Message #1636

From: David Smith <djs314djs314@yahoo.com>
Subject: Klein’s Quartic n-order formula and specific values
Date: Sun, 01 May 2011 17:09:26 -0700

Hi everyone,

Okay, I can now claim I fully understand the Klein’s Quartic puzzle. I apologize
for my earlier 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 realize 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 appear
immediately made obvious to me the nature of the ‘corners’ and ‘edges’. 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 >= 3:

And here are individual values:

Order-2:

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

123048752649957426169448861190331245749810829958743981704307326752100311 522831667036160000000000000

Order-3:

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

993919860351336970116941239153296103939497528293494685545497421465353823 867122389570825760861036003717296401423048183458017237535043916628667848529348 418500952983796478875293990282425667716662641223463686191606729487155200000000 000000000000000000000000

Order-5:

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

153813597602981865296923122878293408297707605840877433021601977114757488 141188456378084938578404431213530399155802876124924501302998795803488994282318 638973392226465765639269482642062480505130632851949992339629771331733910361394 187875621200775783795330499539579105683370865469180184421558825857092857952139 675652574064438504019365838704215066676118844823272520482463359811616584351650 343294082974911191373304636015923982296607487243206304661823823179829496028840 929128708900416627301861209241866736735119716387769997220107174269840158765913 914510643210391000467377130910048027024945418450246083224037558282200707856137 417672596140743196018794757987904441064746702209870189273782494772422622248229 582206349072075076140907820840336618805947675411840837091706167131159031529916 608909239023245119282172250315201115126049303085088200765125878010439058553209 194884800770173580565544960000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000

Order-7:

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

291652081227680814998213157804602335018085364039052280961816073114448730 392894920023356437101146476284320603185810624757021755947665298890077989550883 885311343466371999384057110413096574296849941236724269152103544645442486218695 076600252096301812306832372577853207239998648950663571317985724006765904739680 977105203817627232866878630402546853197300928140997055230021724053848990847216 551957669431605329014717773898577089777619452520575734359293791029055047460576 974564133092157579429829165447260989008971819817294651225865111987625423736784 401171396455752024547584118386324881866238870218023662586959246641024767000358 285518158311830911257029022642565363726603369418008829918092525995884379417405 670910582899355979014578770936069119778606566775357202755594741592869227544047 178229603580102844151585056303034806574605033952482681047551433264000114310716 924001031918988612472952919237611441342665686598541080344137343237544467454063 349911222948500021047225513530557140871503559750837011458585002994770311031339 781109128549029418138339924851610383224600629274310641062928263251950995560148 937777274448616949465037469723128521933851772717618059366344850213381098526318 319060550935249803574921929117917238519906930661961947706955966404872626429782 132804861306667749534406026729935529881538323975563828330896791908785507675059 773775794278700459656226381696124358922920273604368715840236421715579658507382 365453517371351486086458623588131038391141321230610350870074067038457996241667 720900561881878754648098316208849769503988442111398599945245303613729339183849 167767601179210373127434178110063536344436887129279839469942450678228480290417 935016209791318697331932192833279890869534215446438165799082847867421585376427 125195520939078272991998436471331374584526239824202581947561332987759818220178 449625990739452026138843213446477424511640590394442357826468773763181183467355 539723948286105864976159089388473584954233058380887860737526988800000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000

Order-9:

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

677582869828578066590514174426477151611690434506950459808287370437586411 589712092377745815252610930902313658103766880255574538687753843375713294897632 229604198139428110010773285646728126823387215212255969547186353882369700028967 357420423630846160741822346537314280930026018987102915805246070856404614500425 080097246212719291213975788892366066186385076310698582572122488831893325275338 022200576603148637405090301054731728037990100651517592252935578389346975932765 393031787471498515172826138520961440317355862707925533637995776970273715363414 763966001363948789113428479431707402741209524218143534272382729149389093065277 738776510908735266022840504909278164416307388596961383581457574835426043053858 567130861051011559090208050156911206052149351505039750810224644148029574393183 643117846586265363506172950816736400283897320748166942865332312858367774914685 400969693697720325524219880682507206742371460341267136120269318225507969219890 147215159865406150475078930492604815977957514374625438016561144829700944961469 903841769287163517353646630553103279651164614135572024499483587396729352167073 314647757488867863523612624916957271732409936033435795555768007982838821934759 529312703400982479192577976332369625248478885795751249576247219706313028822865 251158674424998015916461969114262493126387933357244586468480986348677384231084 051947741834028586096264224079871413353793237542286834143546953887276596438016 154110548089321896312300275346346285309747824057951560619284748297160126997018 555164003753708299148367101333402034431705143711815830435474957991495425111612 359055249286259249512637496443116519132276856611864187351528116367034140077649 228256600937975605450350463096018689689494317504186472028713476549889531616376 265218640002711615030585980771460377839679256037218151469434036491261624917999 789690125252651753140985992078803105585299781688200576431175253129002734686056 692936038452950797255495408609243046821111989281772950244181499940165613599069 752993108229555644132121597169167501944449522582579165541347370340485791216208 810170273407800901109543642032049395922018609103137870253925946466865040950825 824546115634620984823219064591293138556739533342518609429381273237731039456346 033862977346326049590625286483487039215891508292461576449677325801713807841621 978580747913527131242794907636091311463666958399217772206627407877191823447182 972778655397576328357440033690392795699207683940192423828998570474496739832672 019477652114590214018791934706322664952454443925279628796082488146236072268592 846976676880487114045448401559927701979774401207562058551583086165782131087375 511914244510236734437321250625678331088571958029689058768955878732093815869906 096544425748818588441839626888167846642644935534200259360142428511293980509753 769444013986800173687430173032163302215452990618380458170730647098036066945049 985220622514584487675630337487016621327879705553585461019820126404131837827513 491519322013117934471257069495199166734762494535604267339796974779665699893326 026066490637551663922691469515745734461065858146408933969356951124432041596128 272794916637229823948248584501592347729992029532594860584427611186151786254334 823503820350363849469150455890670118740902566788239032536444855114742741037803 655098630204492498417146997924096033291795233394232609566695291939284728181124 371349676631081443458321783163024319460537095544807147503134516825792732519141 759413323311576933662237655040000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000

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

All the best,
David