The New Sudoku Players' Forum

[JPF]

For Mike:

Code: Select all

question B
+---+---+---+   +---+---+---+
|...|...|154|   |.37|.9.|154|
|.1.|...|826|   |915|437|826|
|...|1..|793|   |48.|15.|793|
+---+---+---+   +---+---+---+
|1..|...|537|   |19.|.4.|537|
|...|.1.|962|   |874|315|962|
|...|...|481|   |.53|.79|481|
+---+---+---+   +---+---+---+
|761|983|245|   |761|983|245|
|528|764|319|   |528|764|319|
|349|521|678|   |349|521|678|
+---+---+---+   +---+---+---+
4 solutions
49 clues        71 clues - 22 placed


For coloin:

Code: Select all

question C
+---+---+---+
|1..|...|...|
|...|1..|7.8|
|...|...|.95|
+---+---+---+
|.1.|8..|...|
|...|9.7|...|
|...|654|.2.|
+---+---+---+
|5..|...|...|
|.97|..2|...|
|.83|...|..2|
+---+---+---+   21C


JPF

[coloin]

Nice one

as it turns out - out of all the many millions of B1245 [4-boxes] [2802 classes also ?] I have only found 44 ED which solve with 2 clues
It was only necessary to examine one each of the 2802 classes of B36789

[EDIT - this figure is 2865] Not having access to this file [ but see below] I randomly selected 10000 [39 obtained] and 50000 B36789 [44 obtained] and added 2 clues

4 have the 2 clues in diagonal boxes, and these 2 clues are dissimilar

Hidden Text: Show

Code: Select all

849675...273189...516432...934826...652791...187543..............................#1A
495786...872913...361452...624137...583694...917528..............................#2A
874965...921873...653412...138729...495638...762541..............................#3A
674859...981723...352461...126538...598674...743192..............................#4A

40 have the 2 clues in the same band

Hidden Text: Show

Code: Select all

467895...829173...513246...682314...194657...735928.............................. #01B
567984...398721...214653...831576...729418...645392.............................. #02B
568974...397128...214356...731685...849712...625493.............................. #03B
678459...319827...254631...163274...927586...485193.............................. #04B
745698...389721...216453...158379...967214...423586.............................. #05B
945678...378921...216453...859317...167294...423586.............................. #06B
549678...387921...216453...958317...761294...423586.............................. #07B
549678...378921...216453...851397...967214...423586.............................. #08B
846975...917328...532146...394782...768531...125694.............................. #09B
846975...791328...532146...984732...367581...125694.............................. #00B
946587...817932...523461...789625...234198...165374.............................. #11B
946587...817932...523416...789625...234198...165374.............................. #12B
894675...731829...652413...976381...143256...528794.............................. #13B
547689...398721...216435...461273...753918...829546.............................. #14B
846975...793821...251436...634192...572683...918754.............................. #15B
657498...398712...124365...719623...536184...482579.............................. #16B
659874...387219...124563...712396...536481...498725.............................. #17B
574698...891723...632451...926534...743186...185279.............................. #18B
968574...731298...425136...143625...657849...892713.............................. #19B
594678...871932...623415...739126...248597...165384.............................. #20B
964578...871932...523614...236197...798425...145386.............................. #21B
964578...871932...523641...236197...798425...145386.............................. #22B
486579...379182...521634...957826...863451...142793.............................. #23B
485679...379182...621534...957826...863451...142793.............................. #24B
549687...187392...623415...491563...768924...352871.............................. #25B
569478...781932...324651...952817...847369...136524.............................. #26B
648597...739182...521436...953824...162973...487651.............................. #27B
796584...128793...534621...261935...859467...347218.............................. #28B
549876...782319...136245...253798...918634...467521.............................. #39B
946875...728139...513642...465283...132794...879516.............................. #30B
456987...789123...132546...925671...314258...867439.............................. #31B
846597...379821...512643...167389...928415...453276.............................. #32B
485976...739128...216453...897312...523694...641785.............................. #33B
765489...981723...432516...859634...146257...327891.............................. #34B
964785...827913...531264...642571...318496...795832.............................. #35B
965478...873921...412536...346259...159783...728614.............................. #36B
945678...873921...612435...324569...159783...768214.............................. #37B
765984...982317...431562...359278...216495...847631.............................. #38B
846975...972138...513246...738514...169782...254693.............................. #39B
469587...378912...125463...842739...913654...657128.............................. #40B

Code: Select all

849675...273189...516432...934826...652791...187543.............................. - #1A
......123......456......798......517......384......269368214975425967831791358642 - one of the 280368 solutions pf #1A  [280368/36 = 7788]
...6..123......456......798..4...517......384......269368214975425967831791358642 - each one of those solutions is completed by the two clues

Code: Select all

+---+---+---+
|...|6..|123|
|...|...|456|
|...|...|798|
+---+---+---+
|..4|...|517|
|...|...|384|
|...|...|269|
+---+---+---+
|368|214|975|
|425|967|831|
|791|358|642|
+---+---+---+


[Serg]

Hi!
I 've done exhaustive search for puzzles, containing the same digit in one-clue boxes B1, B2, B4, B5:

Code: Select all

+-----+-----+-----+
|. . .|. . .|x x x|
|. . .|. . 1|x x x|
|. . 1|. . .|x x x|
+-----+-----+-----+
|. . .|. . .|x x x|
|. . .|. 1 .|x x x|
|. 1 .|. . .|x x x|
+-----+-----+-----+
|x x x|x x x|x x x|
|x x x|x x x|x x x|
|x x x|x x x|x x x|
+-----+-----+-----+

Such puzzles don't exist.

Serg

[JPF]

Hi Serg,
Thank you for this confirmation.
Can you also confirm that the minimum number of solutions is 4?

JPF

[Serg]

< withdrown >

[coloin]

Very good Serg.

Delving back into the archives, im struggling to find blue's calculation of the ED B12347 and B5689... will find it soon

However I have found this snippit from here

The correct sum of 6670903752021072936960 was obtained,
so it should be correct this time, correcting the earlier calculation
with 2802 classes.

A file with one B2347 puzzle (B1 is forced) from each class is at
http://magictour.free.fr/2865.gz

The file of 2865 [not 2802] "similar tuple" B12347 is still available there

Maybe this can be used to prove that minimum solution count is 4

Adding 2 clues to the 2865 gave these 64 puzzles

Hidden Text: Show

Code: Select all

...357169...148257...2693484125.....735......689......153......827......964.8....
...357169...148257...269348412...5..735......689......153......827......964....8.
...258176...149358...367249241....3.3856.....679......813......462......597......
...147258...259376...3681492415.....853......967......139..6...582......674......
...147258...259376...368149241.7....853......967......1398.....582......674......
...147258...259376...368149241...5..853......967......139.....7582......674......
...147258...259376...368149241....6.853......967......139...8..582......674......
...368259...257146...1493784215.....736......859......318..6...245......697......
...368259...257146...149378421...5..736......859......318.....7245......697......
...257349...169258...3481764126.....763......958......831......627......549....8.
...257349...169258...348176412...7..763......958......831......627......549.8....
...257349...169258...348176412......763......958..4...831...6..627......549......
...257349...169258...348176412......763......958.....48317.....627......549......
...168349...259178...347256421....3.386......975..6...143......792......568......
...168349...259178...347256421......386......975......143.2....792......568.....7
...268149...347256...159378124......6834.....957......731......248......596.7....
...268149...347256...159378124......683......957..6...7314.....248......596......
...247359...169278...358146241......836......975..6...319......524.1....687......
...169278...248356...357149126......345......897..4...461...5..258......973......
...169478...348256...2571391234.....458......976....8.317......542......869......
...359476...267158...148239321..4...845......697.....5183......572......469......
...248139...167258...359476213...5..854......976......391.....7542......687......
...347156...168239...259478123...5..845......769......451......982.....7637......
...259136...147258...368479132......846...3..579......314......728......695.....7
...259136...147258...368479132......846......579......314.2....728......695.....7
...367258...159436...2481792314.....486......579....8.713......642......958......
...257436...348179...169258123......486......597..6...3714.....842......659......
...158276...367459...249138231....4.487..5...569......314......892......756......
...368479...249158...157236123...7..849......657.8....731......264......985......
...368479...249158...157236123...7..849......657.9....731......264......985......
...368479...249158...157236123......849......657.8....7314.....264......985......
...368479...249158...157236123......849......657.9....7314.....264......985......
...349156...258479...1672381524.....834......697......413......726......985..6...
...349156...258479...167238152...3..834......697......413......726......985.....7
...347156...259478...1682395124.....384......679......431......296......857..6...
...358276...167439...2491582154.....348......967......731......426......859..6...
...358276...167439...249158215...3..348......967......731......426......859.....7
...247139...169258...358476152...3..834......796......317......249......685.....7
...167458...248139...3592762514.....736......489......314......862......597..6...
...368259...159476...247138152......873......964......315....2.429..6...687......
...347156...168279...259438521......738......964......315....2.692......847..6...
...358476...149238...2671595127.....783......469......137......824.9....956......
...248179...369258...157436621...3..834......579......143......782......956.....7
...359176...147258...268439621......348......579..6...1534.....294......867......
...258436...167259...349178261......783......945......318......426.1....597....8.
...348176...159238...267459621......893......547......164.2....285......739....8.
...247138...368459...159276821...3..463......597......134......285......976.....5
...368159...259476...147238812...3..346......957......531......428......769.....5
...368159...259476...147238812......346......957......531......428...3..769..5...
...157238...249176...368459182......356......497......813....2.729......645.9....
...359278...248156...167439413......682......957..6...7314.....248......596......
...257139...369458...148276134......6824.....795......471......328......569.7....
...158239...367458...249176431......268......759..6...914......8234.....576......
...359478...268159...147236143....2.286......579..6...791......324......865......
...359478...268159...147236143......286......579......791......324.1....865.....7
...369275...247186...158439123......457...3..986......341......268......795.....8
...369275...247186...158439123......457......986......341.2....268......795.....8
...149236...258479...367185231...7..945......867.9....173......524......689......
...149236...258479...367185231......945......867.9....1734.....524......689......
...368475...249136...157289213......945......786..5...471...3..629......538......
...349186...157239...268475123......549......876..5...714...3..982......635......
...258439...349176...167285125......473......689..3...541.2....862......397......
...149276...268135...3574895124.....793......864....9.137......245......986......
...149276...268135...357489512.....4793......864......137....9.245......986......


but these came from only 43 B12347 [not 44] ... hmmm

Hidden Text: Show

Code: Select all

...147258...259376...368149241......853......967......139......582......674......
...149236...258479...367185231......945......867......173......524......689......
...149276...268135...357489512......793......864......137......245......986......
...157238...249176...368459182......356......497......813......729......645......
...158239...367458...249176431......268......759......914......823......576......
...158276...367459...249138231......487......569......314......892......756......
...167458...248139...359276251......736......489......314......862......597......
...168349...259178...347256421......386......975......143......792......568......
...169278...248356...357149126......345......897......461......258......973......
...169478...348256...257139123......458......976......317......542......869......
...247138...368459...159276821......463......597......134......285......976......
...247139...169258...358476152......834......796......317......249......685......
...247359...169278...358146241......836......975......319......524......687......
...248139...167258...359476213......854......976......391......542......687......
...248179...369258...157436621......834......579......143......782......956......
...257139...369458...148276134......682......795......471......328......569......
...257349...169258...348176412......763......958......831......627......549......
...257436...348179...169258123......486......597......371......842......659......
...258176...149358...367249241......385......679......813......462......597......
...258436...167259...349178261......783......945......318......426......597......
...258439...349176...167285125......473......689......541......862......397......
...259136...147258...368479132......846......579......314......728......695......
...268149...347256...159378124......683......957......731......248......596......
...347156...168239...259478123......845......769......451......982......637......
...347156...168279...259438521......738......964......315......692......847......
...347156...259478...168239512......384......679......431......296......857......
...348176...159238...267459621......893......547......164......285......739......
...349156...258479...167238152......834......697......413......726......985......
...349186...157239...268475123......549......876......714......982......635......
...357169...148257...269348412......735......689......153......827......964......
...358276...167439...249158215......348......967......731......426......859......
...358476...149238...267159512......783......469......137......824......956......
...359176...147258...268439621......348......579......153......294......867......
...359278...248156...167439413......682......957......731......248......596......
...359476...267158...148239321......845......697......183......572......469......
...359478...268159...147236143......286......579......791......324......865......
...367258...159436...248179231......486......579......713......642......958......
...368159...259476...147238812......346......957......531......428......769......
...368259...159476...247138152......873......964......315......429......687......
...368259...257146...149378421......736......859......318......245......697......
...368475...249136...157289213......945......786......471......629......538......
...368479...249158...157236123......849......657......731......264......985......
...369275...247186...158439123......457......986......341......268......795......


EDIT
no panic though,these 64 puzzles have the following recipricol B5678

Hidden Text: Show

Code: Select all

..............................576983...892614...413725...924876...635491...781532
..............................983576...614892...725413...876924...491635...532781
..............................985637...671924...432581...594762...713895...826413
..............................593687...672491...814532...486725...731964...925813
..............................975683...612497...483521...824765...736914...591832
..............................786593...491762...532814...625487...974631...813925
..............................783965...496712...521483...675824...914637...832591
..............................593687...814925...672431...726594...981763...435812
..............................786593...925814...431762...594627...673981...812435
..............................675893...982514...431762...596427...814935...723681
..............................893765...514982...672431...426597...935814...781623
..............................873965...592481...614723...425697...981534...736812
..............................975863...481592...623714...796425...534981...812637
..............................895637...721594...436812...672985...583461...914723
..............................895763...471592...632814...726985...584631...913427
..............................785963...492517...613824...924685...536791...871432
..............................973685...524791...816423...485962...691537...732814
..............................985763...721594...436812...874625...613987...592431
..............................793485...681927...524631...872593...936714...415862
..............................486795...973621...512384...695842...831967...724513
..............................594687...673912...812345...925764...436891...781523
..............................786594...912763...534812...825647...673981...491325
..............................876594...921763...534821...793682...615347...482915
..............................694785...572391...831624...785962...916543...423817
..............................685794...972513...431862...726985...593641...814327
..............................485967...973521...612384...596842...831795...724613
..............................974685...532791...816324...485962...693517...721843
..............................786945...935621...421783...592867...673514...814392
..............................596784...712563...483921...824695...935817...671342
..............................586794...712563...493821...924685...835917...671342
..............................674895...523761...981324...496582...835917...712643
..............................674985...523761...891324...486592...935817...712643
..............................493687...675912...821543...982765...534891...716324
..............................786394...912765...534821...675982...891543...423617
..............................493867...675912...821543...782695...534781...916324
..............................493867...675921...812543...584692...931785...726314
..............................876394...912765...534821...685942...791583...423617
..............................786394...912765...534821...895642...671583...423917
..............................483967...925814...671523...792685...534791...816342
..............................673894...491562...825713...984627...716385...532941
..............................693784...415962...872513...984627...731845...526391
..............................736894...924561...815723...682945...593617...471382
..............................974385...526791...813624...785962...691543...432817
..............................734985...591762...826341...482697...675813...913524
..............................593847...624915...781362...976524...815793...432681
..............................935847...472561...681923...823795...796314...514682
..............................576394...912587...834621...785962...691743...423815
..............................675394...912587...834621...786942...591763...423815
..............................693547...571982...824613...982764...716395...435821
..............................936547...724891...581362...675924...413685...892713
..............................972685...534791...816324...485962...693517...721843
..............................785962...493517...621843...932685...516794...874321
..............................592687...734591...816342...685723...471965...923814
..............................895627...713594...426813...584362...671985...932741
..............................895762...471593...632841...583624...716985...924317
..............................594867...682391...731524...876952...915743...423618
..............................576894...982613...431752...825967...793541...614328
..............................685794...712863...493521...924658...836917...571342
..............................674958...823761...591324...485692...936817...712543
..............................976854...831627...425913...592368...783541...614792
..............................986547...721863...435921...692358...573614...814792
..............................874963...692851...513724...726398...931547...485612
..............................493867...682541...715392...924658...876913...531724
..............................873964...624851...591327...485692...936718...712543


Of these 44 are ED

[Serg]

Hi, JPF!

Can you also confirm that the minimum number of solutions is 4?

Now I've done exhaustive search and can confirm, that minimum number of solutions for puzzles with the same digit in one-clue boxes B1245 - 4.

Really I counted completions, not puzzles, having boxes B1245 (partially) filled and empty boxes B36789. Each such completion produces a family of required puzzles (having B1245 boxes with 1 clue (the same digit placed in each box) and filled (9 clues) B36789 boxes).

Serg

[Edited. I've deleted distribution "Number of solutions"/"Number of completions" with wrong numbers of completions and add some comment.]

[coloin]

looks like 4 solutions is the minimum !

Delving back into the archives, im struggling to find blue's calculation of the ED B12347 and B5689... will find it soon

Code: Select all

B12347 - 18296007 canonical forms  [2865 classes]
B5689  -  5699915 canonical forms 

taken from the hidden text in this post here where its all been done !!!

The minimum number of clues for a B12347 section (a band crossing a stack) is 5.
The minimum number of clues for a B5689 section (a 2x2 set of boxes) is 2.
The minimum clue count distributions, for the respective canonical forms, are below.

B12347 MinClues distribution (for the 18296007 canonical forms)
5: 879381
6: 12332250
7: 4822691
8: 258236
9: 3449

B5689 MinClues distribution (for the 5699915 canonical forms)
2: 44
3: 553591
4: 4603035
5: 527258
6: 15701
7: 284
8: 1
9: 1

[JPF]

Now I've done exhaustive search and can confirm, that minimum number of solutions for puzzles with the same digit in one-clue boxes B1245 - 4.

Excellent! even if I don't understand what the number 72 represents.

JPF

[Serg]

< Withdrown >

[JPF]

I had gotten over 15,000 ed 4-sols-puzzles without extensive research. Hence my question...
The exact number may be interesting, the list less.

JPF

[coloin]

The exact number may be interesting, the list less.JPF

using the list of 2865 add the 2 digits [1-9] in 2 diagonal boxes and you will have your answer !!

[blue]

Code: Select all

+-------+-------+-------+
| . . . | . . . | 7 . . |
| . 1 . | . . . | . . . |
| . . . | 1 . . | 5 . 2 |
+-------+-------+-------+
| 1 . . | . 4 . | . . . |
| . . . | 3 . 6 | . 1 . |
| . . . | . 7 . | . . . |
+-------+-------+-------+
| 5 . 7 | . 6 . | . . . |
| . . . | 2 . . | . 9 . |
| 4 6 . | . . . | . 8 . |
+-------+-------+-------+    19C

[Serg]

Hi, JPF!

I had gotten over 15,000 ed 4-sols-puzzles without extensive research. Hence my question...
The exact number may be interesting, the list less.

More accurate search got 137 ED completions, having B1245 boxes filled by 8 clues (the same digits in the each box are absent). An example of such completion:

Code: Select all

+-----+-----+-----+
|1 2 3|4 5 7|. . .|
|4 5 6|2 8 .|. . .|
|7 8 .|3 1 6|. . .|
+-----+-----+-----+
|3 4 8|5 . 1|. . .|
|5 6 1|7 4 2|. . .|
|2 . 7|6 3 8|. . .|
+-----+-----+-----+
|. . .|. . .|. . .|
|. . .|. . .|. . .|
|. . .|. . .|. . .|
+-----+-----+-----+

The same in line form:

Code: Select all

123457...45628....78.316...3485.1...561742...2.7638..............................

These 137 completions produce 57224 ED 4-solution puzzles having 4 clues in B1245 boxes (each box contains the same digit) and boxes B36789 filled by 9 clues each.

Serg

[JPF]

Thanks for those results, Serg!

JPF