The New Sudoku Players' Forum

by **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 :

Largest number of empty groups ?

Complete Sudoku Collection – Vol. 1

5 empty boxes

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

by **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 . .

by **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.

by 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]

by **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

by **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 | . . . | +-------+-------+-------+

by **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

by **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...

by **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.

by **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...

by **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.

by **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.

Carcul

by **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.

by **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!

by **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 | . . .

JPF

The New Sudoku Players' Forum

Empty Boxes - I

Empty Boxes - I

Re: Empty Boxes - I

Re: Empty Boxes - I

Re: Empty Boxes - I