About Red Ed's Sudoku symmetry group

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

Postby Red Ed » Fri Feb 06, 2009 6:30 am

udosuk, I can't do the easy job I offered for you because I still don't have a precise list of which of the 3359232 validity-preserving transformations should score any points. Unfortunately your previous post didn't help much:( I still don't understand whether or not my "scored" version of eleven's taxonomy fully represents your needs.

So I've put that to a side and gone back to concise presentations of automorphism groups. Here's the revised list, with all grids in a form whose aut group can be described concisely. Generally speaking, those grids should be "nicer" to look at than the ones in the lists of 122 posted previously.

2 157694823824173695963582174549238716716945238238716549691457382482361957375829461 1 134 <BxCx>
2 319246875246875913578319624752693148691428357483157296827531469135964782964782531 1 79 <H>
2 457826913162953847983471625274685139819237456635194278726548391548319762391762584 1 37 <D>
3 246187539137259486589436217758943621924618753613725948861572394375894162492361875 1 32 <RS>
3 275849163843165279169273845587326914326914587914587326732691458691458732458732691 1 30 <R2R3S>
3 347965821659218347182473965596821473473596182218734596931687254864352719725149638 1 7 <R3C>
3 359782146278146593614593782761254938945318627823679451482967315136425879597831264 1 10 <RC>
3 497863125251497638638125974863512497974386251125974863749638512512749386386251749 1 8 <C>
3 587946231246531987931287546654193728793628154128754693469875312875312469312469875 1 28 <R3S>
3 682439517537681429419572683823794165145863792796125834278946351354218976961357248 1 22 <BS>
3 718564392564239718392718645631852974849376521275491863427683159153947286986125437 1 9 <R2R3C>
3 829153764754869123163724859231547698598631247647298531376482915415976382982315476 1 25 <S>
4 279135486684279351153468279348516927516927843927843615792681534435792168861354792 1 79 134 <H,BxCx>
4 592137468674582193831496725146953287957268341328741659483615972719824536265379814 1 79 86 <Q>
4 964285371732194685581376249813952467296417538457863912378549126625731894149628753 1 37 79 <H,D>
6 124386795369751248587492613835974126742163859691528437976215384253847961418639572 1 22 37 <D2,BS>
6 136452978452897136897361452245789361613245897978136245524978613789613524361524789 1 8 79 <H,C>
6 147256389829437516356189247714625938982743651635918724298561473561374892473892165 1 28 134 <RxSx,R3S>
6 198426357763895124425317698819642735637958241542731869981264573376589412254173986 1 22 79 <H,BS>
6 251897463674532918398164752746325189523981647189746325932618574467253891815479236 1 25 134 145 <S,BxCx>
6 256189347347256198189347256521968473473521869968473521612895734734612985895734612 1 32 134 <BxCx,BC>
6 276435819819276435435819276342198567567342198198567342653724981981653724724981653 1 32 134 142 <BxCx,RS>
6 426953187371842956589167342947386521152479638863215794695728413234691875718534269 1 9 79 <H,R1R3C>
6 471965328965832471328471965714596832596283147832147659147659283659328714283714596 1 8 134 135 <C,RxSx>
6 519736284742581639386249751691357842257814396834692517963175428175428963428963175 1 10 37 40 <DR>
6 563712489189453762742689153356271948298346571471598326635127894824935617917864235 1 22 37 43 <D2S>
6 619345872453728619872619453196287534287534196534196287961453728345872961728961345 1 8 134 <C,BxCx>
6 652789143134652987789134562218475396475396218396218475927561834561843729843927651 1 9 134 <FhR2xCx,R1R3C>
6 689523417714689325325417986196274853853196274472358691267931548548762139931845762 1 30 134 143 <R3BxSCx>
6 732859641148236957659741832273985164814623795965174283396517428481362579527498316 1 25 134 <S,RxSx>
6 734568129861927435952413876143295687295786341687341592416872953378659214529134768 1 28 134 144 <BxCx,R2S>
6 759284613284613975136759284568437129491526738327198456842361597613975842975842361 1 7 79 <H,R2C>
6 795821463821463795463795821954638217237914658618257934182346579346579182579182346 1 30 79 <H,R1R3S>
6 876419352549732186312856479954681237681273945237945618495168723168327594723594861 1 30 134 <RxSx,R2R3S>
6 894326715312857496756914832569748321248193657173562984625489173487231569931675248 1 10 37 <D2,RC>
6 973526418416978523528413976265184739184739265739265184897352641341697852652841397 1 25 79 <H,S>
6 976254813253816974814973256138542769542769138769138542681497325397625481425381697 1 28 79 <H,R2S>
6 976841532842536971531972846325719468769428315418365729184253697653197284297684153 1 32 79 <H,RS>
6 985631247623947815741825639158493726396752481472186953814579362539268174267314598 1 10 79 <H,RC>
8 648732951751694832932851647469273185185469273273185469896327514517946328324518796 1 37 79 86 134 <D,BxCx>
9 128375964375964281496128375812496753964537128537281496281649537649753812753812649 1 8 29 <R2R3C1S>
9 187234956625798413349561872432659781978143265516827394263975148894316527751482639 1 10 25 32 <S,RC>
9 259643781178925364643781259925178643781364925364259178896537412412896537537412896 1 7 28 30 <R2S,R3C>
9 265719843971438652384652197526971438197843526438265971652197384719384265843526719 1 8 28 30 32 <C,C3B>
9 358741269926835417417692835692417583741358926835926741174583692583269174269174358 1 7 8 9 10 <C,C3R>
9 549837126126549837837126549481765293293481765765293481314972658972658314658314972 1 9 30 32 <R3RS,R2R3C>
9 618475932932186754754329861861547293475932186329861547293754618547618329186293475 1 8 26 <R3C1S>
9 643917258985234671127568493364791825598423167712856349436179582859342716271685934 1 22 25 32 <S,BC>
9 675283149914567328832491756769825413341976582258134697183752964527649831496318275 1 10 28 30 32 <RC,R3S>
9 692357814573814926148269735926735481357481269814692573269573148735148692481926357 1 8 25 32 <B,C>
9 695174823482369517731258946567412389948736251123895674279681435354927168816543792 1 9 28 32 <R3S,RSC>
9 718436259925871364643592187364925718871643592259187436436718925592364871187259643 1 8 32 <C,RS>
9 782915643364782591159436782827159364436827915591643827245398176973261458618574239 1 7 30 32 <RS,R3C>
9 814956327732148695695327481327569814956481273148273956261835749583794162479612538 1 7 9 <C3R,R3CC>
9 872456193693812457451793862236145978918237645745968231129574386387629514564381729 1 22 32 <BC,RS>
9 953768214781245936246931785475326891369817452812459367137582649524693178698174523 1 22 25 <B,S>
12 127984635356271984498356127712849563563712849849563712271498356635127498984635271 1 8 79 134 135 <H,C,RxSx>
12 358479621429651378671328459783294516294516783516783294867932145942165837135847962 1 25 79 134 145 <H,S,BxCx>
12 518349267642875913379126548836512794427698135951734826795263481164987352283451679 1 22 37 43 79 <D,D2S>
12 716283459529146873483759216295837164837461592164592738648375921952614387371928645 1 28 79 134 144 <H,BxCx,R2S>
12 817692354354817692692354817178265439439178265265439178781546923923781546546923781 1 32 79 134 142 <H,RS,BxCx>
12 856219734219437658734658912921865473568374129473921865192743586347586291685192347 1 30 79 134 143 <SH,R3RxSx>
12 861345927247169385359827164685931742174286539923754816538692471416578293792413658 1 10 37 40 79 <D2,DR>
18 125398467398746251746251398839467512467125983251983746674512839512839674983674125 1 8 31 79 <H,C1RS>
18 137528496825649731469173285974381652318256974652794318596417823741832569283965147 1 10 23 79 <H,R1C1BS>
18 158349726934672815267581493782195364519436278643827951326758149875914632491263587 1 10 25 32 79 <H,S,RC>
18 237154698986372154415986237372541986869723541154869372723415869698237415541698723 1 8 25 32 134 135 142 145 <B,C,RxSx>
18 259184367418736592736925184587349621162857943394261875673592418925418736841673259 1 7 9 79 <H,C2R,CHR2>
18 315692478498375612672418395186954723753126984924783156839567241261849537547231869 1 22 25 79 <H,S,B>
18 354716298167982543298354716639875421875421639421639875543167982716298354982543167 1 7 25 28 79 <H,S,R2C>
18 421657938389421765576893421214389576657214893938765214142938657765142389893576142 1 8 32 134 135 <C,RxSx,RS>
18 471385269853926471692714853926853714714269538538147692147538926385692147269471385 1 8 29 79 <H,R1R3C1S>
18 519623478784195623623478195195236784478519362236784951951362847847951236362847519 1 8 24 79 <H,C1BS>
18 536712498271894356984635172613278549827459631495163287368527914752941863149386725 1 22 32 79 <H,BC,RS>
18 547183926329645781861279435754318692932564178186927543475831269293456817618792354 1 22 25 32 79 <H,S,BC>
18 615293874293748561487561329156932487932874615748615293561329748329487156874156932 1 8 25 32 79 <H,C,B>
18 642319785785642319319785642463921857857463921921857463236194578578236194194578236 1 22 32 134 142 <BxCx,BC,RS>
18 643297518972518364851643972596382741417965823238174659185436297364729185729851436 1 7 30 32 79 <H,RS,R2C>
18 654231798798546312231798546312987465879654231546312987465123879987465123123879654 1 8 27 79 <H,C1S>
18 681593427274168359935274168816742593742935681359681274427816935593427816168359742 1 8 26 79 <H,R2C1S>
18 695321748487956213321748695569132874874569132132874569748213956213695487956487321 1 8 28 134 135 <C,RxSx,R3S>
18 746829513315647928829513746197438265562791834438265197681954372273186459954372681 1 22 25 134 145 <B,S,BxCx>
18 749236158158749623623581497974362815815497362362815749497623581581974236236158974 1 8 28 30 32 79 <C,H,C2B>
18 768124395539876412241953687472368951683519724195247836324795168816432579957681243 1 10 28 30 32 79 <H,RC,R2S>
18 786951243951243786243786951174625839839174625625839174597468312468312597312597468 1 9 30 32 134 142 <BxCx,R1R2SC>
18 815923764476581392923764815647239158239158647158647239764392581392815476581476923 1 8 30 134 135 <C,RxSx,R2R3S>
18 825746931674931582931825746582467193467319258193258674258674319746193825319582467 1 8 28 30 32 134 135 142 143 144 <C,RxSx,C2B>
18 873215946291634785564978132415362897637849521928751463346197258789526314152483679 1 10 22 37 40 <D2C,BS>
18 876349512934125687125687493251934876768512934349768125687493251493251768512876349 1 7 8 9 10 79 <H,C,C2R>
18 934761582582934617176582934761493258258617493493258176825349761617825349349176825 1 8 32 79 <H,C,RS>
18 956341827734682195218579463671938542425167389893254716569413278347826951182795634 1 9 28 32 79 <H,R2S,RSC>
18 963287415847195623215463987396728541784519362521346798639872154478951236152634879 1 22 25 32 134 142 <S,RxSx,BC>
18 975362184184597623362184975759623841418759362623841759597236418841975236236418597 1 8 22 79 <H,C,BS>
18 984253671235176849671894523716532498849761235352948716167325984498617352523489167 1 8 9 134 135 <RxSx,C1C2R>
27 695287134287413695413695728341956287569872341728134956872341569134569872956728413 1 7 8 9 10 28 30 32 <C,C2B,C3R>
27 857612943394578126261439785578261394439785612126943857785126439943857261612394578 1 8 22 25 32 <B,C,RS>
36 178459623659132487423687195817945362965213748342768519781594236596321874234876951 1 22 32 79 134 142 <H,RxSx,BC,RS>
36 279145683638297154145683279927514368863729415514368927792451836386972541451836792 1 22 25 32 79 134 142 145 <H,S,BxCx,BC>
36 294371865361895274875264391752643918613958742948712653487126539136589427529437186 1 22 25 79 134 145 <H,S,B,RxSx>
36 516482973824739165973516482651248397248397651397651248165824739482973516739165824 1 8 25 79 134 135 145 <H,C,S,RxSx>
36 629471835835629471471835629962358714714962358358714962296147583583296147147583296 1 8 32 79 134 135 142 <H,C,RxSx,RS>
36 698427135452183679713569842549812763831796524276354981384971256967245318125638497 1 22 25 79 86 <Q',S>
36 712398564645127983398564712456839271839271456271456839127983645564712398983645127 1 8 28 79 134 135 144 <H,C,BxCx,R2S>
36 716284593428593716593671428372469851649158372851327649167842935284935167935716284 1 7 9 79 86 <Q,C2R>
36 952786413743915862816342759625891374397254681184673295239168547478529136561437928 1 10 22 37 40 43 79 <D2S,DR>
36 953274816274681539816539427168742395395168742742395168427816953681953274539427681 1 8 30 79 134 135 143 <C,SH,R3RxSx>
54 159372468684159237723846159915723684846915372237468915591237846468591723372684591 1 8 22 25 32 134 135 142 145 <B,C,RxSx,RS>
54 165348729834972516297651483748129365912536874653487291329765148576814932481293657 1 10 22 25 32 79 <H,S,B,RC>
54 297361845458972613361845297729613584845729361136458729972136458584297136613584972 1 8 9 28 30 32 134 135 142 143 144 <C,RxSx,C3BR>
54 297415836368729154541683972729541683836972415154368297972154368683297541415836729 1 8 22 25 32 79 <H,C,B,RS>
54 318562749974831256562749318183625497749318562625497183497256831256183974831974625 1 8 9 28 30 32 134 135 <RxSx,R3S,C2C3C3R>
54 512347689473968512896125473251734968347896251689512347125473896734689125968251734 1 8 24 25 31 32 79 <H,B,C1RS>
54 768395412249871536153624987415762398836549271927183654392418765571236849684957123 1 10 22 25 32 37 40 43 <B,DR>
54 864579123231486957579123648486957312123648795957312864648795231312864579795231486 1 8 24 25 27 32 79 <H,B,C1S>
54 951678432432951867867243951786432195195786243243519786324195678678324519519867324 1 7 8 9 10 25 28 30 32 79 <C,H,B,C2R>
72 814237695695148723723956814569372481372481569481569372148723956956814237237695148 1 8 10 37 40 79 86 134 135 <D2,C,BxCx>
72 827541963165329748943867521438675219516932874279418635794186352651293487382754196 1 22 25 37 43 79 86 134 145 <D2,S,RxSx>
108 312564897897312564564897312123456789789123456456789123231645978978231645645978231 1 8 22 25 32 79 134 135 142 145 <H,C,B,RxSx,RS>
108 659178243824365917731492586178243659365917824492586731243659178917824365586731492 1 10 22 25 32 37 40 43 79 <B,D2,CH>
108 674983251251674983983251674125398467398467125467125398746839512512746839839512746 1 8 10 28 30 32 79 134 135 142 143 144 <R,C,H,BxCx,R2S>
162 514238697823769451697514238451823769382976145769451823145382976238697514976145382 1 8 9 22 25 30 32 134 135 142 145 <B,S,RxSx,C1C3C3R>
648 639281475281475639475639281396812754812754396754396812963128547128547963547963128 1 8 10 22 25 32 37 40 43 79 86 134 135 142 145 <D2,C,S,BxCx>
Red Ed
 
Posts: 633
Joined: 06 June 2005

Postby Red Ed » Sat Feb 07, 2009 1:52 am

Reviewing that list, there are some rather unnatural-looking descriptions of automorphism groups.

One of the worst is the following grid ...
Code: Select all
6 5 2 7 8 9 1 4 3
1 3 4 6 5 2 9 8 7
7 8 9 1 3 4 5 6 2
2 1 8 4 7 5 3 9 6
4 7 5 3 9 6 2 1 8
3 9 6 2 1 8 4 7 5
9 2 7 5 6 1 8 3 4
5 6 1 8 4 3 7 2 9
8 4 3 9 2 7 6 5 1
... whose automorphism group is:
(EDIT: oops - wrong - that's in normalised form. Will post non-normalised form when I've put the kids to bed ...)
(EDIT2: ... okay, done it.)
Code: Select all
r:123456789 c:123456789
r:231456897 c:231564897
r:312456978 c:312645978
r:798456132 c:132465798
r:879456213 c:213546879
r:987456321 c:321654987
(Notation: r:ABCDEFGHI means row 1 of the new/transformed grid equals row A from the original grid, and so on.)

My program described that group as < FhR2xCx, R1R3C >, meaning that it is generated by the two operations:
Code: Select all
FhR2xCx =
    invert the order of the columns in all stacks (Cx), then
    exchange rows 4 and 7 (R2x), then
    flip across horizontal axis (Fh)
and
Code: Select all
R1R3C =
    cyclically move the columns in all stacks downwards (C), then
    cyclically move the rows in band 3 downwards (R3), then
    cyclically move the rows in band 1 downwards (R1)

If anyone can come up with a cleaner/nicer description of this group, or even just those two generators, then I'd appreciate it.
Last edited by Red Ed on Sat Feb 07, 2009 1:59 am, edited 2 times in total.
Red Ed
 
Posts: 633
Joined: 06 June 2005

Postby eleven » Sat Feb 07, 2009 1:57 am

Thanks to udosuk for correcting my post, i have edited it.

Thanks for this list, Red Ed. I hardly had time to read the posts last week. For me this presentation of the grids looks very fine. It should not be hard to find the "missing" symmetries manually, but just be interesting sometimes.
This is a lot better than what i could do before with my friends program, namely generate puzzles with up to 2 symmetries and calculate the number of automorphisms with gsf's program then.

As an example i looked at the first grid with 108 automorphisms:
Code: Select all
 +-------+-------+-------+
 | 3 1 2 | 5 6 4 | 8 9 7 |
 | 8 9 7 | 3 1 2 | 5 6 4 |
 | 5 6 4 | 8 9 7 | 3 1 2 |
 +-------+-------+-------+
 | 1 2 3 | 4 5 6 | 7 8 9 |
 | 7 8 9 | 1 2 3 | 4 5 6 |
 | 4 5 6 | 7 8 9 | 1 2 3 |
 +-------+-------+-------+
 | 2 3 1 | 6 4 5 | 9 7 8 |
 | 9 7 8 | 2 3 1 | 6 4 5 |
 | 6 4 5 | 9 7 8 | 2 3 1 |
 +-------+-------+-------+

1 8 22 25 32 79 134 135 142 145
<H,C,B,RxSx,RS>

So we already have HT(79),MR(8),JC(25\),RS(134\) and GR(32) given in the grid:
HT 13 2 49 58 67
MR 123 456 789
JC 123 465 798
RS 1 2 3 47 58 69
GR 1 2 3 4 5 6 7 8 9
RS+JC gives 145\

Additionally we have
CS (134): 13 2 46 5 79 8
and thus CS+MC (135) and CS+GR (142)

So the only missing symmetry is JD (22).
I reordered it to have 123 in the first minirows of boxes 159:
Code: Select all
 +-------+-------+-------+
 | 1 2 3 | 5 6 4 | 7 8 9 |
 | 6 4 5 | 8 9 7 | 2 3 1 |
 | 9 7 8 | 3 1 2 | 4 5 6 |
 +-------+-------+-------+
 | 8 9 7 | 1 2 3 | 6 4 5 |
 | 2 3 1 | 4 5 6 | 9 7 8 |
 | 5 6 4 | 7 8 9 | 3 1 2 |
 +-------+-------+-------+
 | 4 5 6 | 9 7 8 | 1 2 3 |
 | 7 8 9 | 2 3 1 | 5 6 4 |
 | 3 1 2 | 6 4 5 | 8 9 7 |
 +-------+-------+-------+

JD 1 2 3 456 789.

So i am happy for now with the list. Maybe udosuk's criteria will give interesting grids to compare, but i have good stuff to play with for a longer time:)
I changed the link to the 122 automorphism groups in the first post to this list now.

Some thoughts about these generating sets. I already realized 2 things.
The one was, that a symmetry, which implies another one (in the sense, that there is no real grid, which has it and does not have the other one), might generate only a subset of the automorphisms generated by both.

The other was, that the number of automorphisms of an aut group is not always the product of the numbers of automorphisms of the symmetries in the generating set.

By definition you can get all automorphisms by combining the symmetries in the set in arbitrary ways. If applying 2 symmetries is commutative (AB=BA) for all pairs in the generating set, this simplifies to (A^i)(B^j)(C^k) for a set <A,B,C> (i=0,..n-1, where (A^n)=I etc.). But also here its not proved, that the number of auts is not less the product (?).

If you dont have this property, that they are commutative, the generated set can have more automorphisms than the product.

To get a better feeling for that, we calculated a sample set, how the 72 automorphisms of the max. minlex grids can be expressed as combinations of <DM,CS,JR>. We shortened it to D (diagonal reflection), K (sticks) and J (jumping row). We read it from left to right (KD then means first sticks then diagonal reflection), but i think, it does not matter here.
Code: Select all
D K J
KD JD DK JK DJ KJ JJ
DKD JKD DJD KJD JJD KDK JDK DJK KDJ JDJ DKJ DJJ
KDKD JDKD DJKD KDJD JDJD DKJD DJJD JKDK KJDK JJDK KDJK JDJK JKDJ KJDJ JJDJ KDKJ JDKJ KDJJ JDJJ
JKDKD KJDKD JJDKD KDJKD JDJKD JKDJD KJDJD JJDJD KDKJD JDKJD KDJJD JDJJD JKDJK KJDJK JJDJK JKDKJ
  KJDKJ JJDKJ JKDJJ KJDJJ JJDJJ
JKDJKD KJDJKD JJDJKD JKDKJD KJDKJD JJDKJD JKDJJD KJDJJD JJDJJD


Not very intuitive:)
Note that KDKD or DKDK is the same as Half Turn.
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Postby eleven » Sat Feb 07, 2009 2:42 am

Red Ed wrote:If anyone can come up with a cleaner/nicer description of this group, or even just those two generators, then I'd appreciate it.

Since there are only those 2 symmetries in the grid, you could take my representatives R1R2C and CxBx.
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Postby Red Ed » Sat Feb 07, 2009 3:42 am

I don't think that works, eleven. I don't think those two symmetries (R1R2C and CxBx) can coexist in a single grid.
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Postby eleven » Sat Feb 07, 2009 4:41 am

Internet sharing at home can be hard, i know this since an hour, but my friend is playing games:)
To use her 5 minutes pause: Yes, in your grid you have to swap rows 1 and 3 to switch between the symmetries. This would mean R1xCxBxR1x instead of FhR2xCx to describe the group, which does not look easier. But is is for me:)
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Postby udosuk » Sat Feb 07, 2009 4:08 pm

Yes, thanks for the list Ed, I think it's good enough for the time being.

At the moment I can't spend too much time studying these, so have to take it real slow. The best thing about leisure is there is no deadline.:)

If given time, I'll look at the 162-automorphism class and try to integrate all 162 from the 10 different symmetries. Then I'll try to see if it helps on analysing the 648 class (MC grid). I'd also like to study why all order-9 symmetries (FL, WL, BL, FD and the mixed ones) imply HT. There are just too many interesting aspects in this topic.:idea:
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Postby Red Ed » Sat Feb 07, 2009 5:16 pm

udosuk wrote:I'd also like to study why all order-9 symmetries (FL, WL, BL, FD and the mixed ones) imply HT.

That comment interested me, so I checked it. It's not quite right:
    Class 23, "FD"
    18 137528496825649731469173285974381652318256974652794318596417823741832569283965147 1 10 23 79 <H,R1C1BS>

    Class 24, "BL"
    18 519623478784195623623478195195236784478519362236784951951362847847951236362847519 1 8 24 79 <H,C1BS>
    54 512347689473968512896125473251734968347896251689512347125473896734689125968251734 1 8 24 25 31 32 79 <H,B,C1RS>
    54 864579123231486957579123648486957312123648795957312864648795231312864579795231486 1 8 24 25 27 32 79 <H,B,C1S>

    Class 26, "2FR1WR"
    9 618475932932186754754329861861547293475932186329861547293754618547618329186293475 1 8 26 <R3C1S>
    18 681593427274168359935274168816742593742935681359681274427816935593427816168359742 1 8 26 79 <H,R2C1S>

    Class 27, "FL"
    18 654231798798546312231798546312987465879654231546312987465123879987465123123879654 1 8 27 79 <H,C1S>
    54 864579123231486957579123648486957312123648795957312864648795231312864579795231486 1 8 24 25 27 32 79 <H,B,C1S>

    Class 29, "1FR2WR"
    9 128375964375964281496128375812496753964537128537281496281649537649753812753812649 1 8 29 <R2R3C1S>
    18 471385269853926471692714853926853714714269538538147692147538926385692147269471385 1 8 29 79 <H,R1R3C1S>

    Class 31, "WL"
    18 125398467398746251746251398839467512467125983251983746674512839512839674983674125 1 8 31 79 <H,C1RS>
    54 512347689473968512896125473251734968347896251689512347125473896734689125968251734 1 8 24 25 31 32 79 <H,B,C1RS>
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Doing the twist

Postby Red Ed » Sat Feb 07, 2009 11:16 pm

<deleted - contained innumerable mistakes>
Last edited by Red Ed on Sat Feb 07, 2009 10:02 pm, edited 2 times in total.
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Postby eleven » Sun Feb 08, 2009 12:16 am

I will refer to the groups in Red Ed's list by the line number, e.g. i call the one with 10 symmetries and only 18 automorphisms S87. (Note, that the line numbers are different in the first list).
Code: Select all
S87: 3 grids, 18 auts
 +-------+-------+-------+
 | 8 2 5 | 7 4 6 | 9 3 1 |
 | 6 7 4 | 9 3 1 | 5 8 2 |
 | 9 3 1 | 8 2 5 | 7 4 6 |
 +-------+-------+-------+
 | 5 8 2 | 4 6 7 | 1 9 3 |
 | 4 6 7 | 3 1 9 | 2 5 8 |
 | 1 9 3 | 2 5 8 | 6 7 4 |
 +-------+-------+-------+
 | 2 5 8 | 6 7 4 | 3 1 9 |
 | 7 4 6 | 1 9 3 | 8 2 5 |
 | 3 1 9 | 5 8 2 | 4 6 7 |
 +-------+-------+-------+
1 8 28 30 32 134 135 142 143 144
<C,RxSx,C2B>


I shorten the symmetries below by the bold letter.
K: RS 1 24 3 56 78 9
M: MR 193 258 467

G: 32 GC S13, S2 leftwards 1 2 3 4 5 6 7 8 9
[Edit: I had mixed 1 and 2 here]
1: 30 (1 JC, 2 GC): JC S13, GC S2 139 285 476
2: 28 (2 JC, 1 GC): JC S13,GC S2 leftwards, 193 258 467

I verified, what udosuk claimed, that the 18 automorphisms can be defined following (i.e. all these operations lead to different cell numbers):

Code: Select all
I, K,
M, G, 1, 2,
MK, GK, 1K, 2K,
MM, GG, 11, 22,
MMK, GGK, 11K, 22K


Moreover there are a lot of equations, which dont let you have more automorphisms:

Code: Select all
G=M1,1=M2, 2=MG,
KM=MK, KG=GK, K1=1K, K2=2K, MG=GM, M1=1M, G1=1G, G2=2G, 12=21,
G1=22, G2=11, GG=12,
etc.
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Postby eleven » Sun Feb 08, 2009 7:56 am

I forgot to mention, that the generating symmetries <C,RxSx,C2B> are identical with my <M,K,2>.
Since from the first equations line you have 1=M2 and G=M1=MM2 you can express the above automorphisms also with these 3 symmetries.

Using such a notation you can formulate, what udosuk proved here (a grid with DM and Sticks normalized also has HT and QT) with
H..Half Turn
Q..Quarter Turn = DBxRx
K..Column Sticks = BxCx
D..Diagonal Mirror (D\)
Bx..reverse bands
Cx..reverse columns in all stacks
Rx..reverse rows in all bands

This one has to be verified:
DKDK=H (you also have Half Turn normalized)
and this directly follows:
KD=BxCxD <=> BxKDBx=BxBxCxDBx=CxDBx=DRxDDBx=DRxBx=DBxRx=Q (reversing the bands gives a grid with Quarter Turn)
or
DK=DBxCx <=> CxDKCx=CxDBxCxCx=CxDBx=DRxDDBx=DRxBx=DBxRx=Q (reversing the columns in all stacks gives a grid with Quarter Turn)

A problem is, that we hardly can find single symbols for the symmetries we use. We have 26 single symmetries, one for each letter. Often we need the mirrored symmetry and now also the alternative forms for Mini Diagonals and Gliding Rows. Maybe we should use chinese symbols, udosuk ?
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Postby StrmCkr » Sun Feb 08, 2009 6:33 pm

deleted
Last edited by StrmCkr on Sun Dec 11, 2016 9:06 am, edited 1 time in total.
Some do, some teach, the rest look it up.
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Postby eleven » Tue Feb 10, 2009 1:58 am

We have to distinguish between implications given by the cell mapping alone, like CS+DM (<BxCx,D>) => HT and QT, DM+JD (<DBS>) => DM and JD (DBSDBS=BSBS, DBSDBSDBS=D), or FR => MR (SC1SC1SC1 = C), and those by restrictions to possible numberings from the sudoku rules.
An example for that is FR => HT. From the 9 cell mappings defined by FR (when starting with a fixed grid) you cant follow, that an HT mapping always will exist (in a real grid).

Now i dont have a real proof for FR => HT, but it was nice to see all those HT symmetries:

For a FR number mapping (123456789) you only can have these 9 lines, where (to avoid a duplicate in a box) the first 3, second 3 and third 3 have to be distributed to bands 1, 2 and 3.
Code: Select all
147 258 369
714 825 936
471 582 693

258 369 471
825 936 147
582 693 714

369 471 582
936 147 258
693 714 825

Here you can see 9 Half Turn number cycles, where always 4 pairs of lines reflect one another and the single one reflects itself:
Code: Select all
HT 19 28 37 46 5: 1,23,48,57,69
HT 13 2 49 58 67: 2,13,47,59,68
HT 16 25 34 79 8: 3,12,49,58,67
HT 12 39 48 57 6: 4,17,29,38,56
HT 15 24 3 69 78: 5,19,28,37,46
HT 18 27 36 45 9: 6,18,27,39,45
HT 14 23 59 68 7: 7,16,25,34,89
HT 17 26 35 4 89: 8,15,24,36,79
HT 1 29 38 47 56: 9,14,26,35,78

Thus for each possible distribution of the lines in the bands you will find a morph with HT symmetry.

E.g. with 158 in band 1, 267 in band 2 and 349 in band 3 you can reorder the lines to 267/185/934 with HT number cycle 17 26 35 4 89:
Code: Select all
714 825 936
582 693 714
369 471 582

147 258 369
936 147 258
825 936 147

693 714 825
471 582 693
258 369 471

Or you can transform 347/169/258 to 258/619/473:
Code: Select all
714 825 936
825 936 147
936 147 258

582 693 714
147 258 369
693 714 825

258 369 471
369 471 582
471 582 693
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Postby udosuk » Tue Feb 10, 2009 5:22 am

Thanks for the nice insights, eleven.

I actually have worked out something on a similar line to see why FR => HT etc but didn't have the time to write it up nicely (like you did). Note my holiday is officially over now so I'll have to take it *real slow* from now on.:(

Red Ed wrote:
udosuk wrote:I'd also like to study why all order-9 symmetries (FL, WL, BL, FD and the mixed ones) imply HT.

That comment interested me, so I checked it. It's not quite right:

Thanks, I overlooked the 2 extra cases for the mixed ones. What we can conclude though is that all order-9 symmetries coexist with HT in at least one of the 122 classes.
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Postby eleven » Tue Feb 10, 2009 10:54 pm

It was a bit surprising for me, that Red Ed had listed <D2S> as generator of the first group with class 43 (D+BS), where i had expected SBD.
So (reading it from left to right now), my representative DBS must be equivalent to SD2 (and DB).

This is Red Ed's grid:
Code: Select all
S27, 6 auts, 18 grids
3 symmetries 22 37 43: JD (BS), DM (D) and DM+JD (DBS)
 +-------+-------+-------+
 | 5 6 3 | 7 1 2 | 4 8 9 |
 | 1 8 9 | 4 5 3 | 7 6 2 |
 | 7 4 2 | 6 8 9 | 1 5 3 |
 +-------+-------+-------+
 | 3 5 6 | 2 7 1 | 9 4 8 |
 | 2 9 8 | 3 4 6 | 5 7 1 |
 | 4 7 1 | 5 9 8 | 3 2 6 |
 +-------+-------+-------+
 | 6 3 5 | 1 2 7 | 8 9 4 |
 | 8 2 4 | 9 3 5 | 6 1 7 |
 | 9 1 7 | 8 6 4 | 2 3 5 |
 +-------+-------+-------+

Here you cant see the DM and JD symmetry directly, just S2D.

After SSFv we are here: (Fv: mirror at column 5 or flip against vertical axis resp.)
Code: Select all
 +-------+-------+-------+
 | 3 6 5 | 9 8 4 | 2 1 7 |
 | 9 8 1 | 2 6 7 | 3 5 4 |
 | 2 4 7 | 3 5 1 | 9 8 6 |
 +-------+-------+-------+
 | 6 5 3 | 8 4 9 | 1 7 2 |
 | 8 9 2 | 1 7 5 | 6 4 3 |
 | 1 7 4 | 6 2 3 | 8 9 5 |
 +-------+-------+-------+
 | 5 3 6 | 4 9 8 | 7 2 1 |
 | 4 2 8 | 7 1 6 | 5 3 9 |
 | 7 1 9 | 5 3 2 | 4 6 8 |
 +-------+-------+-------+
D 14 25 3 69 7 8
BS 159 264 387

As equation: (x~y, if there is a t with txt'=y, here y=SD2, x=DBS, t=FvS, t'=(FvS)'=S'Fv=SSFv)
FvS(DBS)SSFv = FvSDBFv = FvSDDSDFv = FvSSDFv = FvSSFvFvDFv = SD2
FhD2(SD2)D2Fh = FhD2SFh = DS = DDBD = BD
D(DS)D = SD = DBDD = DB
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