The New Sudoku Players' Forum

[JPF]

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|..4|...|...|
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|..5|16.|.78|
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|.3.|..6|.2.|
|.1.|.2.|9.4|
|2.8|5..|71.|
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maybe it's time to do a more rational search!
... not yet...

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|..2|15.|.67|
+---+---+---+
|.5.|..6|.8.|
|.9.|.2.|4..|
|2.7|9..|.35|
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|.6.|.17|.3.|
|.5.|82.|4..|
|2.7|3.4|.86|
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three last ones for the road:

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|..4|...|...|
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|..5|..6|.78|
+---+---+---+
|.7.|.2.|.8.|
|.9.|5..|4..|
|3..|7.4|.9.|
+---+---+---+


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|..2|..5|.67|
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|.8.|.9.|.3.|
|.6.|.37|4.9|
|4.9|1.8|.75|
+---+---+---+


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|..2|..5|.67|
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|.8.|.5.|.7.|
|.3.|.27|.8.|
|2.9|8.6|43.|
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JPF

[JPF]

Code: Select all

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|..2|...|...|
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|..1|..5|.67|
+---+---+---+
|.8.|.7.|.3.|
|.3.|.98|.7.|
|1..|2..|4.5|
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JPF

[coloin]

Well ... that assumes that you can’t remove clues from band 2
Once we have that then gridchecker can remove clues from band 3 !!
This is not as small as yours though !!!!

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|..5|61.|.7.|
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|.6.|..5|..3|
|.1.|.37|8.4|
|2..|.8.|.19|
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[JPF]

Sorry, I don't understand your(s) point(s).
Can you elaborate, please?

JPF

[Hajime]

From the image above:

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|..4|...|...|
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|..2|..5|.67|
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|.8.|.5.|.7.|
|.3.|.27|.8.|
|2.9|8.6|43.|
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this one is smaller. is it?

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|.3.|.27|.8.|
|.8.|.5.|.7.|
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I just switched some rows.

[JPF]

This thread is about the "smallest" pattern.
Here are the patterns for the two puzzles:

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........1........1.....1.....1........1........1..1.11.1..1..1..1..11.1.1.11.111.
........1........1.....1.....1........1..1.11..1.......1..11.1..1..1..1.1.11.111.


The pattern of the first one is smaller than the second one.

JPF

[coloin]

What I mean is that one has to get the minimum possible clues in band 2 before getting the minimum clues in band 3
We now have 4 clues in row 6 .
A clue in r6c3 is required - and it is highly likely that your r6 is now a minimum. Well found .
Now gridchecker can find all the ways to complete band 3

However essentially we are looking at the minimum ways to complete 2 bands ... which is similar to serg’s work on the crossing bands... and also of course the patterns without puzzles work ...

[coloin]

Using your pattern in the double band

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|.3.|.6.|..8|
|.9.|.87|1..|
|2..|1..|4.5|
+---+---+---+


Must be close to the minimum now !!

[JPF]

after a long and painful exercice of numbers crunching:

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|..2|..5|.67|
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|.8.|.5.|..6|
|.6.|.27|.3.|
|2.3|91.|47.|
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and

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|..1|...|...|
|..4|...|...|
|..2|..5|.67|
+---+---+---+
|.7.|.5.|..6|
|.8.|.27|.3.|
|2.5|81.|4..|
+---+---+---+


JPF

[coloin]

Well done !! I think you did the hard work - as this one was quite close ...

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|.7.|.5.|..8|
|.9.|.26|.3.|
|2.5|.19|.76|
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... and surely that is the minimum, unless r9c3 can be axed !

[JPF]

Unfortunately, with 2 empty columns (c4 and c7), your pattern is not minlex:

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|..x|..x|.xx|
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|.x.|.x.|..x|
|.x.|.xx|.x.|
|x.x|.xx|.xx|
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its minlex:

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|...|...|...|
|...|...|xxx|
|..x|..x|.xx|
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|...|...|...|
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|xx.|..x|x.x|
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JPF

[coloin]

of course ... my mistake !!!

attacking the r9c5

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|.6.|.5.|..8|
|.3.|.29|.7.|
|2.8|6.7|43.|
+---+---+---+


[JPF]

Code: Select all

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


JPF

[Hajime]

Unfortunately, with 2 empty columns (c4 and c7), your pattern is not minlex:

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


its minlex:

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


JPF

The minlex pattern is far more smaller then the first. So why continue with the first pattern?

[JPF]

Hi Hajime,
This thread is probably difficult to follow without reading all the posts since the beginning.
We agreed with coloin that the pattern I posted here is the smallest valid one:

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+---+---+---+   +---+---+---+
|...|...|...|   |...|...|...|
|...|...|..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|
+---+---+---+   +---+---+---+

.................1.....1.....1........1..1..1..1..1.1..1..1...1.1..1..1..1..1..11


As we are not totally certain that it's the smallest one, don't hesitate to post any improvement!

Since then, coloin proposed to explore MinLex patterns not starting with an empty row:
here:

And having established the minimum pattern which necessarily must have 2 clues in the first band
If we are to define a pattern filter
One could find the minimum pattern which has a first band of the min lex pattern with these 3 clues, for example

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|..4|..5|.6.|
|..6|.7.|.8.|
|.7.|.1.|..9|
+---+---+---+
|.5.|...|6..|
|.2.|1.7|...|
|3..|49.|.2.|
+---+---+---+

....

and now, we are hunting the smallest minlex patterns having the fisrt band as above.

JPF