Non-adjacent clues

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

Non-adjacent clues

Postby tso » Sat May 27, 2006 6:48 pm

Here's a couple of marginally interesting patterns. All givens are mutually non-adjacent, orthogonally and diagonally. Also, when represented as a single string, the clues are still non-adjacent.

Code: Select all
. 6 . 3 . 8 . 1 .
. . . . . . . . .
2 . . . 7 . . . 4
. . 6 . . . 2 . .
1 . . . . . . . 7
. . 8 . . . 5 . .
4 . . . 1 . . . 2
. . . . . . . . .
. 1 . 4 . 3 . 6 .



.6.3.8.1..........2...7...4..6...2..1.......7..8...5..4...1...2..........1.4.3.6.



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



1....2.9...4......6...5...8..2...7..4...8...1..3...9..8...6...4..9......5....1.3.
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Postby JPF » Sun May 28, 2006 12:13 am

One 20 clues :
Code: Select all
 . 9 . | 2 . 4 | . 3 .
 . . . | . . . | . . .
 7 . . | . 3 . | 5 . 6
-------+-------+-------
 . . 4 | . . . | . . .
 2 . . | . . . | . 4 .
 . . . | 5 . 1 | . . .
-------+-------+-------
 3 . . | . . . | . . 1
 . . . | . . . | 9 . .
 5 . 2 | . 6 . | . . 7



and a 21 clues :
Code: Select all
 . . 5 | . 1 . | 7 . .
 2 . . | . . . | . . 1
 . . 4 | . . 8 | . . .
-------+-------+-------
 1 . . | . . . | . . 6
 . . 8 | . 7 . | 9 . .
 . . . | . . . | . . .
-------+-------+-------
 6 . . | 1 . 5 | . . 2
 . . . | . . . | . . .
 7 . 3 | . 6 . | 8 . 5

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Postby Ruud » Sun May 28, 2006 12:59 am

This must be the maximum for this type:

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

2.9.3.8.4.4.8.2.1.6.8.9.3.2.9.3.8.6.8.4.7.2.5.2.4.5.9.9.5.8.1.6.7.9.6.8.1.6.4.9.7


It is not minimal.

Here is a nice symmetrical one:

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

.9.4.1.5.7...2...8..6...2..6.......3.3..4..1.1.......5..9...6..2...1...7.4.3.8.2.


Ruud.
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Re: Non-adjacent clues

Postby Cec » Sun May 28, 2006 2:27 am

Just curious - what is the significance of listing all the "clues" or "givens" below each puzzle.?

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Re: Non-adjacent clues

Postby tso » Sun May 28, 2006 3:31 am

Ruud wrote:This must be the maximum for this type ...


Ruud.


Neither of these two puzzles fits the pattern as some (most in the first case) of the givens are diagonally adjacent.

Cec wrote:Just curious - what is the significance of listing all the "clues" or "givens" below each puzzle.?

Cec


Just that the clues are also non-adjacent there as well. Not particularly earth-shattering.
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Re: Non-adjacent clues

Postby Cec » Sun May 28, 2006 5:31 am

tso wrote:Just that the clues are also non-adjacent there as well. Not particularly earth-shattering.

Thanks for that tso- I should have realized from the thread heading "Non-adjacent clues".

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Re: Non-adjacent clues

Postby JPF » Sun May 28, 2006 10:46 pm

tso wrote:
Ruud wrote:This must be the maximum for this type ...

Neither of these two puzzles fits the pattern as some (most in the first case) of the givens are diagonally adjacent.


I think the maximum number of clues is 23.
Here is an example of a 23 clues 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 .


resultant puzzle not found.

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Re: Non-adjacent clues

Postby udosuk » Mon May 29, 2006 2:04 am

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

Obviously the above (25 clues) pattern must yield some unavoidable sets (with no fixed cells) somewhere, but could you give a constructive proof as where the "unavoidable" unavoidable sets will occur?
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Re: Non-adjacent clues

Postby ronk » Mon May 29, 2006 3:25 am

tso wrote:All givens are mutually non-adjacent, orthogonally and diagonally.

Given that two diagonal lines on a grid are either parallel or orthogonal, that is curious terminology.
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Re: Non-adjacent clues

Postby tso » Mon May 29, 2006 4:44 am

ronk wrote:
tso wrote:All givens are mutually non-adjacent, orthogonally and diagonally.

Given that two diagonal lines on a grid are either parallel or orthogonal, that is curious terminology.


I don't know what you are referring to, but here's what I mean:

'a' and 'b' are adjacent orthogonally:
Code: Select all
. . . . .
. ab . .
. . . . .
. . . . .
. . . . .


'c' and 'd' are adjacent diagonally:
Code: Select all
. . . . .
. c . . .
. . d . .
. . . . .
. . . . .

'e', 'f' and 'g' are mutually non-adjacent:
Code: Select all
e . . . .
. . . . .
f . g . .
. . . . .
. . . . .
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Postby JPF » Tue May 30, 2006 10:47 am

Here's a 22 non-adjacent clues puzzle :

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


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Re: Non-adjacent clues

Postby ronk » Tue May 30, 2006 10:51 am

tso wrote:I don't know what you are referring to ...

The terms orthogonal and non-orthogonal refer to the presence or absence of a right angle (90 degree) relationship between lines (or curves). A sudoku cell is not a line. A line between two cells doesn't have an angular relationship to another line.

And the adjective mutually in "mutually non-adjacent" is redundant.

Therefore, I suggest the terms non-adjacent, laterally-adjacent, and diagonally-adjacent.
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Re: Non-adjacent clues

Postby tso » Tue May 30, 2006 1:52 pm

ronk wrote:
tso wrote:I don't know what you are referring to ...

The terms orthogonal and non-orthogonal refer to the presence or absence of a right angle (90 degree) relationship between lines (or curves). A sudoku cell is not a line. A line between two cells doesn't have an angular relationship to another line.


This usage of "orthogonal" in this sense is not uncommon in discussions of abstract games or puzzles, nor unclear, regardless of it current lay dictionary definitions. Diagonal lines run at a 45 degree angle to the edges of the grid. Orthogonal lines run parallell to and at a 90 angle to the edges. "Orthogonally adjacent" means adjacent along an orthogonal line, just as "diagonally adjacent" means adjacent along a diagonal line. We all agree on what "diagonally adjacent" means though some dictionary definitions of "diagonal" don't really support that usage.

For examples, see the usage of "orthogonal" here; in the discription of LOA, "The connections within the group may be either orthogonal or diagonal."

... or here or here.

... or especially here, where the term is used repeatedly in a page made in collaboration with abstract strategy game inventor/expert and senior lecturer at the University of Canterbury.

In the field of cellular automata, no less than John Conway, inventor of LIFE, uses the word orthogonal in this sense. An "orthogonal spaceship" is one that moves up, down, right or left -- not diagonally. The "orthogonal neighborhood" of a cell consists of the four cells that share a side with it.


And the adjective mutually in "mutually non-adjacent" is redundant.


Yes it is. I'm famous for my redundancies. (see above)

Therefore, I suggest the terms non-adjacent, laterally-adjacent, and diagonally-adjacent.


Sounds good to me.
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Re: Non-adjacent clues

Postby Smythe Dakota » Wed May 31, 2006 1:55 am

ronk wrote:.... And the adjective mutually in "mutually non-adjacent" is redundant. ....

Maybe not. When there are three cells, you can have:

1. Each cell is adjacent to each of the other two cells.

2. Each cell is adjacent to at least one of the other two cells, but there is a pair of cells (among the three) which is not adjacent.

3. Each cell is non-adjacent to each of the other two cells.

It might be reasonable to describe case 1 as "mutually adjacent" or "pairwise adjacent", case 2 as "non-adjacent", and case 3 as "mutually non-adjacent" or "pairwise non-adjacent".

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Re: Non-adjacent clues

Postby ronk » Wed May 31, 2006 12:01 pm

tso, after reading your post, orthogonally-adjacent doesn't seem quite so strange to me. Thanks for the explanation.

Smythe Dakota wrote:When there are three cells ...
Interesting, but I think we're considering only pair adjacencies here.
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