## Canonical Form

Everything about Sudoku that doesn't fit in one of the other sections
coloin wrote:If there can be a 2 at r2c4....why cant there be a 3 at r2c4 ?

Good question !
With 4M grids more, I didn't find any...

I will try to post an updated complete table soon.

JPF
Last edited by JPF on Sat Jan 27, 2007 4:07 pm, edited 2 times in total.
JPF
2017 Supporter

Posts: 3755
Joined: 06 December 2005
Location: Paris, France

Here is a reference to the 416 page 4

kjellfp wrote:Before the counting, all 416-band groups were generated, and their symmetry group stored. I also store enough information making it possible to permute any band back to its class representative.

EDIT from page 25 on the sudoku maths kjellfp optimized the 3x3 enumeration
they also discussed automorphism - but i didnt understand it at the time !

I dont think the list has been published as such - i will look

Its here 44-gang
Of note the 44 labelling is different to Red Eds

The 416 labelling mirrors to a degree the min lex normalization, but some are ordered differently - a pity but it probably is worthwhile correcting.

It is likely , if we can correct this glitch, that we wil have a classification system for all essentially different grids.

I am awaiting kjellfp's response

Meanwhile my "bottom" grids out of a million were

Code: Select all
`378 378 414  , 405 405 416   587426193294381756631759428163294587842567319975138264726813945458972631319645872381 408 411  , 400 402 407   264759183978431652513682749186975234395264871427813965851397426632148597749526318386 404 404  , 390 399 402   659217438471983625238465917314798562567321849982654371825146793143579286796832154`

It might be interesting to see what their min lex norm is ?

C
coloin

Posts: 1790
Joined: 05 May 2005

I can't tell from this thread if you(*) understand how bands are grouped into 416 classes. In short: it's just a partition of the 9! x 46656 x 56 possible bands into isomorphism classes. On average, a band is isomorphic to ~6280 others, only one of which is used to represent any given class. For an alternative explanation, see section 2.2 of Bertram and Frazer's paper. Or if you knew all this already then apologies for wasting bandwidth.

There's something else I can't tell from this thread: what are you trying to do? Are you just trying to list all the cells in the whole grid whose values never change in min-lex canonicalisation? If so then a quick enumeration of the 416 band classes in min-lex form tells you that the top band must be
Code: Select all
`123|456|78945.|...|......|...|...`
where '.' means "value not fixed". Beyond that (that is, in the next two bands) I would be surprised if you could fix many cell values.

(* you = any contributors to this thread; I'm not directing this at coloin in particular)
Last edited by Red Ed on Sat Jan 27, 2007 6:48 pm, edited 1 time in total.
Red Ed

Posts: 633
Joined: 06 June 2005

Thanks Ed
Except perhaps not quite....
Possibly there might be at least 5 bands of the 416 which wont appear in the first band ?

The index416 could well be used to classify our different grids - hoewever the glitch where it confuses two different [but similar grids] is caused by an over reduction in those bands with repeating minirows.

These are [probably] the bands as defined by the index416 program
1 123456789456789123789123456
2 123456789456789123789123465
3 123456789456789123789123645
4 123456789456789123789132465
5 123456789456789123789132564
6 123456789456789123789213465
7 123456789456789123789231564
8 123456789456789123789231645
9 123456789456789123798132546
10 123456789456789123798132564
11 123456789456789123798213564
12 123456789456789123798213654
13 123456789456789123798231564
14 123456789456789123798312564
15 123456789456789123879213546
16 123456789456789123897231564
17 123456789456789123897231645
18 123456789456789132789123546
19 123456789456789132789132654
20 123456789456789132789213654
21 123456789456789132789231456
22 123456789456789132789231564
23 123456789456789132789321564
24 123456789456789132798213456
25 123456789456789132798231546
26 123456789456789132879231564
27 123456789456789231789123645
28 123456789456789231789132546
29 123456789456789231789231564
30 123456789456789231789312456
31 123456789456789231798312654
32 123456789457189236689237514
33 123456789457189236689273541
34 123456789457189236689327154
35 123456789457189236689327514
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37 123456789457189236689372451
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41 123456789457189236689723451
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318 123456789457289163896317524
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321 123456789457289163968731542
322 123456789457289316986137452
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328 123456789457289613689371245
329 123456789457289613689371254
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336 123456789457289613896731254
337 123456789457289613968731425
338 123456789457289613986137245
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340 123456789457289613986173542
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344 123456789457289631689713245
345 123456789457289631689731254
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348 123456789457289631698713452
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382 123456789457389126986721543
383 123456789457389162896217435
384 123456789457389162986712345
385 123456789457389162986721354
386 123456789457389216986127453
387 123456789457389261896217534
388 123456789457389261986127534
389 123456789457389261986721354
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391 123456789457389612896172345
392 123456789457389612896172354
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394 123456789457389612896712345
395 123456789457389612896712354
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401 123456789457389621698127345
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404 123456789457389621896172543
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412 123456789457839612896271345
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C
Last edited by coloin on Sat Jan 27, 2007 7:01 pm, edited 2 times in total.
coloin

Posts: 1790
Joined: 05 May 2005

Ah, we crossed in the post. Yes, point taken about bands possibly not appearing as rows 1-3. Hmm. Will ponder.
Red Ed

Posts: 633
Joined: 06 June 2005

OK, done some coding rather than some thinking. It seems likely that all min-lex grids have this form:
Code: Select all
`123|456|78945.|.89|......|...|...---+---+---2..|...|......|...|......|...|...---+---+---...|...|......|...|......|...|...`
Will have a think overnight about how one might prove that that is the best possible (if indeed it is).
Red Ed

Posts: 633
Joined: 06 June 2005

Thanks for that confirmation !

If I'm not mistaken these two grids might have a 3 at r2c4 [according to the readout of 381 and 386]
But the min lex i dont think is fully performed on the index416 representative ? EDIT - It is not unfortunatly
Code: Select all
`381 408 411  , 400 402 407   264759183978431652513682749186975234395264871427813965851397426632148597749526318386 404 404  , 390 399 402   659217438471983625238465917314798562567321849982654371825146793143579286796832154`

EDIT - they dont have a 3 at r2c4 when min lex normalized

I can confirm that r2c4 wont be an 8 [bands 412-416]
EDIT - this has been proved wrong !

C
Last edited by coloin on Sun Jan 28, 2007 7:28 pm, edited 3 times in total.
coloin

Posts: 1790
Joined: 05 May 2005

coloin wrote:Thanks for that confirmation !

If I'm not mistaken these two grids might have a 3 at r2c4 [according to the readout of 381 and 386]
But the min lex i dont think is fully performed on the index416 representative ?
Code: Select all
`381 408 411  , 400 402 407   264759183978431652513682749186975234395264871427813965851397426632148597749526318386 404 404  , 390 399 402   659217438471983625238465917314798562567321849982654371825146793143579286796832154`

I can confirm that r2c4 wont be an 8 [bands 412-416]

C

According to gsf's solver, the canonical form of your grids have [r2c4]=1.

Code: Select all
`123456789457189623689723145235614978764895312891372456342561897576948231918237564123456789457189623698273154235647891769831542841592367384915276512764938976328415`

In search of your elusive [r2c4]=3 scenario, I generated 10^6 random filled grids starting with this pattern:

Code: Select all
`*-----------------------*| 1 2 3 | 4 5 6 | 7 8 9 || 4 5 7 | 3 8 9 | . . . || . . . | . . . | . . . ||-------+-------+-------|| 2 . . | . . . | . . . || . . . | . . . | . . . || . . . | . . . | . . . ||-------+-------+-------|| . . . | . . . | . . . || . . . | . . . | . . . || . . . | . . . | . . . |*-----------------------*`

[r2c4]=3 never survived after canonicalization! It always ended up in [c789].
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

Red Ed wrote:OK, done some coding rather than some thinking. It seems likely that all min-lex grids have this form:
Code: Select all
`123|456|78945.|.89|......|...|...---+---+---2..|...|......|...|......|...|...---+---+---...|...|......|...|......|...|...`
Will have a think overnight about how one might prove that that is the best possible (if indeed it is).

It has already been proved that these 14 cells are fixed in the min-lex grids.
To prove that there are no other fixed cells, here is a list of 6 grids such that for any of the remaining 67 cells, 2 digits can be different.
Code: Select all
`123456789457189236896327514275948361314562897968713425539871642641235978782694153123456789457189326689372415296738541348591672571264893715943268832617954964825137123456789457189236689723514231967458578214693964835127316578942742391865895642371123456789457189236689723154234697815761835492895214673348972561512368947976541328123456789457189263968327145234798651619532874785641392391874526572963418846215937123456789456789132789123546237964851845217963961835427374592618512678394698341275`
[edit : shorter list]

other questions :

Can a min-lex grid have r2c4=3 ?
What are the minimal and the maximal min-lex grids ?

JPF
Last edited by JPF on Sun Jan 28, 2007 5:38 am, edited 2 times in total.
JPF
2017 Supporter

Posts: 3755
Joined: 06 December 2005
Location: Paris, France

JPF wrote:It has already been proved that these 14 cells are fixed in the min-lex grids.
Really? I missed the proof that r2c5,6 = 8,9 ... can you point me to that please.

Can a min-lex grid have r2c4=3 ?
What are the minimal and the maximal min-lex grids ?
Interesting ... at least now I think I understand what this thread is trying to achieve!

The minimum min-lex grid is obviously the canonical grid:
Code: Select all
`123456789456789123789123456231564897564897231897231564312645978645978312978312645`
The maximum min-lex grid is ... mmm, I don't know!
Red Ed

Posts: 633
Joined: 06 June 2005

Red Ed wrote:The minimum min-lex grid is obviously the canonical grid:
Code: Select all
`123456789456789123789123456231564897564897231897231564312645978645978312978312645`

What about this one :
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`123456789456789123789123456214365897365897214897241635531672948642938571978514362`

For the max, my favourite for the moment :
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`123456789457289631698317254245178396731964825986523417312645978574892163869731542`

JPF
JPF
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Location: Paris, France

JPF wrote:What about this one ...
Gaaa... how did I miss that? I'm having a bad time on this thread!

The minimum min-lex grid should be v. easy to find: it's just the global minimum grid. Is that what you've just showed above? (I've not checked)
Red Ed

Posts: 633
Joined: 06 June 2005

daJ95376 wrote:According to gsf's solver, the canonical form of your grids have [r2c4]=1.

thanks, that means the index416 doesnt quite give us the min lex that we require.......

I'm also having doubts if the 6 figure banding code can identify all essentially different grids - assuming the flaw in the repeating minirows can be ironed out.

I am also begining to suspect this maximum min-lex grid is going to be a tough one to prove !

C
coloin

Posts: 1790
Joined: 05 May 2005

Red Ed wrote:The minimum min-lex grid should be v. easy to find: it's just the global minimum grid. Is that what you've just showed above? (I've not checked)

No, it was just a counter-example

JPF
JPF
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Posts: 3755
Joined: 06 December 2005
Location: Paris, France

Right, so here is the minimum min-lex grid:
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` 1 2 3 | 4 5 6 | 7 8 9 4 5 6 | 7 8 9 | 1 2 3 7 8 9 | 1 2 3 | 4 5 6-------+-------+------- 2 1 4 | 3 6 5 | 8 9 7 3 6 5 | 8 9 7 | 2 1 4 8 9 7 | 2 1 4 | 3 6 5-------+-------+------- 5 3 1 | 6 4 2 | 9 7 8 6 4 2 | 9 7 8 | 5 3 1 9 7 8 | 5 3 1 | 6 4 2`
Or, in compact form:
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`min-lex: 123456789456789123789123456214365897365897214897214365531642978642978531978531642    JPF: 123456789456789123789123456214365897365897214897241635531672948642938571978514362                                                          ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^`
You were close, JPF!

Having found this grid, I searched for it on the 'net and found an amazing short Perl program to calculate it.
Red Ed

Posts: 633
Joined: 06 June 2005

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