Investigation of one-crossing-free patterns

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Re: Investigation of one-crossing-free patterns

Postby coloin » Mon Jul 13, 2015 11:14 pm

Short of something to do ... I have further investigated puzzles with a full row and a full box in the same band.
A continuation of the 9plus12 theme ..... except there are 15plux "x" clues
eg
Code: Select all
+-----+-----+-----+       
|. . .|. . .|. . .|
|. . .|. . .|. . .|       
|. . .|. . .|. . .|
+-----+-----+-----+ 
|. . .|. . .|. . .|
|. . .|. . .|. . .|       
|. . .|. . .|. . .|
+-----+-----+-----+     
|. . .|. . .|x x x| 
|. . .|. . .|x x x|   
|x x x|x x x|x x x|
+-----+-----+-----+ add 'x' more clues....

It seems easy [enough] to generate valid puzzles with these clues plus any 10 other clues ....
e.g
Code: Select all
+---+---+---+
|...|1.8|...|
|..4|..2|...|
|..5|...|...|
+---+---+---+
|...|...|5..|
|9..|.3.|...|
|8..|..9|...|
+---+---+---+
|...|...|123|
|...|...|456|
|123|456|789|
+---+---+---+[15plus10puzzle]

Only a very few 15plus9puzzles however [!]
There are also only 3 ED ways to have the 15 clues .....

From looking at Serg's patterns - maybe there are more rules we can apply ...
It might be that there are very few 15plus9 patterns which are valid
And perhaps we may be able to increase our invalid pattern criteria ...

C
coloin
 
Posts: 1598
Joined: 05 May 2005

Re: Investigation of one-crossing-free patterns

Postby Serg » Tue Jul 28, 2015 5:08 pm

Hi, coloin!
Unfortunately, I have no ideas in what way your "15plus9puzzles" approach can help to go further in possible puzzles exploring. But I think it's better to say "nothing", than nothing to say. I encountered problems trying to expand "one-crossing-free" approach to 6 free boxes - too many variants, too much CPU time it's necessary. I'm pondering alternative approaches to exhaustive search methods.

Serg
Serg
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Location: Russia

Re: Investigation of one-crossing-free patterns

Postby coloin » Wed Aug 19, 2015 4:57 pm

Thanks for considering it Serg
Here are the four 4 15plus9s that I found by generating more than 180,000 15plus10s !

Code: Select all
123456789456......789........8...........2..4.7.........1.....6...8........27....
123456789456......789........8...........2..4.7.........1.....6....8.......27....
123456789456......789........8....6....51...............4..9.........5........21.

123456789457......896.......8..........6........1.36........16..............89...


Since its not exhaustive, there may be more ... but there wont be many.

Of the 3 ED ways to have the 15 clues [only 3] the canonical one is by far the most common.

If these 15plus9 puzzles are the only puzzles - a possible use might be in removing patterns which cant have puzzles.

But of course it needs to be proved.

C
coloin
 
Posts: 1598
Joined: 05 May 2005

Re: Investigation of one-crossing-free patterns

Postby Serg » Mon Jan 16, 2017 9:19 pm

Hi, people!
I've done exhaustive search and now can confirm blue's result - published below patterns have no valid puzzles, so one can use it for filtering out invalid patterns.

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     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 | . . .


Serg
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