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
0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA5476103223017654EFCDBA9867453210AB89FEDCAB89FEDC67453210EFCDBA98230176543210AB897654EFCD7654EFCD3210AB89BA982301FEDC6745FEDC6745BA982301 #     0r   256a
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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0123456789ABCDEF456801C3D2EF97AB7DEF92AB546C01839CAB8DEF1073456210372654ABC9D8FE54C63810EFD729BA89BAFEDC23105476D2FEBA796854103C2654D7FE3C01AB983810C9BA7645EF2DBA791032FE8D6C45FEDC5486BA9273016745EF2D0138BAC9AB927301CDFE8654C301AB984526FED7EF8D6C4597BA3210 #   823r    32a
0123456789ABCDEF456801C3D2EFAB977DEF29BA45C638109CABD8FE013754261076549CA8B3F2DE29FE86AB5D4C7103543C107D2F6EB98AD8BA32EF9071654C3754CA061E982FBDBAD29E486CF01735C610F735B42D9EA8FE89BD21735A06C46BC17354EAD280F98F95ABD23704EC61A20DEF89C615437BE3476C10FB89DA52 #   829r    32a
0123456789ABCDEF456801C3D2EFAB977DEF92AB546C38109CAB8DEF10735426103C547D2AB6F98E29BA68FE0DC175435476109CF83EB2DAD8FE23BA9547610C3710FC564B892EADBA89DE2473F016C5C6547A30E1D29FB8FED2B9186C5A07346BC13745AE2D80F98F95ABD23704EC61A20DEF89C615437BE347C601BF98DA52 #   829r    32a
0123456789ABCDEF456809CD3E2FA71BCAB938FE5D170426D7FE2AB14C069538103C5486AB79D2FE5F7A1032C4DE86B9629B7DEF185340AC8DE4AC9B06F25173265FD710E348B9CA3810C65492BAFED79EA7BF23D0C16845B4CD9EA87F65230173016245BA9CEF8DAC458309F7ED1B62E986FBDC21307A54FBD2E17A65843C90 #  2366r    24a
0123456789ABCDEF456809CD3E2FA71BCAE938FB0D175426D7FB2AE14C569038103C5486AB79D2FE5F7A1032C4DE86B9629E7DBF180345AC8DB4AC9E56F20173260FD715B348E9CA3815C60492EAFBD79EA7BF23D0C16845B4CD9EA87F65230173516240EA9CBF8DAC408359F7BD1E62E986FBDC21307A54FBD2E17A65843C90 #  2366r    24a
0123456789ABCDEF45689CD31E0F7B2AABD918EF25C74306E7CF0AB26D3491581C40FE7B568AD293597230A4EBFD6C818EFA61CD0293574BBD368259471CEFA0209E7F4C3851A6BD6F812D9EBA40357C74AB5310DCE628F9C35DB68A9F720E143A05C721F468B9DE981CDB35A02EF467D6E4A9F873B510C2F2B7E406C1D98A35 #  8610r    96a
0123456789ABCDEF45689CD3E12F0A7BA7CF2E0B465D8193DE9BA81F07C3654210375426ABD9FE8C2A5CE7B9F846301DE849DAFC13027B56FBD6103875EC42A93D108654BA972CFE5C8EB9AD2F34176076F432C05D1EA9B8B9A2F17E6C8053D462B57391CEF8D40A83016D4592BAEFC7947ACFE2D061B835CFED0B8A34759621 #  8747r    24a
0123456789ABCDEF45689CD3E12F0A7BDA9BE81F07C63542E7CF2A0B435D8196103DFB86A4795E2C8E5CAD40F2B36917A2497E51C608DBF3FB76C932D5E148A023F467C9581EA0BD5CDEB0A87F6412396810D3F4BA927C5EB9A7512E3CD0F6843DB5869C1EF7240A760132B59D4AEFC8948A1FED203CB765CFE2047A6B8593D1 #  8747r    32a
0123456789ABCDEF458901CDEF23AB6767EFAB23CD450189CDABEF890167234510672345ABCD89FE54CD890167EFBA23AB98FEDC10324576EF32BA769854DC012E0637A4B58C9F1D8A4C9DE026F1573BB9D5C81F730E64A2F371625B4AD9E8C036F074B25C9A1ED8721E563AD4B8F09C9CB4D0F83E16725AD85A1C9EF27036B4 #  9404r     8a
0123456789ABCDEF458910CDEF23AB7667EFBA32DC451089DCABEF890167235412CA085347E96FDB7E58A6FCB21D490394F67BDEA30C8521BD30912456F8E7AC29143D78FABE0C653805F41A7DC69EB2EB7DC2961450F83AFA6C5EB03892D14756B28341CEDF7A908FD129056B7A34CEA3476CEB908152FDC09ED7AF2534B618 # 11456r     1a
0123456789ABCDEF458C019D3E7FA62B69AE3BCF512D8407DB7F82AE406C951310D9547826EAFB3C28BAEDFC1539704654C61032BDF7E98AE3F796BA04C8D1523215C804EA96BF7D7604D915FCB32EA8BA98F7E3D2105C64FCEDBA267854139087402351CFDE6AB99D516C40AB8237FEAE627FDB930148C5CF3BAE89674502D1 # 14140r    96a
0123456789ABCDEF458C019D73EF62AB69AEC3BF512D40877DBF82AE406C519310D954C826BA37FE28EAD7FB153904C63CFB96EA047815D254761032DCFE98BA87402D51BFC3AE6993516C40AE82BF7DAB62EF739D018C45EFCDAB8967452301BA98FEDC32107654C6043915FBD7EA28D2157804EA96FB3CFE37BA26C854D910 # 14176r   768a
0123456789ABCDEF458C019D73EF62AB69AEC3BF512D40877DBF82AE406C519310D954C826BA37FE28EAD7FB153904C63CFB96EA047815D254761032DCFE98BA87402D51EA96FB3C93516C40FBD7EA28AB62EF73C854D910EFCDAB8932107654BA98FEDC67452301C6043915AE82BF7DD2157804BFC3AE69FE37BA269D018C45 # 14176r   256a
0123456789ABCDEF458C019DEF37AB2669AF3CEB512D87047DBE28FAC60451391076548C3EF92ABD28EB96AF1DC530473CFAD7BE2046198554D91023B78AF6CE8B42E3C0D5716F9A9E5D7F46ACB80312A760B9D1F3E24C58F3C18A5294607EDBB218CE356A9FD470CF34A2087BDE9561DA956B740213E8FCE607FD19485CB2A3 # 14197r   192a
0123456789ABCDEF458C239DEF17AB0669BE1CFA350D4287AF7D08EBC642193513927AB864ECF05D5748E91FA0D326BCB6E03DC572F18A94CADF6420B895731E28395604DE7FB1CA7415D23C9B6A0EF8DE0B9FA14C285763FCA6B78E5130D4293DCAF05617BE98428B57A14923C6EFD092F18E730D546CABE064CBD2FA893571 # 17411r    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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Re: Max Rank Found !!

Postby Serg » Tue Apr 13, 2021 12:59 pm

Hi, Mathimagics!
Mathimagics wrote:
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-)

Congratulations with this unexpected discovery! It was hard to beleive for me that it was possible to find maximum minlex 16 x 16 Sudoku grid. I'll try to confirm your statement that band's minlex forms with r1+r2 ranks from 18689 to 18694 don't exist.

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Re: Max Rank Found !!

Postby Mathimagics » Tue Apr 13, 2021 1:34 pm

Serg wrote:I'll try to confirm your statement that band's minlex forms with r1+r2 ranks from 18689 to 18694 don't exist.


Hi Serg!

That would be great, to have that confirmed ...

Once you have done that, you might like to go further and check the full list of 327 ranks for which no minlex band forms exist, by my reckoning. These are listed below (10 per line).

Cheers
MM

Ranks-with-No-Bands: Show
Code: Select all
  18313  18314  18315  18316  18317  18318  18319  18320  18321  18322
  18323  18328  18329  18330  18331  18332  18333  18334  18339  18340
  18344  18345  18346  18347  18348  18349  18350  18363  18365  18369
  18371  18372  18373  18374  18375  18376  18377  18378  18379  18380
  18381  18382  18383  18384  18385  18395  18397  18401  18403  18404
  18405  18406  18407  18408  18409  18410  18411  18412  18415  18416
  18417  18418  18419  18420  18421  18422  18423  18424  18426  18427
  18429  18431  18432  18433  18434  18435  18436  18437  18438  18439
  18440  18441  18442  18443  18444  18445  18446  18447  18448  18449
  18450  18451  18452  18453  18454  18455  18456  18457  18459  18461
  18462  18463  18464  18465  18466  18468  18469  18470  18471  18472
  18473  18474  18475  18476  18477  18478  18479  18480  18481  18482
  18483  18484  18485  18486  18487  18488  18489  18490  18491  18492
  18493  18494  18495  18496  18497  18498  18499  18500  18501  18502
  18503  18504  18505  18506  18507  18508  18509  18510  18511  18512
  18513  18514  18515  18516  18517  18518  18519  18520  18521  18522
  18523  18524  18525  18526  18527  18528  18529  18530  18531  18532
  18533  18534  18535  18536  18537  18538  18539  18540  18541  18542
  18543  18544  18545  18546  18547  18548  18549  18550  18551  18552
  18553  18554  18555  18556  18557  18558  18559  18560  18561  18562
  18563  18564  18565  18566  18567  18568  18569  18570  18571  18572
  18573  18574  18575  18576  18577  18578  18579  18580  18581  18582
  18583  18584  18585  18586  18587  18588  18589  18590  18591  18592
  18593  18594  18595  18596  18597  18598  18599  18600  18602  18604
  18605  18606  18607  18608  18609  18610  18611  18612  18613  18614
  18615  18616  18617  18618  18619  18620  18621  18622  18623  18624
  18625  18626  18627  18628  18629  18630  18631  18632  18633  18634
  18635  18636  18637  18638  18639  18640  18641  18642  18643  18644
  18645  18646  18647  18648  18649  18650  18651  18652  18653  18654
  18655  18656  18657  18658  18659  18660  18661  18662  18663  18664
  18665  18666  18667  18668  18669  18670  18671  18672  18673  18674
  18675  18676  18677  18678  18679  18680  18681  18682  18684  18686
  18687  18689  18690  18691  18692  18693  18694
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Re: Sudoku16: Minlex Forms

Postby Mathimagics » Wed Apr 14, 2021 9:04 am

.
We do appear to have identified both the first and the last of the 16x16 ED grids ...

Just to put this into some sort of cosmic perspective:

Code: Select all
         Number of  grids:  est 6E98
Atoms in visible universe:  est 1E80
       Number of ED grids:  est 2E71
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Re: Max Rank Found !!

Postby Serg » Mon Apr 19, 2021 8:41 pm

Hi, Mathimagics!
Mathimagics wrote:
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-)

Serg wrote:I'll try to confirm your statement that band's minlex forms with r1+r2 ranks from 18689 to 18694 don't exist.

Now I've done exhaustive search for 20 last two-row ranks (18675-18694). Here are results.
Code: Select all
Two-row   r2 row of two-row   Does band's minlex form with such r2 row exist?
rank      minlex form

18675     458C9A3DE1F6B027    No
18676     458C9A3DE1F6B072    No
18677     458C9A3DE1F6B270    No
18678     458C9A3DE1F6B702    No
18679     458C9A3DE1F76B02    No
18680     458C9A3DE1F7B026    No
18681     458C9A3DE1F7B062    No
18682     458C9A3DE1F7B260    No
18683     458C9D23EF06AB17    Yes
18684     458C9D23EF06AB71    No
18685     458C9D23EF06BA71    Yes
18686     458C9D23EF07AB61    No
18687     458C9D23EF07BA61    No
18688     458C9D23EF17BA60    Yes
18689     458C9D2EF037A1B6    No
18690     458C9D2EF037AB16    No
18691     458C9D2EF037BA16    No
18692     458C9D2EF137B0A6    No
18693     458C9D2EF137BA06    No
18694     458C9D2EF317BA06    No

So, I can confirm your statement "... there are no bands, and thus no grids, with a higher rank (18689 to 18694)." and thus I confirm the grid you published is maximum 16 x 16 Sudoku minlex grid.

Serg
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Re: Sudoku16: Minlex Forms

Postby Mathimagics » Tue Apr 20, 2021 5:00 am

Serg wrote:I can confirm your statement "... there are no bands, and thus no grids, with a higher rank (18689 to 18694)." and thus I confirm the grid you published is maximum 16 x 16 Sudoku minlex grid.


That's excellent, thank you, Serg ! 8-)
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Re: Sudoku16: Minlex Forms

Postby Mathimagics » Thu Apr 22, 2021 7:25 pm

Mathimagics wrote:
  • we have found grids for 17,065 different ranks. This includes all ranks in the range [0, 17028]
  • the maximum rank for which a grid example has been found is now 17158.

So a minor milestone reached, all indicators >= 17000 ...


The discovery of very high-ranking 1-ply grids (and some 2/3-ply) by BandKnit has had a good side-effect, such grids tend to have high-ranking neighbours, and so the overall search effort has been greatly boosted. Today's status report is:

  • we have found grids for 17,722 different ranks. This includes all ranks in the range [0, 17580]
  • the maximum rank is now fixed at 18688.
  • with 327 ranks known to have no grids possible, the number of unresolved ranks is down to (18695 - 17722 - 327) = 646

BandKnit is a very expensive process, and so it can't really be used to prove/disprove grid-existence for most unresolved ranks. For some cases a 1-ply search can be done (and has been done), but below a certain point the number of minlex bands is simply too great to even do a 1-ply search.

But for the very top ranks, it has at least given us some resolution. This table lists, in descending order, the top 10 ranks for which minlex bands exist, and what we know about them:

Code: Select all
 Rank     NMB    Grids    Status
 ================================
 18688      1        7    Complete
 18685      1       10    Complete
 18683      1       23    Complete
 18603      4        0    Complete
 ---------------------------------
 18601     13       ??    Open
 ---------------------------------
 18467      1      yes   
 18460      5      yes   
 18458      8      yes   
 18430      3       ??    Open
 18428      7       ??    Open


Notes:

  • the top 4 ranks have been completely tested by BandKnit, and the result is that we now know the 40 highest minlex grids (the "Top 40")
  • rank 18603 is the highest rank for which there are minlex bands, but no minlex grids
  • rank 18601 has been partly tested, and there no 1-ply or 2-ply grids. To complete (ie check all-ply) the search would require approx 90 core-days. This this can be done in a week (on PC JACK) or so, so I do intend to complete that.
  • for the remaining ranks, we have found 1-ply grids and/or grids via the general search for those marked "yes". Otherwise (marked ??) the status is unresolved.
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Mathimagics
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