## Low/Hi Clue Thresholds

Everything about Sudoku that doesn't fit in one of the other sections

### Re: Low/Hi Clue Thresholds

Yes thats excellent coding ....if no DB need 13 clues it is probably academic for the 17-search as you possibly have searched and excluded all grids with bands requiring 6 clues .
champagne wrote:6 of the 416 bands have a minimum of 6 clues in band. (1;224;237;254;307;413 )
Any pair of these bands has a minimum number of clues of 12.
the best chance to pass 12 clues as minimum for 2 bands is with such a pair.....

I agree with that !
I'm a bit more optimistic now ....!

Previously published details on the 4 grids
Hidden Text: Show
Code: Select all
`  1,  1,  1,224,224,224       0,     0,     0,     0,     0,     0       # 18                                   4,224,224,224,224,224     252,     0,     0,     0,     0,     0       #  6                                 224,224,224,254,254,413       0,     0,     0,     0,     0,     0       #  6                                 224,224,224,225,225,414       0,     0,     0,    90,    90,   324       #  2                                                                                                                                               The bands that need 6 clues are  1,224,237,254,307,413                                                  123456789456789123789123456214897365365214897897365214541632978632978541978541632 : blue1  puz20=0 puz21= 7488123456789456789123789132465218967534564213978937548216391875642645321897872694351 : blue2  puz20=0 puz21= 3138123456789457189326689327154216534897745891632938672541361245978574918263892763415 : blue3  puz20=0 puz21= 3894123456789457189326689327154216534897745891632938672541392765418574918263861243975 : blue4  puz20=0 puz21=19682`

Last edited by coloin on Sat Jan 22, 2022 6:56 pm, edited 1 time in total.
coloin

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### Re: Low/Hi Clue Thresholds

coloin wrote:
Previously published details on the 4 grids
224,224,224,254,254,413 0, 0, 0, 0, 0, 0 # 6
224,224,224,225,225,414 0, 0, 0, 90, 90, 324 # 2
123456789457189326689327154216534897745891632938672541361245978574918263892763415 : blue3 puz20=0 puz21= 3894
123456789457189326689327154216534897745891632938672541392765418574918263861243975 : blue4 puz20=0 puz21=19682

Sorry first for all the trouble in the band Id .; Working on index, I have from time to time to adjust the number by +-1. I'll did it badly. I'll adjust the posts.

I was interested in checking the band "224" having the lowest count of "6 clues".

162 *162 remains a big number. I have less valids out of these 162*162 , but still 576 about 2/100 of the raw count
Last edited by champagne on Sun Jan 23, 2022 9:58 am, edited 1 time in total.
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### Re: Low/Hi Clue Thresholds

Yes they are all big numbers .....
I an getting pessimistic to find a DB13 now, although it wouldn't be feasible to check all the potential DB.....
eg all 15 band pairs from (1;224;237;254;307;413 )
coloin

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### Re: Low/Hi Clue Thresholds

coloin wrote:Yes they are all big numbers .....
I an getting pessimistic to find a DB13 now, although it wouldn't be feasible to check all the potential DB.....
eg all 15 band pairs from (1;224;237;254;307;413 )

Yes, in theory it's not to hard, assuming that the 13 is not in another pair (a much lower probability)
I have the code to produce all ED pairs, just missing time and cores to do it.
The number of pairs containing only the six bands is not that big,

BTW, the 900 million pairs have a minimum number of clues in the range 8-12. (forgetting the case 7) and likely a small number of pairs requiring 12 clues.
the pic must be in 9-10 with hundred millions of pairs
champagne
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### Re: Low/Hi Clue Thresholds

I think you have a good appreciation of the task - there is almost certainly DB which require at least 8 [4+4] clues but actually need 9.

983 Million DB ... how many pairs both with 6 clues needed.....

416 x 415 / 2 combinations = 86,320 pairs each equally likely perhaps [?] and we have 15 pairs both with 6 clues required

983 M * 15 / 86,300 = 170,000 pairs

MCN 16 grids might also be a good place to look .... but they may well be included in above ?
-
mathimagics wrote: Another interesting thing about the grids is that only 187 (of the 222,278 found) have non-trivial automorphisms.

from this clever thread no grids with mcn >16

Addendum ... maybe its just not possible to put 13 UAs in 54 clues [cf the above thread] . Certainly not possible to put 13 disjoint U4s .
coloin

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### Re: Low/Hi Clue Thresholds

coloin wrote:I think you have a good appreciation of the task - there is almost certainly DB which require at least 8 [4+4] clues but actually need 9.

983 Million DB ... how many pairs both with 6 clues needed.....

416 x 415 / 2 combinations = 86,320 pairs each equally likely perhaps [?] and we have 15 pairs both with 6 clues required

983 M * 15 / 86,300 = 170,000 pairs

Hi coloin,

The rough count is likely correct.
As I wrote in another post, if we just consider ED pairs, the total is closer to 620 millions pairs.

Anyway, I have the code to produce any of the options. I have just to clean the inline producer of the bands for the search 17.

this could be done in "say" one week if I put it on the top of the to do file.

Champagne
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### Re: Low/Hi Clue Thresholds

coloin wrote:Addendum ... maybe its just not possible to put 13 UAs in 54 clues [cf the above thread] . Certainly not possible to put 13 disjoint U4s .

13 disjoint uas minimum 4x13=52 surely not possible.

13 UAs in 54 clues: I work with hundreds sometimes more than 1000 uas depending on the pair, this with a strict limitation in the UAs size (around 20 clues at maximum). Could you tell more about this

Note : the band set of UAs to get all valid one band solutions can reach 81 UAs (from memory) this is for band 29. Generally, the set of UAs for one band is around 30
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### Re: Minimum clues in a double band [DB]

champagne wrote:....
In the 17 search, we are considering ED bands 12 "minimal against the diagonal view"
This reduces the number of minimal in my case to 610 163 364 but using the 983,959,110 ED double bands would not be a problem....
"
Yes I didn't quite understand the .... " minimal against the diagonal view" .... but finally the penny dropped !!!
It must mean a diagonal reflection in a DB - which maintains equivalence - but the transformation is difficult to visualise !
I think that is a new one for most of us ... but well worth it !!!!

But maybe there is a bar set at 12 - [or it just doesn't exist] which might explain why you haven't found one in the more likely grids
[there is only theoretical space for 12 disjoint U4s in a double band - as there can only be 4 pairs per box][one spare clue per box]
But there are many non disjoint UAs which maybe just don't succumb to our 12 clues - especially with the bigger UAs which we never know about
and there are many possible band pairs for it to happen...

As with many things we try to achieve.... it is because its difficult that we try...
Proving something is impossible is just as valid as finding a needle in a haystack
coloin

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### Re: Minimum clues in a double band [DB]

coloin wrote:
champagne wrote:....
In the 17 search, we are considering ED bands 12 "minimal against the diagonal view"
This reduces the number of minimal in my case to 610 163 364 but using the 983,959,110 ED double bands would not be a problem....
"
Yes I didn't quite understand the .... " minimal against the diagonal view" .... but finally the penny dropped !!!
It must mean a diagonal reflection in a DB - which maintains equivalence - but the transformation is difficult to visualise !
I think that is a new one for most of us ... but well worth it !!!!

Sorry if I did not explain it in the right way.
And I am not so sure that we don't have to use the 983,959,110 ED double bands in your exercise.

In the 17 search, we want to avoid redundancy, so, to make it simple, we process the 5 472 730 538 ED solution grids.
To catch these ED solution grids, we don't process bands having a lower morph in the canonicalization process. 90° rotation or main diagonal symmetry must be used to cover all morphs
Thinking a little more about that, I think that you would like to have all pairs, not only pairs relevant in the catalog of ED solution grids.
champagne
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### Re: Low/Hi Clue Thresholds

hi coloin,
more about the bands requiring 6 clues.

In my 17 search, these bands have index 0 to 5. The reason is that the bands 1 are ordered to have the lowest count of valid 6 clues (as in mladen table) in the solution grid.

These six bands ordered 224, 237, 1, 413, 254, 307 have auto morphs respectively 5,1,107,35,5,5.
If as I think, the average auto morph is somewhere between 1 and 2, we should find less than average pairs.

to give an idea, in this area, a band 1 has more than 5Million bands 2 attached in the process with no auto morph.
band 1 (index2) has only 52 561
band 224 (index 0) 912 650 (5 auto morphs)
band 224(index 1) 2 680 806 (1 auto morph)

The number of pairs out of the 983,959,110 ED double bands could be below 100 000.
champagne
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### Re: Low/Hi Clue Thresholds

Ok ...thats good they all need 12 clues ... and each DB expands to give approx 40 ED solution grids ...

maybe in those 100K DB ... there will be those which have a "ganster44reverse" pattern which isnt solvable in 6 clues ......

there are only 44 "gangster44reverse equivalents" or DBgangsters perhaps is better

this one produced fortuitously - from earlier post
Code: Select all
`...........................123456789456789123789123456214897365365214897897365214   - none are solvable with 5 `

this might well be the only one of the 44 which isnt solvable in 5 / which needs 6 clues

All solutions of those DB which have that DB gangster all need 18C

but maybe those grids will just be combinations of our 6 bands (1;224;237/254;307;413) - and they are already excluded from your 17C search no doubt

Or maybe eg (1;224;237/xxx;yyy;zzz) (horizontal 3 / vertical 3) for example ..... that would be good because those solution grids could be excluded from your 17C search
coloin

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### Re: Low/Hi Clue Thresholds

coloin wrote:Ok ...thats good they all need 12 clues ... and each DB expands to give approx 40 ED solution grids ...

maybe in those 100K DB ... there will be those which have a "ganster44reverse" pattern which isnt solvable in 6 clues ......

there are only 44 "gangster44reverse equivalents" or DBgangsters perhaps is better

Hi coloin

if I got your point, this should be a 17 distribution 6+6+5

here is the list of known for the corresponding band 1 6 clues

.23.........1....6.9..7........9....5....2...8......71.4.....9.....3.2.....6.....;1;63;354;5
.23.........1....6.9..7........9....5...2....8......51.4.....9.....3.2.....5.....;1;137;330;5
1.......9...7...2..8...3..........7.5....1...6...9...8...............5.1.726.....;2;180;340;5
12...6...........3......45..9.5.......7.....1...34....3............97..2..5......;2;204;292;5
12..........78...3..9......5...4.9..6....1...........8.......1....8.2....7.....6.;2;206;209;5
1...5......6....2......34.......7.6.39...8...5..............3....126............8;2;231;410;5
1...5......6....2......34.........6734...8...9.......................3.8..126....;2;294;347;5
1...........8..6.2.9...7...2.8......6............34.7....21..5..4.....3..........;3;329;372;5
.234...........6.........512...1.........89...4.....3.............3...4.8.5..1...;4;31;370;5
12...........8.6.3...7.......8....7...4.3.........2.1......1...........8.46..5...;4;320;334;5
1....6.8...7.....3....2....26...1......9..5.7...........57......3.....6.....9....;5;23;52;5
12...6...............3..45..1......6..45...7.....9......5....9......1..27........;5;122;318;5
12...6...............3..45...45......6..9...2.......7......2..6..5....9.7........;5;160;318;5
..3.....9...1...2..8...7......5......126...........3.5....9.8..6......1......3...;5;172;343;5

here index 0-5 as first band
none of them has the second band with an index 0-5
champagne
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### Re: Low/Hi Clue Thresholds

maybe this explains .....

but maybe you knew this already !!!!!

Code: Select all
`123456789457893612896127345634289157578631924219574863761345298942768531385912476 # {413 168 168  , 238 125 126}  `

This grid solution cant have a 17

Edited to another grid.
Program index416 attributes different band numbers. Band numbers needing 6 clues are { 1,132,145,168,202,413}
This grid has 3 bands which needs 6 clues. However minlexing it will hide it in the mix.
Last edited by coloin on Wed Jan 26, 2022 11:29 pm, edited 1 time in total.
coloin

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### Re: Low/Hi Clue Thresholds

coloin wrote:maybe this explains .....

but maybe you knew this already !!!!!

Code: Select all
`513624789984715326276389154425136897791458632638297541142563978857941263369872415    193 289 284  , 307 307 413 `

This grid solution cant have a 17

I don't know why
but as you got it, from the list above where we have no 5,5 I agree, it can not have one.
champagne
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### Re: Low/Hi Clue Thresholds

32 more solution grids which cant have a 17 ... and most lost in the minlexing

Hidden Text: Show
Code: Select all
`123456789457893612896127345274689153638571924519234867761945238942368571385712496'413 202 202  , 316 399 344123456789457893612896127345274689153638571924519234867961745238742368591385912476'413 202 202  , 382 180 305123456789457893612896127345268319457714568923539274861971645238685932174342781596'413 202 202  , 275 234 361123456789457893612896127345268319457714568923539274861972645138685931274341782596'413 202 202  ,  79 395  89123456789457893612896127345234689157678531924519274863761345298985712436342968571'413 202 202  , 150 398  56123456789457893612896127345274689153638571924519234867761345298985712436342968571'413 202 202  , 260 409  57123456789457893612896127345234689157678531924519274863361945278942768531785312496'413 202 202  ,  49 372 354123456789457893612896127345234689157678531924519274863961345278342768591785912436'413 202 202  , 377 397 107123456789457893612896127345274689153638571924519234867361945278942768531785312496'413 202 202  , 260 401 364123456789457893612896127345274689153638571924519234867961345278342768591785912436'413 202 202  , 408 397 118123456789457893612896127345631278954574619823289534167742361598968745231315982476'413 202 202  , 395 235 149123456789457893612896127345631278954574619823289534167745361298968742531312985476'413 202 202  , 407 374 269123456789457893612896127345671238954534619827289574163742361598968745231315982476'413 202 202  , 396 366  65123456789457893612896127345671238954534619827289574163745361298968742531312985476'413 202 202  , 412 303  50123456789457893612896127345714268953568319427239574861971645238685932174342781596'413 202 202  , 342 200 304123456789457893612896127345714268953568319427239574861972645138685931274341782596'413 202 202  , 129 409 282123456789457893612896127345342689157678512934519374268231945876964738521785261493'413 202 202  , 382 225 216123456789457893612896127345342689157678512934519374268931245876264738591785961423'413 202 202  , 408 194 235123456789457893612896127345542689137678312954319574268231945876964738521785261493'413 202 202  , 192 338  50123456789457893612896127345542689137678312954319574268931245876264738591785961423'413 202 202  , 192 403 264123456789457893612896127345238569471574318926619274853961745238785932164342681597'413 202 202  , 395  93 248123456789457893612896127345238569471574318926619274853962745138785931264341682597'413 202 202  , 299 291 317123456789457893612896127345238579461564318927719264853971645238685932174342781596'413 202 202  , 392  77 156123456789457893612896127345238579461564318927719264853972645138685931274341782596'413 202 202  , 228 386 120123456789457893612896127345238569471574318926619274853741935268962781534385642197'413 202 202  , 385 193 333123456789457893612896127345238569471574318926619274853742935168961782534385641297'413 202 202  , 409 323  75123456789457893612896127345268579431534618927719234856971345268642781593385962174'413 202 202  , 410 126 237123456789457893612896127345268579431534618927719234856972345168641782593385961274'413 202 202  , 354 354  46123456789457893612896127345245689137678315924319274568531942876964738251782561493'413 202 202  , 208 341 120123456789457893612896127345245689137678315924319274568782561493964738251531942876'413 202 202  , 208 341 120123456789457893612896127345245689137678315924319274568782961453564738291931542876'413 202 202  , 342 171 155123456789457893612896127345245689137678315924319274568931542876564738291782961453'413 202 202  , 342 171 155`
Last edited by coloin on Wed Jan 26, 2022 11:36 pm, edited 2 times in total.
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