Thread: "Klein's Quartic n-order formula and specific values"

From: David Smith <djs314djs314@yahoo.com>
Date: Sun, 1 May 2011 17:09:26 -0700 (PDT)
Subject: Klein's Quartic n-order formula and specific values



--0-251550593-1304294966=:84942
Content-Type: text/plain; charset=iso-8859-1
Content-Transfer-Encoding: quoted-printable

Hi everyone,

Okay, I can now claim I fully understand the Klein's Quartic puzzle.=A0 I a=
pologize
for my earlier mistakes, and guarantee that the following formula and calcu=
lations
are accurate, excepting a typo.=A0 I had failed to realize that Klein's Qua=
rtic is very
much like a 3-dimensional puzzle.=A0 It has 'corners' and 'edges'.=A0 I did=
not study the
order-2 puzzle enough to realize that the 'corners' existed.=A0 This is of =
course
obvious to all of you, since you have played with the puzzle more than I ha=
ve, and
many of you have even solved it.=A0 The way the order-3 and higher puzzle a=
ppear
immediately made obvious to me the nature of the 'corners' and 'edges'.=A0 =
The
order-2 puzzle's appearance somewhat disguises its true nature, but that is=
no
excuse for my lack of investigation.

Here is the formula for the number of permutations of the order-n Klein's Q=
uartic
puzzle for odd n >=3D 3:





And here are individual values:

Order-2:

((55!)/2)*(3^53) =3D

123048752649957426169448861190331245749810829958743981704307326752100311\
522831667036160000000000000

Order-3:

(84!)*(56!)*(2^81)*(3^55) =3D

993919860351336970116941239153296103939497528293494685545497421465353823\
867122389570825760861036003717296401423048183458017237535043916628667848529=
348\
418500952983796478875293990282425667716662641223463686191606729487155200000=
000\
000000000000000000000000

Order-5:

(((168!)/((7!)^24))^2)*((168!)/2)*(84!)*(56!)*(2^81)*(3^55) =3D

153813597602981865296923122878293408297707605840877433021601977114757488\
141188456378084938578404431213530399155802876124924501302998795803488994282=
318\
638973392226465765639269482642062480505130632851949992339629771331733910361=
394\
187875621200775783795330499539579105683370865469180184421558825857092857952=
139\
675652574064438504019365838704215066676118844823272520482463359811616584351=
650\
343294082974911191373304636015923982296607487243206304661823823179829496028=
840\
929128708900416627301861209241866736735119716387769997220107174269840158765=
913\
914510643210391000467377130910048027024945418450246083224037558282200707856=
137\
417672596140743196018794757987904441064746702209870189273782494772422622248=
229\
582206349072075076140907820840336618805947675411840837091706167131159031529=
916\
608909239023245119282172250315201115126049303085088200765125878010439058553=
209\
194884800770173580565544960000000000000000000000000000000000000000000000000=
000\
0000000000000000000000000000000000000000000000000000

Order-7:

(((168!)/((7!)^24))^6)*(((168!)/2)^2)*(84!)*(56!)*(2^81)*(3^55) =3D=20

291652081227680814998213157804602335018085364039052280961816073114448730\
392894920023356437101146476284320603185810624757021755947665298890077989550=
883\
885311343466371999384057110413096574296849941236724269152103544645442486218=
695\
076600252096301812306832372577853207239998648950663571317985724006765904739=
680\
977105203817627232866878630402546853197300928140997055230021724053848990847=
216\
551957669431605329014717773898577089777619452520575734359293791029055047460=
576\
974564133092157579429829165447260989008971819817294651225865111987625423736=
784\
401171396455752024547584118386324881866238870218023662586959246641024767000=
358\
285518158311830911257029022642565363726603369418008829918092525995884379417=
405\
670910582899355979014578770936069119778606566775357202755594741592869227544=
047\
178229603580102844151585056303034806574605033952482681047551433264000114310=
716\
924001031918988612472952919237611441342665686598541080344137343237544467454=
063\
349911222948500021047225513530557140871503559750837011458585002994770311031=
339\
781109128549029418138339924851610383224600629274310641062928263251950995560=
148\
937777274448616949465037469723128521933851772717618059366344850213381098526=
318\
319060550935249803574921929117917238519906930661961947706955966404872626429=
782\
132804861306667749534406026729935529881538323975563828330896791908785507675=
059\
773775794278700459656226381696124358922920273604368715840236421715579658507=
382\
365453517371351486086458623588131038391141321230610350870074067038457996241=
667\
720900561881878754648098316208849769503988442111398599945245303613729339183=
849\
167767601179210373127434178110063536344436887129279839469942450678228480290=
417\
935016209791318697331932192833279890869534215446438165799082847867421585376=
427\
125195520939078272991998436471331374584526239824202581947561332987759818220=
178\
449625990739452026138843213446477424511640590394442357826468773763181183467=
355\
539723948286105864976159089388473584954233058380887860737526988800000000000=
000\
000000000000000000000000000000000000000000000000000000000000000000000000000=
000\
000000000000000000000000000000000000000000000000000000000000000000000000000=
000\
00000000000000000000000000000000000000

Order-9:

(((168!)/((7!)^24))^12)*(((168!)/2)^3)*(84!)*(56!)*(2^81)*(3^55) =3D

677582869828578066590514174426477151611690434506950459808287370437586411\
589712092377745815252610930902313658103766880255574538687753843375713294897=
632\
229604198139428110010773285646728126823387215212255969547186353882369700028=
967\
357420423630846160741822346537314280930026018987102915805246070856404614500=
425\
080097246212719291213975788892366066186385076310698582572122488831893325275=
338\
022200576603148637405090301054731728037990100651517592252935578389346975932=
765\
393031787471498515172826138520961440317355862707925533637995776970273715363=
414\
763966001363948789113428479431707402741209524218143534272382729149389093065=
277\
738776510908735266022840504909278164416307388596961383581457574835426043053=
858\
567130861051011559090208050156911206052149351505039750810224644148029574393=
183\
643117846586265363506172950816736400283897320748166942865332312858367774914=
685\
400969693697720325524219880682507206742371460341267136120269318225507969219=
890\
147215159865406150475078930492604815977957514374625438016561144829700944961=
469\
903841769287163517353646630553103279651164614135572024499483587396729352167=
073\
314647757488867863523612624916957271732409936033435795555768007982838821934=
759\
529312703400982479192577976332369625248478885795751249576247219706313028822=
865\
251158674424998015916461969114262493126387933357244586468480986348677384231=
084\
051947741834028586096264224079871413353793237542286834143546953887276596438=
016\
154110548089321896312300275346346285309747824057951560619284748297160126997=
018\
555164003753708299148367101333402034431705143711815830435474957991495425111=
612\
359055249286259249512637496443116519132276856611864187351528116367034140077=
649\
228256600937975605450350463096018689689494317504186472028713476549889531616=
376\
265218640002711615030585980771460377839679256037218151469434036491261624917=
999\
789690125252651753140985992078803105585299781688200576431175253129002734686=
056\
692936038452950797255495408609243046821111989281772950244181499940165613599=
069\
752993108229555644132121597169167501944449522582579165541347370340485791216=
208\
810170273407800901109543642032049395922018609103137870253925946466865040950=
825\
824546115634620984823219064591293138556739533342518609429381273237731039456=
346\
033862977346326049590625286483487039215891508292461576449677325801713807841=
621\
978580747913527131242794907636091311463666958399217772206627407877191823447=
182\
972778655397576328357440033690392795699207683940192423828998570474496739832=
672\
019477652114590214018791934706322664952454443925279628796082488146236072268=
592\
846976676880487114045448401559927701979774401207562058551583086165782131087=
375\
511914244510236734437321250625678331088571958029689058768955878732093815869=
906\
096544425748818588441839626888167846642644935534200259360142428511293980509=
753\
769444013986800173687430173032163302215452990618380458170730647098036066945=
049\
985220622514584487675630337487016621327879705553585461019820126404131837827=
513\
491519322013117934471257069495199166734762494535604267339796974779665699893=
326\
026066490637551663922691469515745734461065858146408933969356951124432041596=
128\
272794916637229823948248584501592347729992029532594860584427611186151786254=
334\
823503820350363849469150455890670118740902566788239032536444855114742741037=
803\
655098630204492498417146997924096033291795233394232609566695291939284728181=
124\
371349676631081443458321783163024319460537095544807147503134516825792732519=
141\
759413323311576933662237655040000000000000000000000000000000000000000000000=
000\
000000000000000000000000000000000000000000000000000000000000000000000000000=
000\
000000000000000000000000000000000000000000000000000000000000000000000000000=
000\
000000000000000000000000000000000000000000000000000000000000000000000000000=
000\
0000000000000000000000000000000000000000000000000000000000000


I have to go now, but thank you all for your patience.=A0 This formula and =
the values
should not have to be corrected this time.

All the best,
David

--0-251550593-1304294966=:84942
Content-Type: text/html; charset=iso-8859-1
Content-Transfer-Encoding: quoted-printable

top" style=3D"font: inherit;">Hi everyone,

Okay, I can now claim I f=
ully understand the Klein's Quartic puzzle.  I apologize
for my ear=
lier mistakes, and guarantee that the following formula and calculations>are accurate, excepting a typo.  I had failed to realize that Klein's=
Quartic is very
much like a 3-dimensional puzzle.  It has 'corners=
' and 'edges'.  I did not study the
order-2 puzzle enough to realiz=
e that the 'corners' existed.  This is of course
obvious to all of =
you, since you have played with the puzzle more than I have, and
many of=
you have even solved it.  The way the order-3 and higher puzzle appea=
r
immediately made obvious to me the nature of the 'corners' and 'edges'=
.  The
order-2 puzzle's appearance somewhat disguises its true natu=
re, but that is no
excuse for my lack of investigation.

Here is t=
he
formula for the number of permutations of the order-n Klein's Quartic
p=
uzzle for odd n >=3D 3:



3D"http://i1029.photobuc=<br/ket.com/albums/y360/djs314/KQ.gif" src=3D"http://i1029.photobucket.com/albu=
ms/y360/djs314/KQ.gif">

And here are individual values:

Order=
-2:

((55!)/2)*(3^53) =3D

123048752649957426169448861190331245=
749810829958743981704307326752100311\
522831667036160000000000000
>Order-3:

(84!)*(56!)*(2^81)*(3^55) =3D

993919860351336970116=
941239153296103939497528293494685545497421465353823\
8671223895708257608=
61036003717296401423048183458017237535043916628667848529348\
41850095298=
3796478875293990282425667716662641223463686191606729487155200000000\
000=
000000000000000000000

Order-5:

(((168!)/((7!)^24))^2)*((168!)=
/2)*(84!)*(56!)*(2^81)*(3^55)
=3D

153813597602981865296923122878293408297707605840877433021601977=
114757488\
1411884563780849385784044312135303991558028761249245013029987=
95803488994282318\
63897339222646576563926948264206248050513063285194999=
2339629771331733910361394\
187875621200775783795330499539579105683370865=
469180184421558825857092857952139\
6756525740644385040193658387042150666=
76118844823272520482463359811616584351650\
34329408297491119137330463601=
5923982296607487243206304661823823179829496028840\
929128708900416627301=
861209241866736735119716387769997220107174269840158765913\
9145106432103=
91000467377130910048027024945418450246083224037558282200707856137\
41767=
2596140743196018794757987904441064746702209870189273782494772422622248229\<=
br>582206349072075076140907820840336618805947675411840837091706167131159031=
529916\
6089092390232451192821722503152011151260493030850882007651258780=
10439058553209\
19488480077017358056554496000000000000000000000000
0000000000000000000000000000\
000000000000000000000000000000000000000000=
0000000000

Order-7:

(((168!)/((7!)^24))^6)*(((168!)/2)^2)*(84=
!)*(56!)*(2^81)*(3^55) =3D


291652081227680814998213157804602335018085364039052280961816073114=
448730\
3928949200233564371011464762843206031858106247570217559476652988=
90077989550883\
88531134346637199938405711041309657429684994123672426915=
2103544645442486218695\
076600252096301812306832372577853207239998648950=
663571317985724006765904739680\
9771052038176272328668786304025468531973=
00928140997055230021724053848990847216\
55195766943160532901471777389857=
7089777619452520575734359293791029055047460576\
974564133092157579429829=
165447260989008971819817294651225865111987625423736784\
4011713964557520=
24547584118386324881866238870218023662586959246641024767000358\
28551815=
8311830911257029022642565363726603369418008829918092525995884379417405\
=
670910582899355979014578770936069119778606566775357202755594741592869227544=
047\
1782296035801028441515850563030348065746050339524826810475514332640=
00114310716\
924001031918988612472952919237611441342665686598541
080344137343237544467454063\
3499112229485000210472255135305571408715035=
59750837011458585002994770311031339\
78110912854902941813833992485161038=
3224600629274310641062928263251950995560148\
937777274448616949465037469=
723128521933851772717618059366344850213381098526318\
3190605509352498035=
74921929117917238519906930661961947706955966404872626429782\
13280486130=
6667749534406026729935529881538323975563828330896791908785507675059\
773=
775794278700459656226381696124358922920273604368715840236421715579658507382=
\
3654535173713514860864586235881310383911413212306103508700740670384579=
96241667\
72090056188187875464809831620884976950398844211139859994524530=
3613729339183849\
167767601179210373127434178110063536344436887129279839=
469942450678228480290417\
9350162097913186973319321928332798908695342154=
46438165799082847867421585376427\
12519552093907827299199843647133137458=
4526239824202581947561332987759818220178\
4496259907394520261388
43213446477424511640590394442357826468773763181183467355\
53972394828610=
5864976159089388473584954233058380887860737526988800000000000000\
000000=
000000000000000000000000000000000000000000000000000000000000000000000000\r>0000000000000000000000000000000000000000000000000000000000000000000000000=
00000\
00000000000000000000000000000000000000

Order-9:

(((=
168!)/((7!)^24))^12)*(((168!)/2)^3)*(84!)*(56!)*(2^81)*(3^55)
=3D

677582869828578066590514174426477151611690434506950459808287370=
437586411\
5897120923777458152526109309023136581037668802555745386877538=
43375713294897632\
22960419813942811001077328564672812682338721521225596=
9547186353882369700028967\
357420423630846160741822346537314280930026018=
987102915805246070856404614500425\
0800972462127192912139757888923660661=
86385076310698582572122488831893325275338\
02220057660314863740509030105=
4731728037990100651517592252935578389346975932765\
393031787471498515172=
826138520961440317355862707925533637995776970273715363414\
7639660013639=
48789113428479431707402741209524218143534272382729149389093065277\
73877=
6510908735266022840504909278164416307388596961383581457574835426043053858\<=
br>567130861051011559090208050156911206052149351505039750810224644148029574=
393183\
6431178465862653635061729508167364002838973207481669428653323128=
58367774914685\
40096969369772032552421988068250720674237146034126
7136120269318225507969219890\
147215159865406150475078930492604815977957=
514374625438016561144829700944961469\
9038417692871635173536466305531032=
79651164614135572024499483587396729352167073\
31464775748886786352361262=
4916957271732409936033435795555768007982838821934759\
529312703400982479=
192577976332369625248478885795751249576247219706313028822865\
2511586744=
24998015916461969114262493126387933357244586468480986348677384231084\
05=
194774183402858609626422407987141335379323754228683414354695388727659643801=
6\
154110548089321896312300275346346285309747824057951560619284748297160=
126997018\
5551640037537082991483671013334020344317051437118158304354749=
57991495425111612\
35905524928625924951263749644311651913227685661186418=
7351528116367034140077649\
228256600937975605450350463096018689689494317=
504186472028713476549889531616376\
2652186400027116150305859807714603778=
39679256037218151469434036491261624917999\
789690125252651753140
985992078803105585299781688200576431175253129002734686056\
6929360384529=
50797255495408609243046821111989281772950244181499940165613599069\
75299=
3108229555644132121597169167501944449522582579165541347370340485791216208\<=
br>810170273407800901109543642032049395922018609103137870253925946466865040=
950825\
8245461156346209848232190645912931385567395333425186094293812732=
37731039456346\
03386297734632604959062528648348703921589150829246157644=
9677325801713807841621\
978580747913527131242794907636091311463666958399=
217772206627407877191823447182\
9727786553975763283574400336903927956992=
07683940192423828998570474496739832672\
01947765211459021401879193470632=
2664952454443925279628796082488146236072268592\
846976676880487114045448=
401559927701979774401207562058551583086165782131087375\
5119142445102367=
34437321250625678331088571958029689058768955878732093815869906\
09654442=
5748818588441839626888167846642644935534200259360142428511293980509
753\
7694440139868001736874301730321633022154529906183804581707306470980=
36066945049\
98522062251458448767563033748701662132787970555358546101982=
0126404131837827513\
491519322013117934471257069495199166734762494535604=
267339796974779665699893326\
0260664906375516639226914695157457344610658=
58146408933969356951124432041596128\
27279491663722982394824858450159234=
7729992029532594860584427611186151786254334\
823503820350363849469150455=
890670118740902566788239032536444855114742741037803\
6550986302044924984=
17146997924096033291795233394232609566695291939284728181124\
37134967663=
1081443458321783163024319460537095544807147503134516825792732519141\
759=
413323311576933662237655040000000000000000000000000000000000000000000000000=
\
0000000000000000000000000000000000000000000000000000000000000000000000=
00000000\
00000000000000000000000000000000000000000000000000000000000000=
0000000000000000\
0000000000000000000000000000000000000000000000
00000000000000000000000000000000\
00000000000000000000000000000000000000=
00000000000000000000000


I have to go now, but thank you all for =
your patience.  This formula and the values
should not have to be c=
orrected this time.

All the best,
David

--0-251550593-1304294966=:84942--



From: David Smith <djs314djs314@yahoo.com>
Date: Sun, 1 May 2011 19:30:53 -0700 (PDT)
Subject: Re: [MC4D] Klein's Quartic n-order formula and specific values








=C2=A0



=20=20


=20=20=20=20
=20=20=20=20=20=20
=20=20=20=20=20=20
Hi everyone,

Okay, I can now claim I fully understand the Klein's Quartic puzzle.=C2=A0 =
I apologize
for my earlier mistakes, and guarantee that the following formula and calcu=
lations
are accurate, excepting a typo.=C2=A0 I had failed to realize that Klein's =
Quartic is very
much like a 3-dimensional puzzle.=C2=A0 It has 'corners' and 'edges'.=C2=A0=
I did not study the
order-2 puzzle enough to realize that the 'corners' existed.=C2=A0 This is =
of course
obvious to all of you, since you have played with the puzzle more than I ha=
ve, and
many of you have even solved it.=C2=A0 The way the order-3 and higher puzzl=
e appear
immediately made obvious to me the nature of the 'corners' and 'edges'.=C2=
=A0 The
order-2 puzzle's appearance somewhat disguises its true nature, but that is=
no
excuse for my lack of investigation.

Here is the
formula for the number of permutations of the order-n Klein's Quartic
puzzle for odd n >=3D 3:





And here are individual values:

Order-2:

((55!)/2)*(3^53) =3D

123048752649957426169448861190331245749810829958743981704307326752100311\
522831667036160000000000000

Order-3:

(84!)*(56!)*(2^81)*(3^55) =3D

993919860351336970116941239153296103939497528293494685545497421465353823\
867122389570825760861036003717296401423048183458017237535043916628667848529=
348\
418500952983796478875293990282425667716662641223463686191606729487155200000=
000\
000000000000000000000000

Order-5:

(((168!)/((7!)^24))^2)*((168!)/2)*(84!)*(56!)*(2^81)*(3^55)
=3D

153813597602981865296923122878293408297707605840877433021601977114757488\
141188456378084938578404431213530399155802876124924501302998795803488994282=
318\
638973392226465765639269482642062480505130632851949992339629771331733910361=
394\
187875621200775783795330499539579105683370865469180184421558825857092857952=
139\
675652574064438504019365838704215066676118844823272520482463359811616584351=
650\
343294082974911191373304636015923982296607487243206304661823823179829496028=
840\
929128708900416627301861209241866736735119716387769997220107174269840158765=
913\
914510643210391000467377130910048027024945418450246083224037558282200707856=
137\
417672596140743196018794757987904441064746702209870189273782494772422622248=
229\
582206349072075076140907820840336618805947675411840837091706167131159031529=
916\
608909239023245119282172250315201115126049303085088200765125878010439058553=
209\
19488480077017358056554496000000000000000000000000
0000000000000000000000000000\
0000000000000000000000000000000000000000000000000000

Order-7:

(((168!)/((7!)^24))^6)*(((168!)/2)^2)*(84!)*(56!)*(2^81)*(3^55) =3D
=20

291652081227680814998213157804602335018085364039052280961816073114448730\
392894920023356437101146476284320603185810624757021755947665298890077989550=
883\
885311343466371999384057110413096574296849941236724269152103544645442486218=
695\
076600252096301812306832372577853207239998648950663571317985724006765904739=
680\
977105203817627232866878630402546853197300928140997055230021724053848990847=
216\
551957669431605329014717773898577089777619452520575734359293791029055047460=
576\
974564133092157579429829165447260989008971819817294651225865111987625423736=
784\
401171396455752024547584118386324881866238870218023662586959246641024767000=
358\
285518158311830911257029022642565363726603369418008829918092525995884379417=
405\
670910582899355979014578770936069119778606566775357202755594741592869227544=
047\
178229603580102844151585056303034806574605033952482681047551433264000114310=
716\
924001031918988612472952919237611441342665686598541
080344137343237544467454063\
349911222948500021047225513530557140871503559750837011458585002994770311031=
339\
781109128549029418138339924851610383224600629274310641062928263251950995560=
148\
937777274448616949465037469723128521933851772717618059366344850213381098526=
318\
319060550935249803574921929117917238519906930661961947706955966404872626429=
782\
132804861306667749534406026729935529881538323975563828330896791908785507675=
059\
773775794278700459656226381696124358922920273604368715840236421715579658507=
382\
365453517371351486086458623588131038391141321230610350870074067038457996241=
667\
720900561881878754648098316208849769503988442111398599945245303613729339183=
849\
167767601179210373127434178110063536344436887129279839469942450678228480290=
417\
935016209791318697331932192833279890869534215446438165799082847867421585376=
427\
125195520939078272991998436471331374584526239824202581947561332987759818220=
178\
4496259907394520261388
43213446477424511640590394442357826468773763181183467355\
539723948286105864976159089388473584954233058380887860737526988800000000000=
000\
000000000000000000000000000000000000000000000000000000000000000000000000000=
000\
000000000000000000000000000000000000000000000000000000000000000000000000000=
000\
00000000000000000000000000000000000000

Order-9:

(((168!)/((7!)^24))^12)*(((168!)/2)^3)*(84!)*(56!)*(2^81)*(3^55)
=3D

677582869828578066590514174426477151611690434506950459808287370437586411\
589712092377745815252610930902313658103766880255574538687753843375713294897=
632\
229604198139428110010773285646728126823387215212255969547186353882369700028=
967\
357420423630846160741822346537314280930026018987102915805246070856404614500=
425\
080097246212719291213975788892366066186385076310698582572122488831893325275=
338\
022200576603148637405090301054731728037990100651517592252935578389346975932=
765\
393031787471498515172826138520961440317355862707925533637995776970273715363=
414\
763966001363948789113428479431707402741209524218143534272382729149389093065=
277\
738776510908735266022840504909278164416307388596961383581457574835426043053=
858\
567130861051011559090208050156911206052149351505039750810224644148029574393=
183\
643117846586265363506172950816736400283897320748166942865332312858367774914=
685\
40096969369772032552421988068250720674237146034126
7136120269318225507969219890\
147215159865406150475078930492604815977957514374625438016561144829700944961=
469\
903841769287163517353646630553103279651164614135572024499483587396729352167=
073\
314647757488867863523612624916957271732409936033435795555768007982838821934=
759\
529312703400982479192577976332369625248478885795751249576247219706313028822=
865\
251158674424998015916461969114262493126387933357244586468480986348677384231=
084\
051947741834028586096264224079871413353793237542286834143546953887276596438=
016\
154110548089321896312300275346346285309747824057951560619284748297160126997=
018\
555164003753708299148367101333402034431705143711815830435474957991495425111=
612\
359055249286259249512637496443116519132276856611864187351528116367034140077=
649\
228256600937975605450350463096018689689494317504186472028713476549889531616=
376\
265218640002711615030585980771460377839679256037218151469434036491261624917=
999\
789690125252651753140
985992078803105585299781688200576431175253129002734686056\
692936038452950797255495408609243046821111989281772950244181499940165613599=
069\
752993108229555644132121597169167501944449522582579165541347370340485791216=
208\
810170273407800901109543642032049395922018609103137870253925946466865040950=
825\
824546115634620984823219064591293138556739533342518609429381273237731039456=
346\
033862977346326049590625286483487039215891508292461576449677325801713807841=
621\
978580747913527131242794907636091311463666958399217772206627407877191823447=
182\
972778655397576328357440033690392795699207683940192423828998570474496739832=
672\
019477652114590214018791934706322664952454443925279628796082488146236072268=
592\
846976676880487114045448401559927701979774401207562058551583086165782131087=
375\
511914244510236734437321250625678331088571958029689058768955878732093815869=
906\
096544425748818588441839626888167846642644935534200259360142428511293980509
753\
769444013986800173687430173032163302215452990618380458170730647098036066945=
049\
985220622514584487675630337487016621327879705553585461019820126404131837827=
513\
491519322013117934471257069495199166734762494535604267339796974779665699893=
326\
026066490637551663922691469515745734461065858146408933969356951124432041596=
128\
272794916637229823948248584501592347729992029532594860584427611186151786254=
334\
823503820350363849469150455890670118740902566788239032536444855114742741037=
803\
655098630204492498417146997924096033291795233394232609566695291939284728181=
124\
371349676631081443458321783163024319460537095544807147503134516825792732519=
141\
759413323311576933662237655040000000000000000000000000000000000000000000000=
000\
000000000000000000000000000000000000000000000000000000000000000000000000000=
000\
000000000000000000000000000000000000000000000000000000000000000000000000000=
000\
0000000000000000000000000000000000000000000000
00000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000000000000


I have to go now, but thank you all for your patience.=C2=A0 This formula a=
nd the values
should not have to be corrected this time.

All the best,
David


=20=20=20=20
=20=20=20=20=20

=20=20=20=20
=20=20=20=20


=20



=20=20




--0-693570556-1304303453=:42929
Content-Type: text/html; charset=utf-8
Content-Transfer-Encoding: quoted-printable

top" style=3D"font: inherit;">Well, I made a typo.  I am having the wo=
rst time with this!!  It was just a typo,
but it affected the order=
-2 value (as the typo was in the program I used to calculate
the value),=
which is fortunately the least important value and not handled by the
f=
ormula.  Believe it or not, I was actually foolish enough to rush the =
email, as I
had to be somewhere.  Sometimes I make poor decisions, =
such as when I left
the group.  There is a reason for this, but I'd=
rather not elaborate.  Suffice it to say
I can get very excited an=
d will post without thinking things through.  I'm quite
frustrated =
about this whole mess, and am beginning to once again doubt my place
her=
e.  I appreciate that I somehow seem to be accepted.  Thank you a=
ll for your
infinite patience with my less than acceptable
behavior.

Order-2:

((55!)/2)*(3^54) =3D

3691462579498=
72278508346583570993737249432489876231945112921980256300934\
56849500110=
8480000000000000

All the best,
David

--- On Sun, 5/1/11=
, David Smith <djs314djs314@yahoo.com> wrote:
te style=3D"border-left: 2px solid rgb(16, 16, 255); margin-left: 5px; padd=
ing-left: 5px;">
From: David Smith <djs314djs314@yahoo.com>
Sub=
ject: [MC4D] Klein's Quartic n-order formula and specific values
To: 4D_=
Cubing@yahoogroups.com
Date: Sunday, May 1, 2011, 8:09 PM

=3D"yiv1275103581">





 




=20=20=20=20=20=20
=20=20=20=20=20=20

>ble>




=20=20=20=20=20



=20


Hi everyone,

Okay, I ca=
n now claim I fully understand the Klein's Quartic puzzle.  I apologiz=
e
for my earlier mistakes, and guarantee that the following formula and =
calculations
are accurate, excepting a typo.  I had failed to reali=
ze that Klein's Quartic is very
much like a 3-dimensional puzzle.  =
It has 'corners' and 'edges'.  I did not study the
order-2 puzzle e=
nough to realize that the 'corners' existed.  This is of course
obv=
ious to all of you, since you have played with the puzzle more than I have,=
and
many of you have even solved it.  The way the order-3 and high=
er puzzle appear
immediately made obvious to me the nature of the 'corne=
rs' and 'edges'.  The
order-2 puzzle's appearance somewhat disguise=
s its true nature, but that is no
excuse for my lack of
investigation.

Here is the
formula for the number of permutations of the order-n Klein's Quartic
p=
uzzle for odd n >=3D 3:



3D"http://i1029.photobuc=<br/ket.com/albums/y360/djs314/KQ.gif" src=3D"http://i1029.photobucket.com/albu=
ms/y360/djs314/KQ.gif">

And here are individual values:

Order=
-2:

((55!)/2)*(3^53) =3D

123048752649957426169448861190331245=
749810829958743981704307326752100311\
522831667036160000000000000
>Order-3:

(84!)*(56!)*(2^81)*(3^55) =3D

993919860351336970116=
941239153296103939497528293494685545497421465353823\
8671223895708257608=
61036003717296401423048183458017237535043916628667848529348\
41850095298=
3796478875293990282425667716662641223463686191606729487155200000000\
000=
000000000000000000000

Order-5:

(((168!)/((7!)^24))^2)*((168!)=
/2)*(84!)*(56!)*(2^81)*(3^55)
=3D

153813597602981865296923122878293408297707605840877433021601977=
114757488\
1411884563780849385784044312135303991558028761249245013029987=
95803488994282318\
63897339222646576563926948264206248050513063285194999=
2339629771331733910361394\
187875621200775783795330499539579105683370865=
469180184421558825857092857952139\
6756525740644385040193658387042150666=
76118844823272520482463359811616584351650\
34329408297491119137330463601=
5923982296607487243206304661823823179829496028840\
929128708900416627301=
861209241866736735119716387769997220107174269840158765913\
9145106432103=
91000467377130910048027024945418450246083224037558282200707856137\
41767=
2596140743196018794757987904441064746702209870189273782494772422622248229\<=
br>582206349072075076140907820840336618805947675411840837091706167131159031=
529916\
6089092390232451192821722503152011151260493030850882007651258780=
10439058553209\
19488480077017358056554496000000000000000000000000
0000000000000000000000000000\
000000000000000000000000000000000000000000=
0000000000

Order-7:

(((168!)/((7!)^24))^6)*(((168!)/2)^2)*(84=
!)*(56!)*(2^81)*(3^55) =3D


291652081227680814998213157804602335018085364039052280961816073114=
448730\
3928949200233564371011464762843206031858106247570217559476652988=
90077989550883\
88531134346637199938405711041309657429684994123672426915=
2103544645442486218695\
076600252096301812306832372577853207239998648950=
663571317985724006765904739680\
9771052038176272328668786304025468531973=
00928140997055230021724053848990847216\
55195766943160532901471777389857=
7089777619452520575734359293791029055047460576\
974564133092157579429829=
165447260989008971819817294651225865111987625423736784\
4011713964557520=
24547584118386324881866238870218023662586959246641024767000358\
28551815=
8311830911257029022642565363726603369418008829918092525995884379417405\
=
670910582899355979014578770936069119778606566775357202755594741592869227544=
047\
1782296035801028441515850563030348065746050339524826810475514332640=
00114310716\
924001031918988612472952919237611441342665686598541
080344137343237544467454063\
3499112229485000210472255135305571408715035=
59750837011458585002994770311031339\
78110912854902941813833992485161038=
3224600629274310641062928263251950995560148\
937777274448616949465037469=
723128521933851772717618059366344850213381098526318\
3190605509352498035=
74921929117917238519906930661961947706955966404872626429782\
13280486130=
6667749534406026729935529881538323975563828330896791908785507675059\
773=
775794278700459656226381696124358922920273604368715840236421715579658507382=
\
3654535173713514860864586235881310383911413212306103508700740670384579=
96241667\
72090056188187875464809831620884976950398844211139859994524530=
3613729339183849\
167767601179210373127434178110063536344436887129279839=
469942450678228480290417\
9350162097913186973319321928332798908695342154=
46438165799082847867421585376427\
12519552093907827299199843647133137458=
4526239824202581947561332987759818220178\
4496259907394520261388
43213446477424511640590394442357826468773763181183467355\
53972394828610=
5864976159089388473584954233058380887860737526988800000000000000\
000000=
000000000000000000000000000000000000000000000000000000000000000000000000\r>0000000000000000000000000000000000000000000000000000000000000000000000000=
00000\
00000000000000000000000000000000000000

Order-9:

(((=
168!)/((7!)^24))^12)*(((168!)/2)^3)*(84!)*(56!)*(2^81)*(3^55)
=3D

677582869828578066590514174426477151611690434506950459808287370=
437586411\
5897120923777458152526109309023136581037668802555745386877538=
43375713294897632\
22960419813942811001077328564672812682338721521225596=
9547186353882369700028967\
357420423630846160741822346537314280930026018=
987102915805246070856404614500425\
0800972462127192912139757888923660661=
86385076310698582572122488831893325275338\
02220057660314863740509030105=
4731728037990100651517592252935578389346975932765\
393031787471498515172=
826138520961440317355862707925533637995776970273715363414\
7639660013639=
48789113428479431707402741209524218143534272382729149389093065277\
73877=
6510908735266022840504909278164416307388596961383581457574835426043053858\<=
br>567130861051011559090208050156911206052149351505039750810224644148029574=
393183\
6431178465862653635061729508167364002838973207481669428653323128=
58367774914685\
40096969369772032552421988068250720674237146034126
7136120269318225507969219890\
147215159865406150475078930492604815977957=
514374625438016561144829700944961469\
9038417692871635173536466305531032=
79651164614135572024499483587396729352167073\
31464775748886786352361262=
4916957271732409936033435795555768007982838821934759\
529312703400982479=
192577976332369625248478885795751249576247219706313028822865\
2511586744=
24998015916461969114262493126387933357244586468480986348677384231084\
05=
194774183402858609626422407987141335379323754228683414354695388727659643801=
6\
154110548089321896312300275346346285309747824057951560619284748297160=
126997018\
5551640037537082991483671013334020344317051437118158304354749=
57991495425111612\
35905524928625924951263749644311651913227685661186418=
7351528116367034140077649\
228256600937975605450350463096018689689494317=
504186472028713476549889531616376\
2652186400027116150305859807714603778=
39679256037218151469434036491261624917999\
789690125252651753140
985992078803105585299781688200576431175253129002734686056\
6929360384529=
50797255495408609243046821111989281772950244181499940165613599069\
75299=
3108229555644132121597169167501944449522582579165541347370340485791216208\<=
br>810170273407800901109543642032049395922018609103137870253925946466865040=
950825\
8245461156346209848232190645912931385567395333425186094293812732=
37731039456346\
03386297734632604959062528648348703921589150829246157644=
9677325801713807841621\
978580747913527131242794907636091311463666958399=
217772206627407877191823447182\
9727786553975763283574400336903927956992=
07683940192423828998570474496739832672\
01947765211459021401879193470632=
2664952454443925279628796082488146236072268592\
846976676880487114045448=
401559927701979774401207562058551583086165782131087375\
5119142445102367=
34437321250625678331088571958029689058768955878732093815869906\
09654442=
5748818588441839626888167846642644935534200259360142428511293980509
753\
7694440139868001736874301730321633022154529906183804581707306470980=
36066945049\
98522062251458448767563033748701662132787970555358546101982=
0126404131837827513\
491519322013117934471257069495199166734762494535604=
267339796974779665699893326\
0260664906375516639226914695157457344610658=
58146408933969356951124432041596128\
27279491663722982394824858450159234=
7729992029532594860584427611186151786254334\
823503820350363849469150455=
890670118740902566788239032536444855114742741037803\
6550986302044924984=
17146997924096033291795233394232609566695291939284728181124\
37134967663=
1081443458321783163024319460537095544807147503134516825792732519141\
759=
413323311576933662237655040000000000000000000000000000000000000000000000000=
\
0000000000000000000000000000000000000000000000000000000000000000000000=
00000000\
00000000000000000000000000000000000000000000000000000000000000=
0000000000000000\
0000000000000000000000000000000000000000000000
00000000000000000000000000000000\
00000000000000000000000000000000000000=
00000000000000000000000


I have to go now, but thank you all for =
your patience.  This formula and the values
should not have to be c=
orrected this time.

All the best,
David

--0-693570556-1304303453=:42929--