Sudoku16: Minlex Forms

Programs which generate, solve, and analyze Sudoku puzzles

Re: Sudoku16: Minlex Forms

Postby m_b_metcalf » Mon May 10, 2021 8:26 am

Mathimagics wrote:.

Meanwhile, here is a grid in minlex form, which has rank 1, and which is composed of 64 x UA4's. It has 18,432 automorphisms, which is believed to be the maximum possible:

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



Mathimagics, My program counts 256 UA4s. What's up? Regards,

Mike
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Re: Sudoku16: Minlex Forms

Postby Mathimagics » Mon May 10, 2021 1:24 pm

m_b_metcalf wrote:My program counts 256 UA4s. What's up?

My arithmetic, for a start! :roll:
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Re: Sudoku16: Minlex Forms

Postby Mathimagics » Tue Jun 01, 2021 6:47 am

The current state of play :

Code: Select all
   Status         #Ranks
   ---------------------
   Grid exists     17812
   No band exists    327
   No grid exists      2   (*)
   
   Unknown (LBC)      33
   Unknown (HBC)     521
                   -----
                   18695


  • (*) ranks 18601 and 18603 have been exhaustively searched (by BandKnit) and no grids found
  • the missing ranks are divided into two categories, LBC for those ranks with low minlex band counts, and HBC for high minlex band counts.


Ranks greater than 18312 are LBC. These typically have 0 to 100 minlex bands. HBC ranks are 18312 and below, with band counts typically 100,000 and up.

The LBC missing ranks have all been tested for 1-ply grids, but full treatment with BandKnit is out of the question, for reasons given above. Proving rank 18601 had no grids took nearly 100 core-days. :(

BandKnit is essentially a permutation cruncher, and it just might be possible to parallelise the computations, so a GPU might be able to make more rank proofs possible. That's just a germ of an idea at this stage ...
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