5
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Find the largest number using digits 1,2,3,4. Allowed operations are addition, subtraction, multiplication, division, exponents, brackets and decimal point. No other digits besides one of each of 1, 2, 3, and 4 are allowed.

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2
  • 4
    $\begingroup$ Concatenation allowed (e.g. $43^{21}$)? $\endgroup$
    – z100
    Dec 25, 2022 at 22:45
  • 5
    $\begingroup$ What kind of operation is "decimal point"? Division by 10? Concatenation with a decimal point in between?... $\endgroup$
    – GOTO 0
    Dec 26, 2022 at 18:44

4 Answers 4

15
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My answer is

$\large{.3^{-.2^{-.1^{-4}}}}$

which is approximately equal to

8.2868580201393671 × 10262086317397329836950303705682257163703149058949801218729493738441081598107528212748648366556650639675637494834651527009111141529869553576563913474335144955245264754635782473137984138347137952222638587964313205619452868837383718224634189273547166482909758564678494081744342572499686332340283615712581599021968814570900261444208208929835178424692667808106080151784111892423283390244145242205430052948581301571055543827374667243547881546403168164256188568714616762528088966293537298281238136576979323677929910645315459531637368751080803173205102267159826018600957959640410062625909165800472087302204963540888240564840302750650240671426732699878134259276410912317928891377617777223758778463382382989565963629022860722966911068954608859273028192166051552188580748979849856353251278350155772482582741438334143652808512554106991133426578977315310016893382680601367236364084069749470853443372254750811042507482354057432410918072658245044727765974298088591966668845706028688485051186137975830957602719621770168245055731886715877988832778550404172876713201400626562113916859758503503572420850070382232653128175849958311906700033232287139899003408154625886599216875941009235625509735126594114998258345932048819865655098527930205860927170520898038642117761923498823378971868312904039581254514465902510080892393070829681344739389363741514397040092807009060910726875881384453141825033744104371971395737958626485122501355695379269195842561891300017940444007250843146820361922480751344116392255182941972803187740676380024972345500168720227172189949951860691398299398246182652471366029534374277502048202948135938069081003223661885217857165791753402969255926344169840499123321398708367849182966199587193351349701644514591422555770850315692644641571136329357676395384520945095819221313693113684815506633593297676946168808642863322194517958045547364131780483725049561364697828657935781222305378834189377105661966377091045314454118991031173193087912526897924892725160951529275788563515672220734346956396124580755195580737457208360330881698151150791281872981262824377757789388341314980234145096156743757137201608490271854073822699451807255412734980722522303923139699762630368196630273552286404778994356367186061509595169377391573831369366134581576852453834993464418486217647419063673896681944867898325972456231118592156982127682613245205919782212425302910167840826741063795872684191799995638895308957306891473710500615210723412961659166781986309014404583165906231102152376490342694958370772428651943018869371546229938417838030831640879222189814068049552852589928907904591512743620040242808226590755531639440474663587641583800442621697554893546303902475959432324564759328308633824603584364339546342426161944332919172661328751080766229426908974593575585594168751064122788288092970253110658025684620997932513513219016668348297372724670250463149737295309424720786806708373124828283533811649602811031549933077505719956123312473154763822358203354801083985391411380608647780315375184717889256426123881207826390856345667111087689549622297407014462564174827119086347075989403507398000446125814552260075930162803405191806571447051409515497234495906016852549900173449556022389261758330267597204479791260583751907084710023809663361150126798564574057651142063002126991201159541621775747453943960587843516042014061406344966241606812930445217144682486086239368164217096006694745966605077509824323316797350950803804890505252234540221261101717910889591913189012339997477799796213987139095979757436145623547701243809793324172469809035704392769537488590959754389766183420932857901554122280925066746908599401937569816742285545111251592218253108111831727236692103860359621084009490388877562290554929571984362562733488627462231557530889105022573563269755733906859490935411758586806725925566105583558406294236768213439681427435576356791496700020362584018145112514239866314523702612468426799243565874008251179795267820537055010513171904177107071953525134844444495085869067400340417014939117577063323464811135473991684188389774661885570535165414453111905051057117762104024480560427545927774193983919296047749377009310095034445970086509219765655517182047636586898443029903574234907473202661011203519378243032135094498673114416728146832596419034385663347357286813384988802936465607518179263898564670830743534635557764445817221560926548683524064281301373916413742068383589904430799748515969511038139397752962623760591119565128466028587643663877607033547466074995287085501250361046627570403743516803756281175571424646751675075355430905311392812384858796465341180927564728230634864747358694779915237032894025096970602371119193145800691274104728239715968069577401427649972875251086212443501572694123978428610616687291266725924118089529191570893750730572913152468572735797013293401548799051887549891469843057650306642160835888005379626809823311046391271689708582581786934263545271733175792982830583146160684634782026649400081657564243160252661053564861475565803077758704474992854774551586087323115312658624598321291437604178283274755573428740068102992042469934397053699399101981821335068607223415848562736332181234163823637101583933379544760705182720254328728855929802480966343843038730682669036318565947032347132480266960760212224585892883430573813441300900393490420196944322150292816339562785581710774374405951533436102041940514920175391437138957198377507432485151084771662502213556842039872772869402189286108250827388812376641082424699924056799953688097484143563580661681665090199034432581305718286046370727352310428980483442863698295838444678162603956872707672090337142987462861572455017406776634167819204750450275014013314616427798257187784140636086610841675719307416844860024962012805792708220598973909966351658496116212950967811900195584203886443349118174509186009598866023812404566944755687630488526466906459286344989197324551550838713686837000494880954251127891396585348022329867252736515009125102581898635711727987075271957476336839910426448801079659595572080173867315695789193139602623783526986709363458967575393453038979209898602810549149748342992398209171634000593553337709809924387703661350497174059539051424465341091765834005754316328157119171193430939418798285763841841342703877019088833117493399675628961828118020154003179834205271296495195424061337690158159076912390817153729108777477044881898967960755005542454728625547788938414932385939675027867347404285193605493150361963663458098942306312305045463749732110178660476632100360595221063033351145026420887319278232407800857305895900151986620657940231577015953805160224495568959694773472003123082020164997346486815010640694126853615556814660978925971027243086781330014313175470395381973366600001901892614165953743752166113769545226887009638697761924898626887270622176946526115776965143570387623290385518435560413107341098952588081314382750314761660040803109467910493108892201917644912969279136292346432514092631270940261553763876594905792408796128088804513077170050614072545933280818798163894155516861094216658342247999221150471192185848196913397474821106037462358766664418137

The r.h.s. exponent has $6990$ places meaning the value itself has more than $10^{6989}$ places.

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5
  • 2
    $\begingroup$ Subtraction is allowed but I don't believe making negative numbers is equivalent to subtraction as there is no zeros permitted. $\endgroup$ Dec 26, 2022 at 4:38
  • 1
    $\begingroup$ @TakingNotes I'd argue that -x is subtraction from an implied 0 which would appear to be perfectly legal, but ultimately OP will have to clarify. $\endgroup$ Dec 26, 2022 at 4:52
  • $\begingroup$ Normally I'd say the same but the OPs insistence on no other digits makes me think otherwise. Hopefully OP does get back to us on this matter though. $\endgroup$ Dec 26, 2022 at 5:39
  • $\begingroup$ Yes this solution using -x is allowed as explained by Albert.Lang. However I totally agree with TakingNotes, it should have been claried in the question itself. $\endgroup$
    – ThomasL
    Dec 26, 2022 at 18:48
  • 4
    $\begingroup$ not sure whether the exponent is interesting enough to be written in full... i'd just write it as ~$10^{10^{6989.42}}$ (source) $\endgroup$
    – somebody
    Dec 27, 2022 at 7:57
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I believe the largest possible number is:

115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936

obtainable by:

$2^{4^{3+1}}$

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2
  • $\begingroup$ Why not 31 instead of 3+1? $\endgroup$
    – RHS
    Dec 27, 2022 at 13:10
  • 1
    $\begingroup$ The OP did not include concatenation as an allowed operation. $\endgroup$
    – richarjt
    Dec 27, 2022 at 16:52
4
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If concatenation allowed it is of the length of about 1.09795×10^19 decimal digits:

$2^{3^{41}}$

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4
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My answer is

$\infty$

obtained from

$\frac{2}{(4 - 3 - 1)}$

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10
  • 1
    $\begingroup$ But this is not a number :P $\endgroup$
    – ACB
    Dec 26, 2022 at 17:28
  • 2
    $\begingroup$ @ACB this is a festive answer. $\endgroup$ Dec 26, 2022 at 17:39
  • $\begingroup$ 2/0 is undefined, not infinity - if $a=\frac{2}{0}$, multiply both sides by 0: $0 \times a = 2$ - it cannot possibly be infinity because even for infinity, $\infty \times 0 = 0$ $\endgroup$
    – somebody
    Dec 27, 2022 at 7:49
  • $\begingroup$ @somebody multiplying $\frac{2}{0}$ by $0$ gives $0$, not $2$. Multiplying anything by $0$ gives $0$. $\endgroup$ Dec 27, 2022 at 8:11
  • $\begingroup$ @WeatherVane right. but here we're assuming it cancels out the $0$ on the denominator, instead of being multiplied with the numerator - which should be equivalent $\endgroup$
    – somebody
    Dec 27, 2022 at 8:14

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