The New Sudoku Players' Forum

by **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 !

by **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

by **Serg** » Fri Apr 02, 2021 7:40 pm

Hi, Mathimagics!

Mathimagics wrote:... HB-grid, which has rank 17,411, and 34 grid 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 | 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

by **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.

by **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 ...

by **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 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BADCFE54761032DCFE98BA98BADCFE10325476DCFE98BA547610322301AB89FEDC76546745EFCDBA983210AB8923017654FEDCEFCD67453210BA983210FEDC6745AB897654BA982301EFCDBA987654EFCD2301FEDC3210AB896745 # 0r 256a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123016745AB89FEDC67452301EFCDBA98AB89EFCD23017654EFCDAB89674532103210BA98FEDC54767654FEDCBA981032BA9832107654DCFEFEDC7654321098BA # 0r 288a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123016745BA98DCFE67452301FEDC98BAAB89EFCD32105476EFCDAB89765410323210BA986745FEDC7654FEDC2301BA98BA983210EFCD7654FEDC7654AB893210 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123016745BA98DCFE67452301FEDC98BAAB89EFCD32105476EFCDAB89765410323210BA98EFCD76547654FEDCAB893210BA9832106745FEDCFEDC76542301BA98 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123016745BA98FEDC67452301FEDCBA98AB89EFCD32107654EFCDAB89765432103210BA986745DCFE7654FEDC230198BABA983210EFCD5476FEDC7654AB891032 # 0r 16a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123016745BA98FEDC67452301FEDCBA98AB89EFCD32107654EFCDAB89765432103210BA98EFCD54767654FEDCAB891032BA9832106745DCFEFEDC7654230198BA # 0r 16a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123016745FEDC98BA67452301BA98DCFEAB89EFCD76541032EFCDAB89321054763210BA986745FEDC7654FEDC2301BA98BA983210EFCD7654FEDC7654AB893210 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123016745FEDC98BA67452301BA98DCFEAB89EFCD76541032EFCDAB89321054763210BA98EFCD76547654FEDCAB893210BA9832106745FEDCFEDC76542301BA98 # 0r 16a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123016745FEDCBA9867452301BA98FEDCAB89EFCD76543210EFCDAB89321076543210BA98EFCD54767654FEDCAB891032BA9832106745DCFEFEDC7654230198BA # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123017654EFCD98BA67453210AB89DCFEAB89FEDC67451032EFCDBA982301547632106745FEDCBA9876542301BA98FEDCBA98EFCD76543210FEDCAB8932107654 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123017654EFCD98BA67453210AB89DCFEAB89FEDC67451032EFCDBA98230154763210AB897654FEDC7654EFCD3210BA98BA982301FEDC7654FEDC6745BA983210 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123017654EFCD98BA67453210AB89DCFEAB89FEDC67451032EFCDBA98230154763210AB89FEDC76547654EFCDBA983210BA9823017654FEDCFEDC67453210BA98 # 0r 16a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123017654EFCD98BA67453210AB89DCFEAB89FEDC67451032EFCDBA98230154763210EFCD7654BA987654AB893210FEDCBA986745FEDC3210FEDC2301BA987654 # 0r 16a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123017654EFCDBA9867453210AB89FEDCAB89FEDC67453210EFCDBA982301765432106745FEDC98BA76542301BA98DCFEBA98EFCD76541032FEDCAB8932105476 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123017654EFCDBA9867453210AB89FEDCAB89FEDC67453210EFCDBA98230176543210AB897654DCFE7654EFCD321098BABA982301FEDC5476FEDC6745BA981032 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123017654EFCDBA9867453210AB89FEDCAB89FEDC67453210EFCDBA98230176543210AB89FEDC54767654EFCDBA981032BA9823017654DCFEFEDC6745321098BA # 0r 16a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA5476230123017654EFCDBA9867453210AB89FEDCAB89FEDC67453210EFCDBA98230176543210EFCD765498BA7654AB893210DCFEBA986745FEDC1032FEDC2301BA985476 # 0r 16a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA547623012301AB89EFCD54766745EFCDAB891032AB8923016745DCFEEFCD6745230198BA3210BA98FEDC76547654FEDCBA983210BA9832107654FEDCFEDC76543210BA98 # 0r 192a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA547623012301AB89EFCD54766745EFCDAB891032AB8923016745DCFEEFCD6745230198BA3210FEDC7654BA987654BA983210FEDCBA987654FEDC3210FEDC3210BA987654 # 0r 96a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA547623012301AB89EFCD76546745EFCDAB893210AB8923016745FEDCEFCD67452301BA983210BA98FEDC54767654FEDCBA981032BA9832107654DCFEFEDC7654321098BA # 0r 192a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA547623012301AB89EFCD76546745EFCDAB893210AB8923016745FEDCEFCD67452301BA983210FEDC765498BA7654BA983210DCFEBA987654FEDC1032FEDC3210BA985476 # 0r 96a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA547623012301AB89FEDC54766745EFCDBA981032AB8923017654DCFEEFCD6745321098BA3210BA98EFCD76547654FEDCAB893210BA9832106745FEDCFEDC76542301BA98 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA547623012301AB89FEDC54766745EFCDBA981032AB8923017654DCFEEFCD6745321098BA3210FEDC6745BA987654BA982301FEDCBA987654EFCD3210FEDC3210AB897654 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA547623012301AB89FEDC76546745EFCDBA983210AB8923017654FEDCEFCD67453210BA983210BA98EFCD54767654FEDCAB891032BA9832106745DCFEFEDC7654230198BA # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB456701231032547698BAEFCD54761032DCFEAB8998BADCFE10326745DCFE98BA547623012301AB89FEDC76546745EFCDBA983210AB8923017654FEDCEFCD67453210BA983210FEDC674598BA7654BA982301DCFEBA987654EFCD1032FEDC3210AB895476 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301765498BAFEDC67453210DCFEBA98AB89FEDC10327654EFCDBA985476321032106745BA98DCFE76542301FEDC98BABA98EFCD32105476FEDCAB8976541032 # 0r 256a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301765498BAFEDC67453210DCFEBA98AB89FEDC10327654EFCDBA985476321032106745FEDC98BA76542301BA98DCFEBA98EFCD76541032FEDCAB8932105476 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301765498BAFEDC67453210DCFEBA98AB89FEDC10327654EFCDBA98547632103210AB897654DCFE7654EFCD321098BABA982301FEDC5476FEDC6745BA981032 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301765498BAFEDC67453210DCFEBA98AB89FEDC10327654EFCDBA98547632103210AB89FEDC54767654EFCDBA981032BA9823017654DCFEFEDC6745321098BA # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301765498BAFEDC67453210DCFEBA98AB89FEDC10327654EFCDBA98547632103210EFCD765498BA7654AB893210DCFEBA986745FEDC1032FEDC2301BA985476 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301765498BAFEDC67453210DCFEBA98AB89FEDC10327654EFCDBA98547632103210EFCDBA9854767654AB89FEDC1032BA9867453210DCFEFEDC2301765498BA # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA6745230123017654DCFEBA986745321098BAFEDCAB89FEDC54763210EFCDBA981032765432106745FEDC98BA76542301BA98DCFEBA98EFCD76541032FEDCAB8932105476 # 0r 256a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA6745230123017654DCFEBA986745321098BAFEDCAB89FEDC54763210EFCDBA98103276543210AB897654DCFE7654EFCD321098BABA982301FEDC5476FEDC6745BA981032 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA6745230123017654DCFEBA986745321098BAFEDCAB89FEDC54763210EFCDBA98103276543210AB89FEDC54767654EFCDBA981032BA9823017654DCFEFEDC6745321098BA # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA6745230123017654DCFEBA986745321098BAFEDCAB89FEDC54763210EFCDBA98103276543210EFCD765498BA7654AB893210DCFEBA986745FEDC1032FEDC2301BA985476 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA6745230123017654DCFEBA986745321098BAFEDCAB89FEDC54763210EFCDBA98103276543210EFCDBA9854767654AB89FEDC1032BA9867453210DCFEFEDC2301765498BA # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB895476DCFE6745EFCD103298BAAB892301DCFE5476EFCD674598BA10323210BA987654FEDC7654FEDC3210BA98BA983210FEDC7654FEDC7654BA983210 # 0r 512a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB895476DCFE6745EFCD103298BAAB892301DCFE5476EFCD674598BA10323210BA98FEDC76547654FEDCBA983210BA9832107654FEDCFEDC76543210BA98 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB895476DCFE6745EFCD103298BAAB892301DCFE5476EFCD674598BA10323210FEDC7654BA987654BA983210FEDCBA987654FEDC3210FEDC3210BA987654 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB895476DCFE6745EFCD103298BAAB892301DCFE5476EFCD674598BA10323210FEDCBA9876547654BA98FEDC3210BA9876543210FEDCFEDC32107654BA98 # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB895476FEDC6745EFCD1032BA98AB892301DCFE7654EFCD674598BA32103210BA98FEDC54767654FEDCBA981032BA9832107654DCFEFEDC7654321098BA # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB895476FEDC6745EFCD1032BA98AB892301DCFE7654EFCD674598BA32103210FEDC765498BA7654BA983210DCFEBA987654FEDC1032FEDC3210BA985476 # 0r 16a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB895476FEDC6745EFCD1032BA98AB892301DCFE7654EFCD674598BA32103210FEDCBA9854767654BA98FEDC1032BA9876543210DCFEFEDC3210765498BA # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB897654FEDC6745EFCD3210BA98AB892301FEDC7654EFCD6745BA9832103210BA985476DCFE7654FEDC103298BABA983210DCFE5476FEDC765498BA1032 # 0r 256a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB897654FEDC6745EFCD3210BA98AB892301FEDC7654EFCD6745BA9832103210BA98DCFE54767654FEDC98BA1032BA9832105476DCFEFEDC7654103298BA # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB897654FEDC6745EFCD3210BA98AB892301FEDC7654EFCD6745BA9832103210FEDC547698BA7654BA981032DCFEBA987654DCFE1032FEDC321098BA5476 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB897654FEDC6745EFCD3210BA98AB892301FEDC7654EFCD6745BA9832103210FEDC98BA54767654BA98DCFE1032BA9876541032DCFEFEDC3210547698BA # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB89DCFE54766745EFCD98BA1032AB8923015476DCFEEFCD6745103298BA3210FEDC7654BA987654BA983210FEDCBA987654FEDC3210FEDC3210BA987654 # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB89DCFE76546745EFCD98BA3210AB8923015476FEDCEFCD67451032BA983210FEDC765498BA7654BA983210DCFEBA987654FEDC1032FEDC3210BA985476 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB89FEDC76546745EFCDBA983210AB8923017654FEDCEFCD67453210BA983210BA98DCFE54767654FEDC98BA1032BA9832105476DCFEFEDC7654103298BA # 0r 512a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301AB89FEDC76546745EFCDBA983210AB8923017654FEDCEFCD67453210BA983210FEDC547698BA7654BA981032DCFEBA987654DCFE1032FEDC321098BA5476 # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301BA985476FEDC6745FEDC1032BA98AB893210DCFE7654EFCD765498BA32103210AB897654DCFE7654EFCD321098BABA982301FEDC5476FEDC6745BA981032 # 0r 256a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301BA985476FEDC6745FEDC1032BA98AB893210DCFE7654EFCD765498BA32103210AB89FEDC54767654EFCDBA981032BA9823017654DCFEFEDC6745321098BA # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301BA985476FEDC6745FEDC1032BA98AB893210DCFE7654EFCD765498BA32103210EFCD765498BA7654AB893210DCFEBA986745FEDC1032FEDC2301BA985476 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301BA985476FEDC6745FEDC1032BA98AB893210DCFE7654EFCD765498BA32103210EFCDBA9854767654AB89FEDC1032BA9867453210DCFEFEDC2301765498BA # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301BA987654DCFE6745FEDC321098BAAB893210FEDC5476EFCD7654BA9810323210AB895476FEDC7654EFCD1032BA98BA982301DCFE7654FEDC674598BA3210 # 0r 256a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301BA987654DCFE6745FEDC321098BAAB893210FEDC5476EFCD7654BA9810323210AB89DCFE76547654EFCD98BA3210BA9823015476FEDCFEDC67451032BA98 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301BA987654DCFE6745FEDC321098BAAB893210FEDC5476EFCD7654BA9810323210EFCD5476BA987654AB891032FEDCBA986745DCFE3210FEDC230198BA7654 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301BA98DCFE76546745FEDC98BA3210AB8932105476FEDCEFCD76541032BA983210AB89FEDC54767654EFCDBA981032BA9823017654DCFEFEDC6745321098BA # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89EFCD54761032EFCDAB8998BADCFE23016745DCFE98BA674523012301BA98DCFE76546745FEDC98BA3210AB8932105476FEDCEFCD76541032BA983210EFCD765498BA7654AB893210DCFEBA986745FEDC1032FEDC2301BA985476 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301765498BAEFCD67453210DCFEAB89AB89FEDC10326745EFCDBA985476230132106745BA98DCFE76542301FEDC98BABA98EFCD32105476FEDCAB8976541032 # 0r 384a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301765498BAEFCD67453210DCFEAB89AB89FEDC10326745EFCDBA985476230132106745FEDC98BA76542301BA98DCFEBA98EFCD76541032FEDCAB8932105476 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301765498BAEFCD67453210DCFEAB89AB89FEDC10326745EFCDBA98547623013210AB89FEDC54767654EFCDBA981032BA9823017654DCFEFEDC6745321098BA # 0r 48a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA6745321023017654DCFE98BA6745321098BADCFEAB89FEDC54761032EFCDBA981032547632106745FEDCAB8976542301BA98EFCDBA98EFCD76542301FEDCAB8932106745 # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA6745321023017654DCFE98BA6745321098BADCFEAB89FEDC54761032EFCDBA98103254763210AB897654EFCD7654EFCD3210AB89BA982301FEDC6745FEDC6745BA982301 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA6745321023017654DCFE98BA6745321098BADCFEAB89FEDC54761032EFCDBA98103254763210AB89FEDC67457654EFCDBA982301BA9823017654EFCDFEDC67453210AB89 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA6745321023017654DCFE98BA6745321098BADCFEAB89FEDC54761032EFCDBA98103254763210EFCD7654AB897654AB893210EFCDBA986745FEDC2301FEDC2301BA986745 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA6745321023017654DCFE98BA6745321098BADCFEAB89FEDC54761032EFCDBA98103254763210EFCDBA9867457654AB89FEDC2301BA9867453210EFCDFEDC23017654AB89 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA6745321023017654DCFEAB896745321098BAEFCDAB89FEDC54762301EFCDBA981032674532106745FEDC98BA76542301BA98DCFEBA98EFCD76541032FEDCAB8932105476 # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA6745321023017654DCFEAB896745321098BAEFCDAB89FEDC54762301EFCDBA98103267453210AB897654DCFE7654EFCD321098BABA982301FEDC5476FEDC6745BA981032 # 0r 16a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA6745321023017654DCFEAB896745321098BAEFCDAB89FEDC54762301EFCDBA98103267453210AB89FEDC54767654EFCDBA981032BA9823017654DCFEFEDC6745321098BA # 0r 16a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301AB895476DCFE6745EFCD103298BAAB892301DCFE5476EFCD674598BA10323210FEDC7654AB897654BA983210EFCDBA987654FEDC2301FEDC3210BA986745 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301AB895476DCFE6745EFCD103298BAAB892301DCFE5476EFCD674598BA10323210FEDCBA9867457654BA98FEDC2301BA9876543210EFCDFEDC32107654AB89 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301AB895476EFCD6745EFCD1032AB89AB892301DCFE6745EFCD674598BA23013210FEDCBA9854767654BA98FEDC1032BA9876543210DCFEFEDC3210765498BA # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301AB897654DCFE6745EFCD321098BAAB892301FEDC5476EFCD6745BA9810323210FEDC5476AB897654BA981032EFCDBA987654DCFE2301FEDC321098BA6745 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301AB897654DCFE6745EFCD321098BAAB892301FEDC5476EFCD6745BA9810323210FEDC98BA67457654BA98DCFE2301BA9876541032EFCDFEDC32105476AB89 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301AB89FEDC54766745EFCDBA981032AB8923017654DCFEEFCD6745321098BA3210FEDC5476AB897654BA981032EFCDBA987654DCFE2301FEDC321098BA6745 # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA985476DCFE6745FEDC103298BAAB893210DCFE5476EFCD765498BA10323210AB897654EFCD7654EFCD3210AB89BA982301FEDC6745FEDC6745BA982301 # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA985476DCFE6745FEDC103298BAAB893210DCFE5476EFCD765498BA10323210AB89FEDC67457654EFCDBA982301BA9823017654EFCDFEDC67453210AB89 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA985476DCFE6745FEDC103298BAAB893210DCFE5476EFCD765498BA10323210EFCD7654AB897654AB893210EFCDBA986745FEDC2301FEDC2301BA986745 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA985476DCFE6745FEDC103298BAAB893210DCFE5476EFCD765498BA10323210EFCDBA9867457654AB89FEDC2301BA9867453210EFCDFEDC23017654AB89 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA985476EFCD6745FEDC1032AB89AB893210DCFE6745EFCD765498BA23013210AB897654DCFE7654EFCD321098BABA982301FEDC5476FEDC6745BA981032 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA985476EFCD6745FEDC1032AB89AB893210DCFE6745EFCD765498BA23013210AB89FEDC54767654EFCDBA981032BA9823017654DCFEFEDC6745321098BA # 0r 16a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA985476EFCD6745FEDC1032AB89AB893210DCFE6745EFCD765498BA23013210EFCD765498BA7654AB893210DCFEBA986745FEDC1032FEDC2301BA985476 # 0r 16a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA985476EFCD6745FEDC1032AB89AB893210DCFE6745EFCD765498BA23013210EFCDBA9854767654AB89FEDC1032BA9867453210DCFEFEDC2301765498BA # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA987654EFCD6745FEDC3210AB89AB893210FEDC6745EFCD7654BA9823013210AB895476DCFE7654EFCD103298BABA982301DCFE5476FEDC674598BA1032 # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA987654EFCD6745FEDC3210AB89AB893210FEDC6745EFCD7654BA9823013210AB89DCFE54767654EFCD98BA1032BA9823015476DCFEFEDC6745103298BA # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA987654EFCD6745FEDC3210AB89AB893210FEDC6745EFCD7654BA9823013210EFCD547698BA7654AB891032DCFEBA986745DCFE1032FEDC230198BA5476 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA98DCFE54766745FEDC98BA1032AB8932105476DCFEEFCD7654103298BA3210AB89FEDC67457654EFCDBA982301BA9823017654EFCDFEDC67453210AB89 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA98DCFE54766745FEDC98BA1032AB8932105476DCFEEFCD7654103298BA3210EFCD7654AB897654AB893210EFCDBA986745FEDC2301FEDC2301BA986745 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA98DCFE67456745FEDC98BA2301AB8932105476EFCDEFCD76541032AB893210AB89FEDC54767654EFCDBA981032BA9823017654DCFEFEDC6745321098BA # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476AB89FEDC54761032EFCDBA9898BADCFE23017654DCFE98BA674532102301BA98DCFE67456745FEDC98BA2301AB8932105476EFCDEFCD76541032AB893210EFCD765498BA7654AB893210DCFEBA986745FEDC1032FEDC2301BA985476 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476EFCDAB8954761032AB89EFCD98BADCFE67452301DCFE98BA230167452301BA985476FEDC6745FEDC1032BA98AB893210DCFE7654EFCD765498BA32103210AB89FEDC54767654EFCDBA981032BA9823017654DCFEFEDC6745321098BA # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476EFCDAB8954761032AB89EFCD98BADCFE67452301DCFE98BA230167452301BA985476FEDC6745FEDC1032BA98AB893210DCFE7654EFCD765498BA32103210EFCD765498BA7654AB893210DCFEBA986745FEDC1032FEDC2301BA985476 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476EFCDAB8954761032AB89EFCD98BADCFE67452301DCFE98BA230167452301BA985476FEDC6745FEDC1032BA98AB893210DCFE7654EFCD765498BA32103210EFCDBA9854767654AB89FEDC1032BA9867453210DCFEFEDC2301765498BA # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476EFCDAB8954761032AB89EFCD98BADCFE67452301DCFE98BA230167452301BA987654DCFE6745FEDC321098BAAB893210FEDC5476EFCD7654BA9810323210EFCD5476BA987654AB891032FEDCBA986745DCFE3210FEDC230198BA7654 # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476EFCDBA9854761032AB89FEDC98BADCFE67453210DCFE98BA230176542301BA985476DCFE6745FEDC103298BAAB893210DCFE5476EFCD765498BA10323210EFCD7654AB897654AB893210EFCDBA986745FEDC2301FEDC2301BA986745 # 0r 128a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476EFCDBA9854761032AB89FEDC98BADCFE67453210DCFE98BA230176542301BA985476EFCD6745FEDC1032AB89AB893210DCFE6745EFCD765498BA23013210EFCDBA9854767654AB89FEDC1032BA9867453210DCFEFEDC2301765498BA # 0r 192a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476EFCDBA9854761032AB89FEDC98BADCFE67453210DCFE98BA230176542301FEDC547698BA6745BA981032DCFEAB897654DCFE1032EFCD321098BA54763210AB897654EFCD7654EFCD3210AB89BA982301FEDC6745FEDC6745BA982301 # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476EFCDBA9854761032AB89FEDC98BADCFE67453210DCFE98BA230176542301FEDC547698BA6745BA981032DCFEAB897654DCFE1032EFCD321098BA54763210AB89FEDC67457654EFCDBA982301BA9823017654EFCDFEDC67453210AB89 # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310325476EFCDBA9854761032AB89FEDC98BADCFE67453210DCFE98BA230176542301FEDC5476AB896745BA981032EFCDAB897654DCFE2301EFCD321098BA67453210AB89FEDC54767654EFCDBA981032BA9823017654DCFEFEDC6745321098BA # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310326745DCFEAB895476230198BAEFCD98BAEFCD54762301DCFEAB8910326745230198BA7654FEDC6745DCFE3210BA98AB891032FEDC7654EFCD5476BA9832103210BA98EFCD54767654FEDCAB891032BA9832106745DCFEFEDC7654230198BA # 0r 64a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310326745DCFEAB895476230198BAEFCD98BAEFCD54762301DCFEAB8910326745230198BA7654FEDC6745DCFE3210BA98AB891032FEDC7654EFCD5476BA9832103210FEDCAB8954767654BA98EFCD1032BA9876542301DCFEFEDC3210674598BA # 0r 32a 0123456789ABCDEF45670123CDEF89AB89ABCDEF01234567CDEF89AB4567012310326745DCFEBA985476230198BAFEDC98BAEFCD54763210DCFEAB89103276542301DCFE7654AB89674598BA3210EFCDAB895476FEDC2301EFCD1032BA9867453210BA98EFCD54767654FEDCAB891032BA9832106745DCFEFEDC7654230198BA # 0r 96a 0123456789ABCDEF4567018C3DEF29AB89AB3DEF042C1567CDEF29AB156704831032547698BADCFE54D910327CFE68BA68BADCFE403251797CFE98BA51D6403223046715AE89BFCD97158304BFCDAE26AB86EFCD23049715EFCDAB296715830432407651EA98FBDCBA98FED3C2407651D651C240FB73EA98FE7CBA98D6513240 # 39r 48a 0123456789ABCDEF4567018C3DEF29AB89AB3DEF042C1567CDEF29AB1567048310C8547692BAD3FE54761032DCFE98BA92BADCFE40385176D3FE98BA517640C22C046715AB89EF3D67158304EFCDAB29AB89EFCD23016745EF3DAB2967458C0138407651BA92FEDC7651C240FED3BA98BA92FED3C8107654FEDCBA9876543210 # 39r 192a 0123456789ABCDEF4567018C3DEF29AB89AE3DBF012C4567CDBF29AE456701831032547698BADCFE54D910327CFE68BA68EADCFB103254797CFB98EA54D6103223016745AB89EFCD97458301EFCDAB26AB86EFCD23049715EFCDAB296715830432107654BA98FEDCBA98FED3C2407651D654C210FE73BA98FE7CBA98D6513240 # 39r 48a 0123456789ABCDEF4567018C3DEF29AB89AE3DBF012C4567CDBF29AE4567018310C8547692BAD3FE54761032DCFE98BA92EADCFB10385476D3FB98EA547610C22C016745AE89BF3D67458301BFCDAE29AB89EFCD23016745EF3DAB2967458C0138107654EA92FBDC7654C210FBD3EA98BA92FED3C8107654FEDCBA9876543210 # 39r 192a 0123456789ABCDEF4567018CD3EF92AB89EAD3FB01C25476DCFB29EA45761038103854D692BA7CFE54D610C237FE89BA92AE37BF108C456DC7BF98AE546D012323016D45EA98BFC76D458C01FB37AE92BA89FE732C10D654FE7CBA296D54381038107654AE29FBDC7654C210BFD3EA89AB92EF3DC8016745EFCDAB9876452301 # 53r 32a 0123456789ABCDEF4567018CD3EF92AB8AE9DBF30C127456DBFC2AE9475630181CDBE9206574F83A23AF6745B098E1CD67458CAF1E3D0B92E98013DB2FCA5674321A745698B0DCFE7456C81A3DFE29B09E083FBDCA216745CFBD9E027645A3815674F0C8E1D3BA29A89EBD31F20C4567BDC1A29E54678F03F0325674AB891EDC # 53r 16a 0123456789ABCDEF4567018CD3EF92AB8DAB93EF012C54679CEF2DAB456710831A325F7698B0DC4E5F761AC83D4E29B0C94ED2B05F76A138D8B0394E1AC2F576230167F5AE894BDC67F58C01B4D3EA29AE8DB4932C0176F5B49CAE2D67F53801321A7654E098BFCD7654C81AFB3D0E92E0D8FB39C21A6754FBC9E0D27654831A # 53r 16a 0123456789ABCDEF4567028C3DEF91AB8ABED9F3C0126457D9FCEAB1645730281452C076BF98AED39D3AFE1B7C068245C0762438AED5BF19F8EB9D5A21437C062341760F5B89EACD760F8C42EA3D5B91ACD9B1E5F6702834BE85A3D912C4F6703F9D1BCE476A0582521067A498BCD3FE67A43820D5FE19BCEBC85F9D0321476A # 92r 1a 0123456789ABCDEF4567028C3DEF91AB8ABF3D9E01C25476DC9E1ABF4576203810D4583692BA7CFE58362071DCFE49BAA2FBDCE910478563C7E9A4FB5863021D2D01F345B698EAC76E7CB91AF354D820B98A6ECD27103F54F3458702AEDC6B9134107F586B29AEDC7F58C120EA3DB6499BA2E6D3C401F785E6CD9BA47F851302 # 92r 24a 0123456789ABCDEF456801C3D2EF97AB7DEF92AB546C01839CAB8DEF10734562103726549CBAD8FE54C638107DFE29BA89BAFEDC01325476D2FEBA794586103C2654D7FEC310AB983810C9BA6754EF2DBA791032EFD86C45FEDC5486AB2973016745EF2D3801BAC9AB927301FECD8654C301AB982645FED7EF8D6C45BA973210 # 823r 8a 0123456789ABCDEF456801C3D2EF97AB7DEF92AB546C01839CAB8DEF10734562103726549CBAFED854C638107DFEBA2989BAFEDC01327654D2FEBA7945863C102654C9BAEFD810373810D7FEAB2954C6BA795486C310D2FEFEDC1032675489BA6745AB98FECD2301AB926C453801EF7DC301EF2DBA976845EF8D73012645AB9C # 823r 32a 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

by **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

by **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 .....

by **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 ...

by **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).

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.

Serg

by **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

by **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

by **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).

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

by **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-)

by **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.

The New Sudoku Players' Forum

Sudoku16: Minlex Forms

Re: Sudoku16: Minlex Forms

More Automorphic Grids

Re: Sudoku16: Minlex Forms

Re: Sudoku16: Minlex Forms

HB-grids

HB-grids (Complete)

Max Rank Found !!

Re: Sudoku16: Minlex Forms

Re: Sudoku16: Minlex Forms

Re: Max Rank Found !!

Re: Max Rank Found !!

Re: Sudoku16: Minlex Forms

Re: Max Rank Found !!

Re: Sudoku16: Minlex Forms

Re: Sudoku16: Minlex Forms