Grids 8 5 4 64 bit

Reply with quote Re: Sudoku16: Minlex Forms 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 ! coloin Posts: 2532Joined: 05 May 2005Location: Devon Reply with quote More Automorphic Grids 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

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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 = 32Congratulations! 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 Serg 2018 Supporter Posts: 909Joined: 01 June 2010Location: Russia Reply with quote Re: Sudoku16: Minlex Forms 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. Mathimagics 2017 Supporter Posts: 1926Joined: 27 May 2015Location: Canberra Reply with quote HB-grids 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:Perhaps this is only possible with rank 0, because of its unique properties ... Mathimagics 2017 Supporter Posts: 1926Joined: 27 May 2015Location: Canberra Reply with quote HB-grids (Complete) 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 postThe 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);CheersMM Mathimagics 2017 Supporter Posts: 1926Joined: 27 May 2015Location: Canberra Reply with quote Max Rank Found !! 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 |

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