The New Sudoku Players' Forum

[StrmCkr]

Furthermore, except for the grid XM given below, a minlex grid always contains two other elements: r2c5=8 and r2c6=9.

Code: Select all

   +---+---+---+
    |123|456|789|
    |45.|...|...|
    |...|...|...|
    +---+---+---+
    |2..|...|...|
    |...|...|...|
    |...|...|...|
    +---+---+---+
    |...|...|...|
    |...|...|...|
    |...|...|...|
    +---+---+---+

something me and coloin noticed a few years back
r2c123 is a limited permutation of n+89 tuple as the 89 are fixed order thanks to C89 being interchangeable:
if r2c3 = 6, then r2c456 = 7,8,9 :1 order possible.

if r2c3 is = 7 {we can swap it with 8,9} which gives us 2 possible arrangements:

Code: Select all

n,8,9  or 8,9, n :

where n = 1,2,3 as the only plausible cycles

the notation of 8,9 occupying 1 of two arrangements matches

[JPF]

something me and coloin noticed a few years back

interesting ...
Can you provide a valid link to your post mentioning this point?

JPF

[StrmCkr]

sudoku-enumeration-problem-t34371.html

Is the thread we were playing around with how to generate all unique grids and why the first band had such profound limitations to permutations of tuples,

I'd have to see if I still have my word document I had mapped out As I wrote a generator based on the idea of tuples and their sets and permutations to generate all unique grids. (very slow and cumbersum approach) doubt i still have It...

Hind site : Way easier if I would have just did this set up and cycled the 9 digit templates.

[JPF]

Well,
Let's delve into the study of these MinLex solution grids. Everything is in the thread Canonical Form by Mike Barber, Jan 13, 2007:

Jan 23, 2007 coloin notes that on the list of minlex solutions of the 17s puzzles known at the time, r4c1=2.
Jan 25, 2007 Mike Barber acknowledges the point and indicates the rather trivial reason r4c1=2 for any MinLex solution grid.
Jan 27, 2007 Red Ed: initial considerations on r2c56...
Jan 28, 2007 Red Ed finds XM = Max(MinLex) and r2c45=89.

Feb 03, 2007 gsf provides, in ascending order, the list of all MinLex solution grids. From this list, it is observed that r2c56=89 for all grids, except the last one = XM.

JPF

[champagne]

,

Feb 03, 2007 gsf provides, in ascending order, the list of all MinLex solution grids. From this list, it is observed that r2c56=89 for all grids, except the last one = XM.

JPF

An easy explanation to understand why we have 89 in r2c56 can be the following.

Any minlex solution grid has a minlex band 1
From the list of the 416 minlex bands, we can see that only the last four bands don't have 89 in r2c56.

I don't use the minlex canonical form in the 17 clues scan, but something relatively close.
If we build the catalog in minlex form, the first bands (lowest minlex) produce plenty of solution grids. The count per band decrease sharply and at the end

Here below the count per band (not exactly in minlex order so the band number is not the minlex 1-416) in one or my canonical forms.

396 11
397 1
398 7
399 8
400 5
401 3
402 3
403 0
404 0
405 0
406 1
407 1
408 1
409 0
410 1
411 1
412 0
413 0
414 0
415 0
5 472 730 538

The reason is somehow trivial. As the band 1 has the lowest band id, the other bands/stacks must have higher band id.
For the band 416 minlex, the only possibility is to have all bands/stacks with the same band Id.

[StrmCkr]

Thanks jpf for the links always appreciate history lessons.

my 2-cents was that I also concur with it,
I was sure others found it and explained it way better then I ever could. As I've been constantly reminded over the years to stay out of topics dealing with automorphism, isomorphism and lexiconical form as I lack the formal mathmatical technical written skills to talk about it.

like the 2 in r4c1 makes sense as it's the lowest number to isolate band two and three so they cannot directly swap and exchange internal rows. ~