The New Sudoku Players' Forum

[JPF]

About this thread, there is an easy way to order the set of puzzles : consider their Lex representation.
Just to be clear, this puzzle:

Code: Select all

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

has the Lex representation:

Code: Select all

000002008002700100060090020010400009009060700300000050020050040004008500500100003

or:

Code: Select all

.....2..8..27..1...6..9..2..1.4....9..9.6.7..3......5..2..5..4...4..85..5..1....3

where . stands for 0.

What is the smallest (valid) minimal puzzle? i.e. having the minimum Lex representation?

JPF

[coloin]

quick try

Code: Select all

+---+---+---+
|...|...|...|
|...|...|..1|
|...|..2|...|
+---+---+---+
|..1|.3.|..4|
|..5|...|..6|
|.4.|7..|1.8|
+---+---+---+
|.69|..7|..5|
|.2.|..9|.8.|
|1.4|82.|.93|
+---+---+---+

Code: Select all

.................1.....2.....1.3...4..5.....6.4.7..1.8.69..7..5.2...9.8.1.482..93



[JPF]

Hi coloin,

I don't know the exact answer, but I will try to post better ones as long as I can...

Code: Select all

.................1.....2.....1.3...4..5.....6.4.7..1.8.69..7..5.2...9.8.1.482..93    coloin   
.................1.....2.....1.....3..2..3.4...4..562..1..57.8..39.84..6.47.2...9    JPF     

JPF

[Mathimagics]

What is the smallest (valid) minimal puzzle? i.e. having the minimum Lex representation?

So we are just looking at the set of all minimal puzzles in minlex form, without any requirement for them to solve to minlex grids?

[JPF]

So we are just looking at the set of all minimal puzzles in minlex form, without any requirement for them to solve to minlex grids?

right!

JPF

[Mathimagics]

Ok, how's this?

Code: Select all

.................1.....1.2...1.....3..2..4.....4..56...5..7..4..6.3..5...8.9.....

[JPF]

edited

Code: Select all

.................1.....1.2...1.....3..2..4.....4..56...5..7..4..6.3..5...8.9.....      Mathimagics
.................1.....1..2..1.3...4.5..6..7..768......18.5..2..9..8.3..5..2.361.      JPF1
.................1.....1..2..1..2..3..2....4...567.....18.23.54.24..6....3..18.96      JPF2


Code: Select all

+---+---+---+
|...|...|...|
|...|...|..1|
|...|..1|..2|
+---+---+---+
|..1|..2|..3|
|..2|...|.4.|
|..5|67.|...|
+---+---+---+
|.18|.23|.54|
|.24|..6|...|
|.3.|.18|.96|    JPF2
+---+---+---+


JPF

[JPF]

a new one:

Code: Select all

.................1.....1..2..1..2..3..2....4...3.561.7.14..3..8.25.18.7..39.2.5..


Code: Select all

+---+---+---+
|...|...|...|
|...|...|..1|
|...|..1|..2|
+---+---+---+
|..1|..2|..3|
|..2|...|.4.|
|..3|.56|1.7|
+---+---+---+
|.14|..3|..8|
|.25|.18|.7.|
|.39|.2.|5..|   JPF3
+---+---+---+


JPF

[JPF]

last improvement :

Code: Select all

.................1.....1..2..1..2..3..2....4...3.45..6.15.7..8..26.5..9..3.8.6.1.

Code: Select all

+---+---+---+
|...|...|...|
|...|...|..1|
|...|..1|..2|
+---+---+---+
|..1|..2|..3|
|..2|...|.4.|
|..3|.45|..6|
+---+---+---+
|.15|.7.|.8.|
|.26|.5.|.9.|
|.3.|8.6|.1.|    JPF4
+---+---+---+


JPF

[coloin]

Its funny - that we havent done this exercise before ....and it depends on the way you fill in the first 2 bands ...

last improvement : ....

And by that comment I thought it was definitive - and that you had done an exhaustive search after the first 12 clues.... !
Gridchecker checked ...
Maybe it is the smallest minlex puzzle in "2.55 × 10^25 non-essentially-equivalent minimal puzzles".

Code: Select all

.................1.....1..2..1..2..3..2....4...3.45..6.1...3.8..25.7..9..36.54.1.

Code: Select all

+---+---+---+
|...|...|...|
|...|...|..1|
|...|..1|..2|
+---+---+---+
|..1|..2|..3|
|..2|...|.4.|
|..3|.45|..6|
+---+---+---+
|.1.|..3|.7.|
|.25|.8.|.9.|
|.36|.54|.1.|
+---+---+---+   

[JPF]

Congratulations for your finding!

And by that comment I thought it was definitive - and that you had done an exhaustive search after the first 12 clues.... !

Actually, I did it "by hand" for all the bands.
I still don't see obvious reasons why the following patterns would be impossible:

Code: Select all

+---+---+---+   +---+---+---+
|...|...|...|   |...|...|...|
|...|...|..1|   |...|...|..1|
|...|..1|..2|   |...|..1|..2|
+---+---+---+   +---+---+---+
|...|xxx|xxx|   |..1|...|xxx|
|xxx|xxx|xxx|   |xxx|xxx|xxx|
|xxx|xxx|xxx|   |xxx|xxx|xxx|
+---+---+---+   +---+---+---+
|xxx|xxx|xxx|   |xxx|xxx|xxx|
|xxx|xxx|xxx|   |xxx|xxx|xxx|
|xxx|xxx|xxx|   |xxx|xxx|xxx|
+---+---+---+   +---+---+---+


hidden UAs ?

JPF

[Serg]

Hi, JPF!

I still don't see obvious reasons why the following patterns would be impossible:

Code: Select all

+---+---+---+   +---+---+---+
|...|...|...|   |...|...|...|
|...|...|..1|   |...|...|..1|
|...|..1|..2|   |...|..1|..2|
+---+---+---+   +---+---+---+
|...|xxx|xxx|   |..1|...|xxx|
|xxx|xxx|xxx|   |xxx|xxx|xxx|
|xxx|xxx|xxx|   |xxx|xxx|xxx|
+---+---+---+   +---+---+---+
|xxx|xxx|xxx|   |xxx|xxx|xxx|
|xxx|xxx|xxx|   |xxx|xxx|xxx|
|xxx|xxx|xxx|   |xxx|xxx|xxx|
+---+---+---+   +---+---+---+


The first pattern is subset of P123 maximal invalid pattern:

Code: Select all

        P123
+-----+-----+-----+
|. . .|. 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|
+-----+-----+-----+

(See List of 40 maximal pattern.)

Serg

[coloin]

Yes ....that would explain that !!!

But a better one has emerged ...

Code: Select all

.................1.....1..2..1..2..3..2....4...3.45..6.1...3.7..25.6..8..367.4.19

Code: Select all

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