The New Sudoku Players' Forum

[JPF]

I finally managed to get rid of r9c3:

Code: Select all

+---+---+---+
|...|...|..1|
|...|...|..2|
|...|..3|...|
+---+---+---+
|..2|...|...|
|..4|...|...|
|..1|..5|.67|
+---+---+---+
|.6.|.5.|..8|
|.5.|.17|.3.|
|18.|9.6|4..|
+---+---+---+


JPF

[coloin]

Indeed you have ....but its lex. larger

I am now wading through the band 3 possibilities for gridchecker to remove the redundant clues.
from 288 ED full band 3s which allow all of r789c134 to be removed
- gave 0.5M potential puzzles
- gave 3 puzzles of your minimum pattern - below
So I think we have reached the minimum - well done ... Edit ... and no puzzle without a clue at r7c5 clue

Code: Select all

+---+---+---+
|...|...|..1|
|...|...|..2|
|...|..3|...|
+---+---+---+
|..1|...|...|
|..4|...|...|
|..2|..5|.67|
+---+---+---+
|.8.|.1.|..5|
|.6.|.54|.3.|
|1.3|8.9|4..|
+---+---+---+

In the original pattern with 2 clues in band one
according to the theory of serg's patterns it needs 3,2,2 distribution in band 2

whether the clue at r6c8 can go to r6c9 is the only possibility - and this might be provably invalid with serg's

Code: Select all

+---+---+---+   +---+---+---+
|...|...|...|   |...|...|...|
|...|...|..x|   |...|...|..1|
|...|..x|...|   |...|..2|...|
+---+---+---+   +---+---+---+
|..x|...|...|   |..3|...|...|
|..x|..x|..x|   |..1|..4|..5|
|..x|..x|.x.|   |..6|..7|.4.|
+---+---+---+   +---+---+---+
|.x.|.x.|..x|   |.4.|.1.|..3|
|.x.|.x.|.x.|   |.8.|.3.|.9.|
|.x.|.x.|.xx|   |.5.|.2.|.78|
+---+---+---+   +---+---+---+

Code: Select all

+---+---+---+
|...|...|...|
|...|...|..8|
|...|..4|...|
+---+---+---+
|..8|...|..9|
|..9|..1|..6|
|..2|..7|..3|
+---+---+---+
|.23|.46|.81|
|.85|.93|.42|
|.96|.82|.35|
+---+---+---+      Double  Edit - this pattern is valid but needs one more clue removed from r4c9


Which leaves band 3 to have 3 clues in each box - [18C] but we know all the 18C puzzles with 2 clues in band one

- so unless we find a puzzle with r6c8>c9 or 5 clues in box 9 of smaller pattern - it is looking ike it is actually the minimum too

[JPF]

Indeed you have ....but its lex. larger

yeah. After a while eyes are not able to see reality!

JPF

[coloin]

I generated 601 puzzles of this pattern

Code: Select all

+---+---+---+
|...|...|...|
|...|...|..1|
|...|..2|...|
+---+---+---+
|..1|...|..3|
|..2|..4|..5|
|..5|..6|..7|
+---+---+---+
|176|839|254|
|594|627|138|
|823|415|679|
+---+---+---+

and r4c9 could not be removed in any

Refering to serg's invalid maximal patterns investigation-of-one-crossing-free-patterns

Code: Select all

      P110                       P111                        P113                       P114       
-----+-----+-----+        +-----+-----+-----+        +-----+-----+-----+        +-----+-----+-----+
. . .|. . .|. . .|        |. . .|. . .|. . .|        |. . .|. . .|. . .|        |. . .|. . .|. . .|
. . .|. . .|. . 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 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 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|
-----+-----+-----+        +-----+-----+-----+        +-----+-----+-----+        +-----+-----+-----+
 


So in box 9 with our pattern a clue r7c9 is necessary and therefore our patten in box 9 cant be improved on ....

[coloin]

And to confirm
I generated 1568 puzzles with the pattern in Band 1 & 2 and full band 3
[maybe 75% of the total]

Code: Select all

+---+---+---+
|...|...|...|
|...|...|..1|
|...|..2|...|
+---+---+---+
|..3|...|...|
|..1|..4|..5|
|..6|..7|.4.|
+---+---+---+
|647|918|253|
|182|735|694|
|359|426|178|
+---+---+---+

Gridchecker removed the clues and gave us 4.9M puzzles
There were 8 puzzles of the most minimal pattern that you found ...

Code: Select all

+---+---+---+
|...|...|...|
|...|...|..1|
|...|..2|...|
+---+---+---+
|..3|...|...|
|..1|..4|..5|
|..6|..7|.4.|
+---+---+---+
|.4.|.1.|..3|
|.8.|.3.|.9.|
|.5.|.2.|.78|
+---+---+---+

nearly 170,000 different patterns , and this was the largest min lex one !!

+---+---+---+
|...|...|...|
|...|...|..1|
|...|..2|...|
+---+---+---+
|..3|...|...|
|..1|..4|..5|
|..6|..7|.2.|
+---+---+---+
|.38|745|.9.|
|96.|.28|3..|
|71.|93.|.58|
+---+---+---+

there were three 28C puzzles - with 19 clues in band 3 !!!

Code: Select all

+---+---+---+
|...|...|...|
|...|...|..1|
|...|..2|...|
+---+---+---+
|..3|...|...|
|..1|..9|..7|
|..6|..4|.2.|
+---+---+---+
|.38|.17|29.|
|.67|9.8|1.4|
|.19|426|.78|
+---+---+---+


[JPF]

My first conjecture is that this pattern:

Code: Select all

+---+---+---+
|...|...|...|
|...|...|..x|
|...|..x|...|
+---+---+---+
|..x|...|...|
|..x|..x|..x|
|..x|..x|..x|
+---+---+---+
|xxx|xxx|xxx|
|xxx|xxx|xxx|
|xxx|xxx|xxx|
+---+---+---+


has puzzles with 0 or at least 25 solutions and then is invalid.
Here is an example:

Code: Select all

+---+---+---+
|...|...|...|
|...|...|..1|
|...|..2|...|
+---+---+---+
|..3|...|...|
|..4|..5|..6|
|..1|..3|..7|
+---+---+---+
|458|317|692|
|327|694|815|
|169|528|734|
+---+---+---+  25 sol.


The second conjecture is that the pattern is invalid maximal (i.e. adding a clue anywhere makes it valid)

JPF

[coloin]

Yes .... you are probably right ... serg never did get round to doing the maximal invalid patterns for the double band - but we have made a start !
Im sure the patterns that he has made will help though

[JPF]

this pattern,

Code: Select all

+---+---+---+ 
|...|...|...| 
|...|...|..x| 
|...|..x|...| 
+---+---+---+ 
|..x|...|...| 
|..x|..x|..x| 
|..x|..x|.x.| 
+---+---+---+ 
|.x.|.x.|..x| 
|.x.|.x.|.x.| 
|.x.|.x.|.xx| 
+---+---+---+ 


supposed to be the smallest valid pattern, has only 13 valid puzzles:

Code: Select all

.................1.....2.....3........1..4..5..4..6.7..8..1...6.5..3..9..6..2..54
.................1.....2.....3........1..4..5..5..6.7..7..1...3.8..2..6..4..3..98
.................1.....2.....3........1..4..5..5..6.7..8..1...3.7..2..9..4..3..67
.................1.....2.....3........1..4..5..6..7.2..5..1...8.9..3..7..8..2..34
.................1.....2.....3........1..4..5..6..7.2..5..3...6.8..1..7..9..2..34
.................1.....2.....3........1..4..5..6..7.2..7..1...3.8..4..5..2..3..84
.................1.....2.....3........1..4..5..6..7.2..7..1...3.8..4..5..2..3..89
.................1.....2.....3........1..4..5..6..7.2..8..3...6.9..1..7..5..2..34
.................1.....2.....3........1..4..5..6..7.4..4..1...3.8..3..9..5..2..78
.................1.....2.....3........1..4..5..6..7.4..5..1...8.9..2..3..8..3..27
.................1.....2.....3........1..4..5..6..7.4..5..1...8.9..3..2..8..2..37
.................1.....2.....3........1..4..5..6..7.4..5..3...6.8..1..2..9..2..37
.................1.....2.....3........1..4..5..6..7.4..8..3...6.9..1..2..5..2..37


JPF

[coloin]

Indeed, that is about right !

Code: Select all

    #201
+---+---+---+     +---+---+---+
|...|...|...|     |...|...|...|
|...|...|..x|     |...|...|..1|
|...|..x|...|     |...|..2|...|
+---+---+---+     +---+---+---+
|..x|...|...|     |..3|...|...|
|..x|..x|..x|     |..1|..4|..5|
|..x|..x|.x.|     |..6|..7|.4.|
+---+---+---+     +---+---+---+
|.x.|.x.|..x|     |.4.|.1.|..3|
|.x.|.x.|.x.|     |.8.|.3.|.9.|
|.x.|.x.|.xx|     |.5.|.2.|.78|
+---+---+---+     +---+---+---+      smallest pattern - 2 clues in band 1

Code: Select all

+---+---+---+
|...|...|...|       
|...|...|..1|       
|...|..1|..2|       
+---+---+---+       
|..1|..2|..3|       
|..2|...|.4.|       
|..3|.45|..6|       
+---+---+---+       
|.1.|..3|.7.|       
|.25|.6.|.8.|       
|.36|7.4|.19|       
+---+---+---+     Smallest minlex puzzle found

Patterns with 3 clues in Band 1

Code: Select all

    #301
+---+---+---+   +---+---+---+           
|...|...|...|   |...|...|...|           
|...|...|..x|   |...|...|..1|           
|...|..x|..x|   |...|..2|..3|           
+---+---+---+   +---+---+---+           
|xxx|xxx|xxx|   |..3|...|...|           
|xxx|xxx|xxx|   |..2|..4|.5.|           
|xxx|xxx|xxx|   |..6|..5|72.|           
+---+---+---+   +---+---+---+           
|xxx|xxx|xxx|   |.4.|.1.|8..|           
|xxx|xxx|xxx|   |.7.|.9.|..6|           
|xxx|xxx|xxx|   |.1.|.26|.45|           
+---+---+---+   +---+---+---+      Initial attemp at this  - almost cerainly can be improved !!

Code: Select all

    #302         
+---+---+---+     +---+---+---+     
|...|...|..x|     |...|...|..1|     
|...|...|..x|     |...|...|..2|     
|...|..x|...|     |...|..3|...|     
+---+---+---+     +---+---+---+     
|xxx|xxx|xxx|     |..1|...|...|     
|xxx|xxx|xxx|     |..4|...|...|     
|xxx|xxx|xxx|     |..2|..5|.67|     
+---+---+---+     +---+---+---+     
|xxx|xxx|xxx|     |.8.|.1.|..5|     
|xxx|xxx|xxx|     |.6.|.54|.3.|     
|xxx|xxx|xxx|     |1.3|8.9|4..|     
+---+---+---+     +---+---+---+ established as  smallest valid pattern which doesnt have an empty row

so to continue the theme, ive genetated a few 3plus5plus27s and there were 5 ways to fill a band found [ no #301 no #302]
So we can attempt to approach defining the smallest valid patterns for a double band for a given band 1 with 3 clues
Serg's pattern will help in the end !

Code: Select all

    #303
+---+---+---+     +---+---+---+
|...|...|...|     |...|...|...|
|...|...|..x|     |...|...|..1|
|...|..x|.x.|     |...|..2|.3.|
+---+---+---+     +---+---+---+
|.xx|xxx|xxx|     |...|...|...|
|.xx|xxx|xxx|     |...|..4|25.|
|.xx|xxx|xxx|     |..6|.1.|...|
+---+---+---+     +---+---+---+
|.xx|xxx|xxx|     |261|378|594|
|.xx|xxx|xxx|     |574|169|823|
|.xx|xxx|xxx|     |938|245|176|
+---+---+---+     +---+---+---+


    #304
+---+---+---+     +---+---+---+                   +---+---+---+
|...|...|...|     |...|...|...|                   |...|...|...|
|...|...|..x|     |...|...|..1|                   |...|...|..1|
|...|.xx|...|     |...|.23|...|                   |...|.23|...|
+---+---+---+     +---+---+---+                   +---+---+---+
|xxx|xxx|xxx|     |...|...|...|                   |...|...|...|
|xxx|xxx|xxx|     |..1|4..|..5|                   |...|4..|..5|
|xxx|xxx|xxx|     |.2.|...|.6.|                   |.26|...|...|
+---+---+---+     +---+---+---+                   +---+---+---+
|xxx|xxx|xxx|     |135|782|694|                   |412|598|637|
|xxx|xxx|xxx|     |284|936|157|                   |563|742|189|
|xxx|xxx|xxx|     |697|154|238|                   |978|136|254|
+---+---+---+     +---+---+---+  3plus5plus27     +---+---+---+   3plus4plus27 [one pattern exists]

For later

Code: Select all

        #305                       
    +---+---+---+     +---+---+---+
    |...|...|...|     |...|...|...|
    |...|...|..x|     |...|...|..1|
    |..x|..x|...|     |..2|..3|...|
    +---+---+---+     +---+---+---+
    |.xx|xxx|xxx|     |...|...|...|
    |.xx|xxx|xxx|     |...|..2|34.|
    |.xx|xxx|xxx|     |.5.|.1.|...|
    +---+---+---+     +---+---+---+
    |.xx|xxx|xxx|     |124|367|895|
    |.xx|xxx|xxx|     |578|294|136|
    |.xx|xxx|xxx|     |693|158|724|
    +---+---+---+     +---+---+---+
                                   
        #306                       
    +---+---+---+     +---+---+---+
    |...|...|..x|     |...|...|..1|
    |...|...|.x.|     |...|...|.2.|
    |...|..x|...|     |...|..3|...|
    +---+---+---+     +---+---+---+
    |xxx|xxx|xxx|     |...|...|...|
    |xxx|xxx|xxx|     |...|14.|..5|
    |xxx|xxx|xxx|     |..6|...|.3.|
    +---+---+---+     +---+---+---+
    |xxx|xxx|xxx|     |347|816|592|
    |xxx|xxx|xxx|     |915|274|368|
    |xxx|xxx|xxx|     |628|359|174|
    +---+---+---+     +---+---+---+
                                   
        #307                       
    +---+---+---+     +---+---+---+
    |...|...|..x|     |...|...|..1|
    |...|..x|...|     |...|..2|...|
    |..x|...|...|     |..3|...|...|
    +---+---+---+     +---+---+---+
    |xxx|xxx|xxx|     |...|...|...|
    |xxx|xxx|xxx|     |...|.3.|.4.|
    |xxx|xxx|xxx|     |52.|...|..6|
    +---+---+---+     +---+---+---+
    |xxx|xxx|xxx|     |257|381|694|
    |xxx|xxx|xxx|     |314|796|582|
    |xxx|xxx|xxx|     |869|245|713|
    +---+---+---+     +---+---+---+


#301 is the one i made a start on - as it needs the most clues in band 2..

[JPF]

Let's start with #301.
Your puzzle doesn't have a minlex pattern ; just swap rows 7 and 8.

Here is my first proposal:

Code: Select all

#301
+---+---+---+
|...|...|...|
|...|...|..1|
|...|..2|..3|
+---+---+---+
|..4|...|...|
|..1|..3|..2|
|..5|..6|..7|
+---+---+---+
|.2.|.8.|.36|
|.6.|.4.|.75|
|.3.|.2.|.19|
+---+---+---+


and yes, the puzzle is minimal.

JPF

[coloin]

Code: Select all

+---+---+---+
|...|...|...|
|...|...|..1|
|...|..3|..2|
+---+---+---+
|..4|...|...|
|..1|..5|...|
|..6|..7|53.|
+---+---+---+
|.8.|.5.|.9.|
|.7.|.12|..3|
|.5.|.38|1.6|
+---+---+---+


[JPF]

Code: Select all

#301
+---+---+---+
|...|...|...|
|...|...|..1|
|...|..2|..3|
+---+---+---+
|..4|...|...|
|..1|..5|...|
|..6|..3|.7.|
+---+---+---+
|.8.|.2.|..9|
|.5.|.1.|..6|
|.7.|.4.|135|
+---+---+---+


JPF

[JPF]

Code: Select all

#301 -JPF 3
+---+---+---+
|...|...|...|
|...|...|..1|
|...|..2|..3|
+---+---+---+
|..4|...|...|
|..1|..3|...|
|..5|..6|.7.|
+---+---+---+
|.6.|.8.|...|
|4..|1..|7.6|
|5.8|.67|129|
+---+---+---+


JPF

[JPF]

Code: Select all

#301 - JPF 4
+---+---+---+
|...|...|...|
|...|...|..1|
|...|..2|..3|
+---+---+---+
|..4|...|...|
|..1|..5|...|
|..6|..3|.7.|
+---+---+---+
|.8.|.4.|...|
|4..|9..|1.8|
|6.2|.1.|9.5|
+---+---+---+


JPF

[coloin]

Wow thats quite a jump
Using your puzzle - this pattern must be close

Code: Select all

#301
+---+---+---+
|...|...|...|
|...|...|..1|
|...|..2|..3|
+---+---+---+
|..4|...|...|
|..1|..5|...|
|..6|..3|.7.|
+---+---+---+
|.8.|.4.|...|
|.5.|.2.|.6.|
|6..|91.|24.|
+---+---+---+