Sudoku16: Minlex Forms

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Re: Sudoku16: Minlex Forms

Postby coloin » Thu Apr 01, 2021 2:59 pm

Very good !!
I did think it an increasingly unlikely scenario that a grid has all the 48 row pairs of high rank - and that an automorphic grid with all similar high ranking might be more likely !!! Well found !
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More Automorphic Grids

Postby Mathimagics » Fri Apr 02, 2021 2:38 pm

.
I have found some more high-ranking automorphic grids. These are not H-grids, but they are close relations, and were found while searching for H-grids.

Pursuing my "band knitting" analogy, we need a term for the number of distinct minlex bands (horizontal and vertical) in a grid. The vast majority of grids will have 8 different bands, and H-grids by definition only have one, but any number in between is possible.

I suggest the term k-ply. So H-grids are 1-ply, and the most common grids (with all different bands) are 8-ply. I also conjecture that low-ply grids are far more likely to be automorphic than normal grids.

The first example is 3-ply, it has rank 17371 and 8 automorphisms:

Code: Select all
+---------+---------+---------+---------+
| 0 1 2 3 | 4 5 6 7 | 8 9 A B | C D E F |
| 4 5 8 C | 2 3 9 D | 7 1 E F | A B 6 0 |
| 6 9 B F | 1 C E A | 3 5 0 D | 4 2 7 8 |
| A E 7 D | 0 8 F B | 2 4 6 C | 1 9 5 3 |
+---------+---------+---------+---------+
| 1 2 0 E | C 6 3 5 | D A B 9 | 8 F 4 7 |
| 7 3 D A | F E 8 9 | 4 C 1 2 | 5 6 0 B |
| B C 5 8 | D 0 2 4 | F 6 7 E | 9 3 A 1 |
| F 6 4 9 | B A 7 1 | 0 3 5 8 | E C D 2 |
+---------+---------+---------+---------+
| 2 8 E 7 | 5 1 C 3 | 9 F 4 6 | 0 A B D |
| 3 0 A 6 | E 9 D F | B 8 C 5 | 2 7 1 4 |
| 9 B 1 5 | 8 2 4 0 | A 7 D 3 | 6 E F C |
| D F C 4 | 7 B A 6 | 1 E 2 0 | 3 8 9 5 |
+---------+---------+---------+---------+
| 5 7 6 B | 9 F 1 E | C 2 8 4 | D 0 3 A |
| 8 4 F 0 | 3 D 5 C | 6 B 9 A | 7 1 2 E |
| C A 3 1 | 6 4 0 2 | E D F 7 | B 5 8 9 |
| E D 9 2 | A 7 B 8 | 5 0 3 1 | F 4 C 6 |
+---------+---------+---------+---------+


The minlex bands are shown here, tagged with their rank, and a code that indicates which bands are distinct:

Code: Select all
Minlex Bands:
 1: 0123456789ABCDEF458C239D71EFAB6069BF1CEA350D4278AE7D08FB246C1953 # 17371  A
 2: 0123456789ABCDEF458C239DEF17AB0669FAC1EB420D8735EB7D80FAC6354219 # 17411  B
 3: 0123456789ABCDEF458C239DEF17AB0669FAC1EB420D8735EB7D80FAC6354219 # 17411  B
 4: 0123456789ABCDEF458C239DEF17AB0669FAC1EB420D3587EB7D80FAC6351942 # 17411  C

Minlex Stacks:
 1: 0123456789ABCDEF458C239D71EFAB6069BF1CEA350D4278AE7D08FB246C1953 # 17371  A
 2: 0123456789ABCDEF458C239DEF17AB0669FAC1EB420D8735EB7D80FAC6354219 # 17411  B
 3: 0123456789ABCDEF458C239DEF17AB0669FAC1EB420D8735EB7D80FAC6354219 # 17411  B
 4: 0123456789ABCDEF458C239DEF17AB0669FAC1EB420D3587EB7D80FAC6351942 # 17411  C


Here is a second example. Like the grid above it has rank 17371 and has 8 automorphisms, and is 3-ply, but with slightly different composition:

2nd example: Show
Code: Select all

+---------+---------+---------+---------+
| 0 1 2 3 | 4 5 6 7 | 8 9 A B | C D E F |
| 4 5 8 C | 2 3 9 D | 7 1 E F | A B 6 0 |
| 6 9 F B | C 1 E A | 2 4 0 D | 5 3 7 8 |
| E A 7 D | 8 0 F B | 3 5 6 C | 1 9 4 2 |
+---------+---------+---------+---------+
| 1 3 5 9 | 6 7 A 2 | 0 E D 4 | 8 C F B |
| 7 2 4 8 | 9 D 5 F | B C 3 6 | E A 0 1 |
| B C D A | 3 E 8 0 | 1 F 7 2 | 9 6 5 4 |
| F 6 0 E | B 4 C 1 | A 8 5 9 | 3 2 D 7 |
+---------+---------+---------+---------+
| 2 8 C F | A B 1 9 | D 7 4 5 | 6 0 3 E |
| 9 4 E 6 | D C 3 5 | F B 1 0 | 7 8 2 A |
| A 0 1 5 | E F 7 8 | 9 6 2 3 | B 4 C D |
| D B 3 7 | 0 6 2 4 | E A C 8 | F 5 1 9 |
+---------+---------+---------+---------+
| 3 E 9 1 | 7 2 4 6 | 5 0 8 A | D F B C |
| 5 F 6 0 | 1 A B C | 4 D 9 E | 2 7 8 3 |
| 8 7 B 4 | 5 9 D 3 | C 2 F 1 | 0 E A 6 |
| C D A 2 | F 8 0 E | 6 3 B 7 | 4 1 9 5 |
+---------+---------+---------+---------+


Minlex Bands:
 1: 0123456789ABCDEF458C239D71EFAB6069FBC1EA240D5378EA7D80FB356C1942 # 17371  A
 2: 0123456789ABCDEF458C239D71EFAB6069FE1CAB540D2378BA7D08EF326C1945 # 17371  B
 3: 0123456789ABCDEF458C9D23EF06BA716DEB81AF25C73490AF97EB0C1D438625 # 18655  C
 4: 0123456789ABCDEF458C9D23EF06BA716DEB81AF25C73490AF97EB0C1D438625 # 18655  C

Minlex Stacks:
 1: 0123456789ABCDEF458C239D71EFAB6069FE1CAB540D2378BA7D08EF326C1945 # 17371  B
 2: 0123456789ABCDEF458C9D23EF06BA716DEB81AF25C73490AF97EB0C1D438625 # 18655  C
 3: 0123456789ABCDEF458C239D71EFAB6069FBC1EA240D5378EA7D80FB356C1942 # 17371  A
 4: 0123456789ABCDEF458C9D23EF06BA716DEB81AF25C73490AF97EB0C1D438625 # 18655  C
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Re: Sudoku16: Minlex Forms

Postby Serg » Fri Apr 02, 2021 7:40 pm

Hi, Mathimagics!
Mathimagics wrote:... HB-grid, which has rank 17,411, and 34 grid automorphisms! 8-)
Code: Select all
+---------+---------+---------+---------+
| 0 1 2 3 | 4 5 6 7 | 8 9 A B | C D E F |
| 4 5 8 C | 2 3 9 D | E F 1 7 | A B 0 6 |
| 6 9 B E | 1 C F A | 3 5 0 D | 4 2 8 7 |
| A F 7 D | 0 8 E B | C 6 4 2 | 1 9 3 5 |
+---------+---------+---------+---------+
| 1 3 9 2 | 7 A B 8 | 6 4 E C | F 0 5 D |
| 5 7 4 8 | E 9 1 F | A 0 D 3 | 2 6 B C |
| B 6 E 0 | 3 D C 5 | 7 2 F 1 | 8 A 9 4 |
| C A D F | 6 4 2 0 | B 8 9 5 | 7 3 1 E |
+---------+---------+---------+---------+
| 2 8 3 9 | 5 6 0 4 | D E 7 F | B 1 C A |
| 7 4 1 5 | D 2 3 C | 9 B 6 A | 0 E F 8 |
| D E 0 B | 9 F A 1 | 4 C 2 8 | 5 7 6 3 |
| F C A 6 | B 7 8 E | 5 1 3 0 | D 4 2 9 |
+---------+---------+---------+---------+
| 3 D C A | F 0 5 6 | 1 7 B E | 9 8 4 2 |
| 8 B 5 7 | A 1 4 9 | 2 3 C 6 | E F D 0 |
| 9 2 F 1 | 8 E 7 3 | 0 D 5 4 | 6 C A B |
| E 0 6 4 | C B D 2 | F A 8 9 | 3 5 7 1 |
+---------+---------+---------+---------+
Grid rank = 17411, automorphisms = 32

Congratulations! I could not even imagine that such HB-grids exist. Very interesting! (Please, correct a typo - "34 grid automorphisms" - of course there must be "32" automorphisms.)

Well done!

Serg
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Re: Sudoku16: Minlex Forms

Postby Mathimagics » Sat Apr 03, 2021 5:31 am

Thanks, Serg!

I have corrected that typo.

We already did have one example of a HB-grid from the start, of course, the "most automorphic" grid. And I will look further for "low-rank" HB-grids, as well as continuing the search from the high-rank end.
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HB-grids

Postby Mathimagics » Sat Apr 03, 2021 1:12 pm

.
In a HB-grid, every one of the 48 row/col pairs has the same rank.

If A and B are both HB-grids, and rank(A) = rank(B), does it then follow that A and B are isomorphic? :?:

Remarkably (to me anyway), the answer is "not necessarily" :!:

Here we have 3 ED grids, all of which are HB-grids with rank 0:

Examples, rank 0: Show
Code: Select all
+---------+---------+---------+---------+  +---------+---------+---------+---------+  +---------+---------+---------+---------+
| 0 1 2 3 | 4 5 6 7 | 8 9 A B | C D E F |  | 0 1 2 3 | 4 5 6 7 | 8 9 A B | C D E F |  | 0 1 2 3 | 4 5 6 7 | 8 9 A B | C D E F |
| 4 5 6 7 | 0 1 2 3 | C D E F | 8 9 A B |  | 4 5 6 7 | 0 1 2 3 | C D E F | 8 9 A B |  | 4 5 6 7 | 0 1 2 3 | C D E F | 8 9 A B |
| 8 9 A B | C D E F | 0 1 2 3 | 4 5 6 7 |  | 8 9 A B | C D E F | 0 1 2 3 | 4 5 6 7 |  | 8 9 A B | C D E F | 0 1 2 3 | 4 5 6 7 |
| C D E F | 8 9 A B | 4 5 6 7 | 0 1 2 3 |  | C D E F | 8 9 A B | 4 5 6 7 | 0 1 2 3 |  | C D E F | 8 9 A B | 4 5 6 7 | 0 1 2 3 |
+---------+---------+---------+---------+  +---------+---------+---------+---------+  +---------+---------+---------+---------+
| 1 0 3 2 | 5 4 7 6 | 9 8 B A | D C F E |  | 1 0 3 2 | 5 4 7 6 | 9 8 B A | D C F E |  | 1 0 3 2 | 5 4 7 6 | A B 8 9 | F E D C |
| 5 4 7 6 | 1 0 3 2 | D C F E | 9 8 B A |  | 5 4 7 6 | 1 0 3 2 | D C F E | 9 8 B A |  | 5 4 7 6 | 1 0 3 2 | E F C D | B A 9 8 |
| 9 8 B A | D C F E | 1 0 3 2 | 5 4 7 6 |  | 9 8 B A | D C F E | 1 0 3 2 | 5 4 7 6 |  | 9 8 B A | D C F E | 2 3 0 1 | 7 6 5 4 |
| D C F E | 9 8 B A | 5 4 7 6 | 1 0 3 2 |  | D C F E | 9 8 B A | 5 4 7 6 | 1 0 3 2 |  | D C F E | 9 8 B A | 6 7 4 5 | 3 2 1 0 |
+---------+---------+---------+---------+  +---------+---------+---------+---------+  +---------+---------+---------+---------+
| 2 3 0 1 | 6 7 4 5 | A B 8 9 | E F C D |  | 2 3 0 1 | 6 7 4 5 | F E D C | B A 9 8 |  | 2 3 0 1 | B A 9 8 | 7 6 5 4 | E F C D |
| 6 7 4 5 | 2 3 0 1 | E F C D | A B 8 9 |  | 6 7 4 5 | 2 3 0 1 | B A 9 8 | F E D C |  | 6 7 4 5 | F E D C | 3 2 1 0 | A B 8 9 |
| A B 8 9 | E F C D | 2 3 0 1 | 6 7 4 5 |  | A B 8 9 | E F C D | 7 6 5 4 | 3 2 1 0 |  | A B 8 9 | 3 2 1 0 | F E D C | 6 7 4 5 |
| E F C D | A B 8 9 | 6 7 4 5 | 2 3 0 1 |  | E F C D | A B 8 9 | 3 2 1 0 | 7 6 5 4 |  | E F C D | 7 6 5 4 | B A 9 8 | 2 3 0 1 |
+---------+---------+---------+---------+  +---------+---------+---------+---------+  +---------+---------+---------+---------+
| 3 2 1 0 | 7 6 5 4 | B A 9 8 | F E D C |  | 3 2 1 0 | 7 6 5 4 | E F C D | A B 8 9 |  | 3 2 1 0 | A B 8 9 | 5 4 7 6 | D C F E |
| 7 6 5 4 | 3 2 1 0 | F E D C | B A 9 8 |  | 7 6 5 4 | 3 2 1 0 | A B 8 9 | E F C D |  | 7 6 5 4 | E F C D | 1 0 3 2 | 9 8 B A |
| B A 9 8 | F E D C | 3 2 1 0 | 7 6 5 4 |  | B A 9 8 | F E D C | 6 7 4 5 | 2 3 0 1 |  | B A 9 8 | 2 3 0 1 | D C F E | 5 4 7 6 |
| F E D C | B A 9 8 | 7 6 5 4 | 3 2 1 0 |  | F E D C | B A 9 8 | 2 3 0 1 | 6 7 4 5 |  | F E D C | 6 7 4 5 | 9 8 B A | 1 0 3 2 |
+---------+---------+---------+---------+  +---------+---------+---------+---------+  +---------+---------+---------+---------+
        automorphisms = 18432                        automorphisms = 2048                        automorphisms = 128


Perhaps this is only possible with rank 0, because of its unique properties ...
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HB-grids (Complete)

Postby Mathimagics » Thu Apr 08, 2021 4:37 pm

.

I have improved the "Band Knit" process considerably, and over the past 5 days have used it to test all the 226,142 known H-bands to see if they can be knitted into a 1-ply grid. Any such result is clearly an HB-grid.

I found just 155 HB-grids, of which 129 have rank 0.

[ EDIT ] this set is actually far from complete, see the following post

The full list is given below, each grid tagged with its rank ("r") and automorphism count ("a").

In the post above, I wondered whether any ranks other than 0 might have multiple HB-grids, and the answer is "yes". You will find in the list several instances of different HB-grids with the same rank, eg rank 823, 829 etc...

I have only just noticed, also, that only 2 grids in the list don't have (non-trivial) automorphisms (rank = 92, and rank = 11456);


Cheers
MM

HB-grids: Show
Code: Select all
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745AB89EFCD67452301EFCDAB89AB89EFCD23016745EFCDAB896745230132107654BA98FEDC76543210FEDCBA98BA98FEDC32107654FEDCBA9876543210 #     0r 18432a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745AB89EFCD67452301EFCDAB89AB89EFCD23016745EFCDAB896745230132107654FEDCBA9876543210BA98FEDCBA98FEDC76543210FEDCBA9832107654 #     0r   768a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745AB89EFCD67452301EFCDAB89AB89EFCD23016745EFCDAB89674523013210BA98FEDC76547654FEDCBA983210BA9832107654FEDCFEDC76543210BA98 #     0r  1152a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745AB89FEDC67452301EFCDBA98AB89EFCD23017654EFCDAB896745321032107654FEDCAB8976543210BA98EFCDBA98FEDC76542301FEDCBA9832106745 #     0r   128a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745AB89FEDC67452301EFCDBA98AB89EFCD23017654EFCDAB89674532103210BA98FEDC67457654FEDCBA982301BA9832107654EFCDFEDC76543210AB89 #     0r    96a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745BA98FEDC67452301FEDCBA98AB89EFCD32107654EFCDAB897654321032107654AB89EFCD76543210EFCDAB89BA98FEDC23016745FEDCBA9867452301 #     0r  1024a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745BA98FEDC67452301FEDCBA98AB89EFCD32107654EFCDAB897654321032107654EFCDAB8976543210AB89EFCDBA98FEDC67452301FEDCBA9823016745 #     0r   256a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745BA98FEDC67452301FEDCBA98AB89EFCD32107654EFCDAB89765432103210BA986745EFCD7654FEDC2301AB89BA983210EFCD6745FEDC7654AB892301 #     0r   128a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745BA98FEDC67452301FEDCBA98AB89EFCD32107654EFCDAB89765432103210BA98EFCD67457654FEDCAB892301BA9832106745EFCDFEDC76542301AB89 #     0r   128a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745EFCDAB8967452301AB89EFCDAB89EFCD67452301EFCDAB89230167453210BA987654FEDC7654FEDC3210BA98BA983210FEDC7654FEDC7654BA983210 #     0r   256a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745EFCDAB8967452301AB89EFCDAB89EFCD67452301EFCDAB89230167453210BA98FEDC76547654FEDCBA983210BA9832107654FEDCFEDC76543210BA98 #     0r   128a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745EFCDBA9867452301AB89FEDCAB89EFCD67453210EFCDAB89230176543210BA987654EFCD7654FEDC3210AB89BA983210FEDC6745FEDC7654BA982301 #     0r    64a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745EFCDBA9867452301AB89FEDCAB89EFCD67453210EFCDAB89230176543210BA98FEDC67457654FEDCBA982301BA9832107654EFCDFEDC76543210AB89 #     0r    32a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745FEDCBA9867452301BA98FEDCAB89EFCD76543210EFCDAB893210765432107654EFCDAB8976543210AB89EFCDBA98FEDC67452301FEDCBA9823016745 #     0r  2048a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745FEDCBA9867452301BA98FEDCAB89EFCD76543210EFCDAB89321076543210BA986745EFCD7654FEDC2301AB89BA983210EFCD6745FEDC7654AB892301 #     0r   128a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223016745FEDCBA9867452301BA98FEDCAB89EFCD76543210EFCDAB89321076543210BA98EFCD67457654FEDCAB892301BA9832106745EFCDFEDC76542301AB89 #     0r   128a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223017654EFCDBA9867453210AB89FEDCAB89FEDC67453210EFCDBA982301765432106745FEDCAB8976542301BA98EFCDBA98EFCD76542301FEDCAB8932106745 #     0r   512a
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0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223017654EFCDBA9867453210AB89FEDCAB89FEDC67453210EFCDBA98230176543210AB89FEDC67457654EFCDBA982301BA9823017654EFCDFEDC67453210AB89 #     0r   128a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223017654EFCDBA9867453210AB89FEDCAB89FEDC67453210EFCDBA98230176543210EFCD7654AB897654AB893210EFCDBA986745FEDC2301FEDC2301BA986745 #     0r   128a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA547610322301AB89EFCD67456745EFCDAB892301AB8923016745EFCDEFCD67452301AB893210BA98FEDC76547654FEDCBA983210BA9832107654FEDCFEDC76543210BA98 #     0r  1536a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA547610322301AB89EFCD67456745EFCDAB892301AB8923016745EFCDEFCD67452301AB893210FEDC7654BA987654BA983210FEDCBA987654FEDC3210FEDC3210BA987654 #     0r   768a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA547610322301AB89EFCD76546745EFCDAB893210AB8923016745FEDCEFCD67452301BA983210BA98FEDC67457654FEDCBA982301BA9832107654EFCDFEDC76543210AB89 #     0r   384a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA547610322301AB89EFCD76546745EFCDAB893210AB8923016745FEDCEFCD67452301BA983210FEDC7654AB897654BA983210EFCDBA987654FEDC2301FEDC3210BA986745 #     0r   192a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA547610322301AB89FEDC76546745EFCDBA983210AB8923017654FEDCEFCD67453210BA983210BA98EFCD67457654FEDCAB892301BA9832106745EFCDFEDC76542301AB89 #     0r   512a
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0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA98DCFE67456745FEDC98BA2301AB8932105476EFCDEFCD76541032AB893210EFCD765498BA7654AB893210DCFEBA986745FEDC1032FEDC2301BA985476 #     0r    32a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476EFCDAB8954761032AB89EFCD98BADCFE67452301DCFE98BA230167452301BA985476FEDC6745FEDC1032BA98AB893210DCFE7654EFCD765498BA32103210AB89FEDC54767654EFCDBA981032BA9823017654DCFEFEDC6745321098BA #     0r   128a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476EFCDAB8954761032AB89EFCD98BADCFE67452301DCFE98BA230167452301BA985476FEDC6745FEDC1032BA98AB893210DCFE7654EFCD765498BA32103210EFCD765498BA7654AB893210DCFEBA986745FEDC1032FEDC2301BA985476 #     0r    64a
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476EFCDAB8954761032AB89EFCD98BADCFE67452301DCFE98BA230167452301BA985476FEDC6745FEDC1032BA98AB893210DCFE7654EFCD765498BA32103210EFCDBA9854767654AB89FEDC1032BA9867453210DCFEFEDC2301765498BA #     0r    32a
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Max Rank Found !!

Postby Mathimagics » Mon Apr 12, 2021 4:40 pm

.
Well, it's been quite a day! I found a bug in BandKnit which only became apparent when I tried to re-knit a test grid using its minlex bands/stacks.

(And so the HB-grid list posted above is far from complete!)

It was only testing 1/576 of the possible combinations in its outermost loop. Having fixed that, I then restarted the HB-grid search, finding that there are indeed grids with rank 18688.

There are 7 of them, in fact, and are listed below. In minlex order, the last is this one:

Code: Select all
+---------+---------+---------+---------+
| 0 1 2 3 | 4 5 6 7 | 8 9 A B | C D E F |
| 4 5 8 C | 9 D 2 3 | E F 1 7 | B A 6 0 |
| 6 9 F A | E B C 1 | 0 D 4 2 | 8 7 3 5 |
| E B 7 D | 8 0 F A | 3 5 C 6 | 2 4 9 1 |
+---------+---------+---------+---------+
| 1 C B E | A F 9 6 | 7 8 3 5 | D 0 4 2 |
| 3 2 D 9 | C 8 5 4 | A B 6 0 | F E 1 7 |
| 7 6 5 4 | 3 2 1 0 | D C E F | 9 8 A B |
| A F 0 8 | D 7 B E | 4 2 9 1 | 5 3 C 6 |
+---------+---------+---------+---------+
| 2 4 9 1 | 6 C 3 5 | F A 0 8 | E B 7 D |
| 8 7 3 5 | 2 4 0 D | C 1 B E | 6 9 F A |
| B A 6 0 | 7 1 E F | 2 3 D 9 | 4 5 8 C |
| C D E F | B A 8 9 | 6 7 5 4 | 0 1 2 3 |
+---------+---------+---------+---------+
| 5 3 C 6 | 1 9 4 2 | B E 7 D | A F 0 8 |
| 9 8 A B | F E D C | 1 0 2 3 | 7 6 5 4 |
| D 0 4 2 | 5 3 7 8 | 9 6 F A | 1 C B E |
| F E 1 7 | 0 6 A B | 5 4 8 C | 3 2 D 9 |
+---------+---------+---------+---------+
Rank = 18688, 1-ply, aut =   384


We can state with some confidence, that the grid above is the maximum minlex grid, since there are no bands, and thus no grids, with a higher rank (18689 to 18694). 8-)

Rank18688-All7: Show
Code: Select all
0123456789ABCDEF458C9D23EF17BA6069FAEBC10D428735EB7D80FA35C6249117352406C8BED9FA2498DC35FA01EB76BAD078EF2369451CC6EFBA19D754082332670154ABDCFE8980BEAF7D91356C429D54328C60EF71ABAFC169BE4278530D53061742BE9DAFC878ABFED01C239654DC42539876FA10BEFE19C6AB548032D7
0123456789ABCDEF458C9D23EF17BA6069FAEBC10D428735EB7D80FA35C62491173569B2C8FAD04E24985F7DBE01A3C6BAD03E8C5469F217C6EF01A4D723985B3267BAD9105F4E8C80BE1C354A7D69F29D5476EF2B8C01A3AFC12408963E5B7D5306D74EF2981CBA78ABF2106CE435D9DC42A39671B5EF08FE19C85BA3D07624
0123456789ABCDEF458C9D23EF17BA6069FAEBC10D428735EB7D80FA35C624911735DC4296FAEB0824985306BE7D1CFABAD0FE19548C7623C6EF78AB102345D9326719B5DCEF08A480BE3ADC42915F769D542F78AB603E1CAFC106E47835D9B25306A49DC1BEF28778ABC15F23D9604EDC42673EFA08915BFE19B2806754A3CD
0123456789ABCDEF458C9D23EF17BA6069FAEBC10D428735EB7D80FA35C6249117425306C8FAD9BE5398DC42BE01AF76C6ABFE19D7230854FED078AB5469321C2FC619B54A7DE3083A1706E42B8CF5D998E43ADC105F762BD0B52F78963E1C4A7D5FB2806CE491A38C3EA49D71B560F2A401673EF2985BCDB269C15FA3D04E87
0123456789ABCDEF458C9D23EF17BA6069FAEBC10D428735EB7D80FA35C624911CB25F96783AD04E3ED9C8A45B60F2175F08D7B24E91A3C676A43E10DC2F985B23C61945BA7DEF0898EBFADC10537624D045237896FE1CBAFA1706EB248C35D9873EA40DC1B569F2A4916C3EF2085B7DB260715FA3D94E8CCD5FB28967E401A3
0123456789ABCDEF458C9D23EF17BA6069FAEBC10D428735EB7D80FA35C624911CBE240D783569FA32D971EFAB60458C7654BA89DCEF0123AF086C354291EB7D2491D7BEFA0853C68735AF96C1BED042BA60C85423D9FE17CDEF3210675498AB53C61942BE7DAF0898ABFEDC10237654D042537896FA1CBEFE1706AB548C32D9
0123456789ABCDEF458C9D23EF17BA6069FAEBC10D428735EB7D80FA35C624911CBEAF967835D04232D9C854AB60FE1776543210DCEF98ABAF08D7BE429153C624916C35FA08EB7D8735240DC1BE69FABA6071EF23D9458CCDEFBA896754012353C61942BE7DAF0898ABFEDC10237654D042537896FA1CBEFE1706AB548C32D9
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Re: Sudoku16: Minlex Forms

Postby coloin » Mon Apr 12, 2021 7:48 pm

Very good .... consider yourself very much "stabbed in the back" by those sudoku-16 solution grid "gods" !!!
But pleasing to have the answer definitively !!!
And an automorphic grid did turn out to be the max .....
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Re: Sudoku16: Minlex Forms

Postby Mathimagics » Tue Apr 13, 2021 4:24 am

coloin wrote:And an automorphic grid did turn out to be the max .....

Yes, this set of 7 grids seems functionally "equivalent" to band 416 ...

This is also the very first instance of a complete 16x16 ED band enumeration ...
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