | 24     37     9      |
 *--------------------------------------------------------------------*


I ran Simple Sudoku at this point. It completed the solution with the following reductions.

Colors (*2) to get [r4c5]<>9 and [r8c6]<>9

Multi-Colors to get [r5c1]<>4

Colors to get [r2c1]<>4, [r4c5]<>4, [r5c3]<>4, [r6c6]<>4, and [r8c6]<>4

From here, Naked Singles complete the solution.
Last edited by daj95376 on Fri Jun 30, 2006 12:23 pm, edited 1 time in total.
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Postby udosuk » Fri Jun 30, 2006 12:20 pm

daj95376 wrote:Now, I don't know if it makes any difference in rating the difficulty of tso's puzzle, but it can be solved with: basic techniques, Colors*3, and Multi-Colors. (Two of the Colors could be viewed as one operation.)

A demonstration would be appreciated...
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Postby tso » Fri Jun 30, 2006 3:53 pm

Just to make it clear, that puzzle wasn't mine. I suggested a puzzle with 3 empty rows, 3 empty columns and 3 empty boxes here. The puzzle was too easy. Gordon used the pattern to make 150 more in a wide range of difficulties here. I took one of them and transposed some of the columns and rows to make it symmetrical to make the one posted above.

"Forcing chain" is often used collectively to include xy-chains, xy-wing, xy-ring, x-chain, coloring, etc -- just as "cows" can refer collectively to both bulls and cows.
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Postby Carcul » Fri Jun 30, 2006 4:23 pm

Udosuk wrote:Perfectly diabolical that is... I needed forcing chains to solve that one...


Daj95376 wrote:but it can be solved with: basic techniques, Colors*3, and Multi-Colors. (Two of the Colors could be viewed as one operation.)


One forcing chain is enough to solve the puzzle.

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Re: Empty Boxes - I

Postby tso » Fri Jun 30, 2006 6:31 pm

JPF wrote:Here are some variations on empty boxes.
Easy puzzles ; more patterns than puzzles.

3 empty boxes, 18 clues
Code: Select all
 . . . | 2 1 . | . . .
 . . 3 | 6 . . | . . .
 . 8 4 | . . . | . . .
-------+-------+-------
 2 1 . | . . . | . . .
 6 . . | . . . | . . 4
 . . . | . . . | . 8 3
-------+-------+-------
 . . . | . . . | 5 2 .
 . . . | . . 8 | 1 . .
 . . . | . 3 7 | . . .


From Nikoli Issue #111, #4, 2005 (level 3 of 4):

Code: Select all
 . . . | . 7 6 | . . .
 . . . | . . 9 | 4 . .
 . . . | . . . | 8 3 .
-------+-------+------
 . . . | . . . | . 7 9
 8 . . | . . . | . . 6
 4 3 . | . . . | . . .
-------+-------+------
 . 6 2 | . . . | . . .
 . . 7 | 5 . . | . . .
 . . . | 3 8 . | . . .


Both puzzles solve similarly, with *very* few naked singles around 30 give or take hidden singles. I think this reinfornces the idea that different masks have the tendency to have different tactics.
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Re: Empty Boxes - I

Postby udosuk » Sat Jul 01, 2006 4:23 am

Thanks daj95376 for the demonstration. So you used "Unique Rectangle" as a move... I wouldn't call it a basic technique... But thanks again!:)
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Re: Empty Boxes - I

Postby JPF » Sun Jul 02, 2006 12:00 am

tso wrote:Both puzzles solve similarly, with *very* few naked singles around 30 give or take hidden singles. I think this reinfornces the idea that different masks have the tendency to have different tactics.

Here's one with more than pure singles :
Code: Select all
 . . . | 3 7 . | . . .
 . . 9 | 4 . . | . . .
 . 5 2 | . . . | . . .
-------+-------+-------
 4 6 . | . . . | . . .
 3 . . | . . . | . . 9
 . . . | . . . | . 2 1
-------+-------+-------
 . . . | . . . | 3 5 .
 . . . | . . 2 | 6 . .
 . . . | . 8 9 | . . .


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