7/29/2019 Puzzles and Paradoxes

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^uzzjcs aeh ^arahixcs= Xbc Fies`stcet Tijc io Agd`ku`ty Avcrs`ie

Dy

Orch _`ijcI__I O`eaef`aj Vystcgs

orch.v`ijcNkga`j.fig

Aeh

Hav`h Eawrifl`_ jjaeiva Se`vcrs`ty

_ jjaeiva Vfbiij io Dus`ecss2>> Jaefastcr Avceuc

_`jjaeiva, ^A 10>23 SVA?1>-310-8646

hav`h.eawrifl`Nv`jjaeiva.chu

mailto:fred.viole@gmail.commailto:fred.viole@gmail.commailto:david.nawrocki@villanova.edumailto:david.nawrocki@villanova.edumailto:david.nawrocki@villanova.edumailto:fred.viole@gmail.com

7/29/2019 Puzzles and Paradoxes

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^uzzjcs aeh ^arahixcs= Xbc Fies`stcet Tijc io Agd`ku`ty Avcrs`ie

Adstraft

B`stir`faj puzzjcs aeh parahixcs bavc dcce affcptch dy praft`t`iecrs aeh rcscarfbcrs as cv`hcefc io`rrat`ieaj`ty ie tbc part io tbc `eh`v`huaj. Biwcvcr, wbce fjass`oy`ek tbcsc aeigaj`cs `eti s`g`jarstrufturaj agd`ku`t`cs wc eitc tbc fies`stcefy io `eh`v`huajs' h`stastc oir agd`ku`ty. Asyggctr`fajpayiut strufturcs ioocr figpcesat`ie oir affcpt`ek sigc jcvcj io agd`ku`ty. [bce v`cwch tbriukb tbcjces io agd`ku`ty avcrs`ie, tbc `eh`v`huaj's aft`ies arc qu`tc rat`ieaj.

7/29/2019 Puzzles and Paradoxes

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Ab, pcipjc asl`ek qucst ies jist `e fieous`ie[cjj @ tcjj tbcg tbcrc's ei pridjcg, iejy sijut`ies...

Mibe Jceeie

@. @etrihuft`ie

[bce cxag`e`ek b`stir`faj parahixcs aeh hcv`at`ies orig cxpcftch ut`j`ty tbciry, wc o`eh tbcfiggieaj`ty io agd`ku`ty avcrs`ie. Agd`ku`ty avcrs`ie fae aeh hics cx`st autieigiusjy orig jiss-avcrs`ie. Hae`cj Cjjsdcrk (10?1) ioocrs a hco`e`t`ie,

Agd`ku`ty `s a sudmcft`vc var`adjc, dut `t sbiujh dc piss`djc ti `hcet`oy idmcft`vcjysigc s`tuat`ies j`lcjy ti prcscet b`kb agd`ku`ty, dy eit`ek s`tuat`ies wbcrc ava`jadjc`eoirgat`ie `s sfaety ir idv`iusjy uercj`adjc ir b`kbjy fieoj`ft`ek< ir wbcrc cxprcsschcxpcftat`ies io h`oocrcet `eh`v`huajs h`oocr w`jhjy< ir wbcrc cxprcssch fieo`hcefc `ecst`gatcs tcehs ti dc jiw.

[bce figpar`ek s`g`jar cxpcftch vajucs io h`oocrcet kagdjcs, wc fae `sijatc tbc agd`ku`ty avcrs`ie `e

rcaft`ie ti strufturaj uefcrta`et`cs v`a tbc prcg`ugs ti cxpcftch vajucs ioocrch. ^cipjc wbi purfbascjittcry t`flcts aeh `esuraefc pij`f`cs oir `estaefc, bavc fjcarjy figc ti sigc jiss affcptaefc jcvcj. @etbc rcjat`vcjy prcf`scjy hco`ech cxpcftch vajucs io jittcry aeh `esuraefc cehcavirs, tbc `eh`v`huajrcaj`zcs tbcy w`jj eit j lcjy w`e tbc jittcry eir fijjcft ie tbc `esuraefc pij`fy. Biwcvcr, tbcasyggctry io tbc payioo figpcesatcs oir tb`s jiss affcptaefc. Oir jittcr`cs, `t `s tbc ippirtue`ty tifatapujt tbcgscjvcs ti a krcatcr jcvcj io wcajtb aeh cfj`psc tbc`r ^crsieaj Fiesugpt`ie Vat`at`ie (^FV)pi`et `hcet`o`ch `e _`ijc aeh Eawrifl` (4>11a, 4>16). [`tb `esuraefc, `t `s tbc drcal-cvce coocft io azcri rcture aeajyzch `e _ ijc aeh Eawrifl` (4>16). Ditb iutfigcs arc pis`t`vc ut`j`ty kcecratirs oirtbc `eh`v`huaj.

Xb`s papcr a`gs ti fjass`oy b`stir`faj parahixcs aeh puzzjcs `eti fatckir`cs dasch ie var`ius

fcrta`ety `e hco`e`ek pridad`j`t`cs aeh cxpcftch vajucs. [bce pripcrjy fietcxtuaj`zch, tbc parahixcsaeh puzzjcs arc fies`stcet w`tb `eh`v`huaj prcocrcefcs ti avi`h agd`ku`ty.

@@. B@VXIT@FAJ ^ATAHIRCV

a. Vt. ^ctcrsdurk ^arahix

Xbc Vt. ^ctcrsdurk ^arahix `s a jittcry w`tb ae `eo`e`tc cxpcftch vajuc payioo. Xbc parahix

stcgs orig `eh`v`huajs' rcjuftaefc ti ioocr ae agiuet io giecy figgcesuratc w`tb ae `eo`e`tccxpcftch vajuc. Orig Fagcrcr (4>>3),

Fies`hcr tbc fjass`f Vt. ^ctcrsdurk kagdjc= A oa`r gcgiryjcss fi`e `s oj`ppch uet`j abcahs tures up. @o tbc bcah `s oj`ppch ie tbc e-tb tr`aj, tbc kagdjc pays $4e. Xbckagdjc tbcrcoirc pays $4 w`tb pridad`j`ty .3 (a bcah figcs up r`kbt away), $8 w`tbpridad`j`ty .43 (a ta`j, tbce tbc oatcouj bcah), $2 w`tb pridad`j`ty .143, aeh si ie. Xbccxpcftch vajuc io tbc kagdjc `s 1+1+1+.., rcpcatch `eo`e`tcjy, si `ts cxpcftch vajuc `s`eo`e`tc. Dut wbce aslch biw gufb tbcy wiujh pay ti duy sufb a kagdjc, pcipjc

7/29/2019 Puzzles and Paradoxes

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riut`ecjy rcpirt sugs ariueh $4>. (As Bafl`ek (102>) writc, "ocw io us wiujh paycvce $43 ti cetcr sufb a kagc."). Dcreiujj` fiefjuhch orig tb`s (dasch ie`etrispcft`ie) tbat tbc `eo`e`tc sug io cxpcftch ut`j`t`cs io giecy iutfigcs fiujh dco`e`tcaeh rcfief`jadjc w`tb a jiw duy`ek pr`fco ut`j`ty was suoo`f`cetjy fiefavc(`.c., u(x):>).

d.

Cjjsdcrk ^arahix

Aeitbcr cxagpjc io `eh`v`huajs cxb`d`t`ek agd`ku`ty avcrs`ie `s wbce uefcrta`ety `s figparch`e ajtcreat`vc fbi`fcs. Xbc parahix `s as oijjiws as prcscetch `e Fbiw aeh Var`e (4>>1)=

Xbcrc arc twi daks, jadcjch as Dak A aeh Dak D ie tbc tadjc. Dak A `s o`jjch w`tbcxaftjy 3> ycjjiw dajjs aeh 3> wb`tc dajjs. Dak D `s o`jjch w`tb 1>> dajjs tbat arc ycjjiwaeh wb`tc dut yiu hi eit leiw tbc`r rcjat`vc pripirt`ie.

Dak A Dak D3> ycjjiw dajjs 9 ycjjiw dajjs3> wb`tc dajjs 9 wb`tc dajjs1>> titaj dajjs 1>> titaj dajjs

O`rst, yiu arc ti kucss a fijir (wb`tc ir ycjjiw). Ecxt, w`tbiut jiil`ek, yiu arc ti hrawa dajj orig c`tbcr Dak A ir Dak D. @o yiu hraw tbc dajj gatfb`ek tbc fijir yiu kucssch,tbce yiu w`jj w`e $1>>< itbcrw`sc yiu w`e eitb`ek. [bat `s tbc sgajjcst agiuet iogiecy tbat yiu wiujh affcpt ratbcr tbae pjay tb`s kagc w`tb Dak A9 w`tb Dak D9

Xbc parahix ar`scs orig `eh`v`huajsprcocrcefc ti dct ie tbc iutfigc io Dak A ratbcr tbae Dak D hucti Dak Ds ueleiwe pripirt`ie io ycjjiw aeh wb`tc dajjs.

Aehrcw Ji, `e ae `etcrcst`ek rcfirhch jcfturc, `eoirgajjy rcfrcatcs tbc parahix w`tb b`s auh`cefc,eit`ek tbc rcjuftaefc ti pay oir a kagdjc huc ti tbc uefcrta`ety `emcftch `eti tbc scfieh sct iokagdjcs.1

f. Ajja`s' ^arahix

Ajja`s' (1036) parahix was tbc o`rst v`ijat`ie io cxpcftch ut`j`ty tbciry hifugcetch. Xbcparahix `evijvcs a part`f`paet affcpt`ek jcss tbae cxpcftch vajuc oir ajtcreat`vc fbi`fcs fieta`e`ek afcrta`e iutfigc. [bce ei fcrta`e ajtcreat`vc `s prcscetch, `eh`v`huajs gax`g`zc tbc cxpcftch vajuc.Xbc parahix `s as oijjiws=

CR^CT@GCEX 1 (Gist part`f`paets fbiisc A)V@XSAX@IE A V@XSAX@IE D$1g w`tb fcrta`ety 20% fbaefc io $1g< 1>% fbaefc io $3g< 1% fbaefc io eitb`ek

1.>>S($1 G) 5 >.20S($1 G) + >.>1S($> G) + >.1S($3 G)

1 bttp=//g`twirjh.g`t.chu/v`hci/;08

http://mitworld.mit.edu/video/794http://mitworld.mit.edu/video/794http://mitworld.mit.edu/video/794http://mitworld.mit.edu/video/794

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CR^CT@GCEX 4 (Gist part`f`paets fbiisc D)V@XSAX@IE A V@XSAX@IE D20% fbaefc io eitb`ek< 11%fbaefc io $1g 0>% fbaefc io eitb`ek< 1>% fbaefc io $3g

1.>>S($1 G) : >.20S($1 G) + >.>1S($> G) + >.1S($3 G)

h.

Cqu`ty ^rcg`ug ^uzzjc

Xbc Cqu`ty ^rcg`ug ^uzzjc `s a gae`ocstat`ie io Ajja`s' parahix. @t was `etrihufch `e 1023 dyGcbra aeh ^rcsfitt `e irhcr ti hcsfr`dc `evcstir dcbav`ir frcat`ek tbc prcg`ug io cqu`t`cs ti tbc`rfcrta`e ajtcreat`vc, SV Xrcasur`cs. [bce agd`ku`ty avcrs`ie `s pripcrjy figpcesatch oir e tbc ivcrajjjiss-avcrs`ie ficoo`f`cet , tbc cqu`ty prcg`ug puzzjc aeh firrcspieh`ek r`sl orcc ratc puzzjc asprcscetch `e Gcbra aeh ^rcsfitt (1023) `s sijvch. _`ijc aeh Eawrifl` (4>11d) ioocr suppirt`ekcv`hcefc ti O`sbcr Djafl's (1021) sijut`ie w`tb Ajja`s' parahix< aeh Gael`w aeh ]cjhcs' (1001)sijut`ie v`a Hcaj ir Ei Hcaj cv`hcefc ir`k`eajjy hcsfr`dch `e ^ist ct aj. (4>>2).

c.

@esuraefc aeh Jittcr`cs

@eh`v`huajs' dcbav`ir ti part`f`patc `e ditb `esuraefc aeh jittcr`cs bas fieoiuehch cfieig`stsoir hcfahcs. Xbc afts arc tbiukbt ti rcprcscet tbc ippis`tc spcftrug io r`sl avcrs`ie aeh s`gujtaeciuspart`f`pat`ie, `ecxpj`fadjc. Biwcvcr, wbce hcfiestruftch, `esuraefc aeh jittcr`cs arc ditb b`kbjyhco`ech asyggctr`faj iutfigc cvcets `eh`v`huajs pay a prcg`ug ti cxpcftch vajuc oir. Ditb iutfigcsarc ajsi pis`t`vc ut`j`ty kcecratirs. [`ee`ek tbc jittcry bas ae idv`ius pis`t`vc ut`j`ty fieeitat`ie,w`tb tbc piss`d`j`ty io cfj`ps`ek tbc `eh`v`huaj's ^FV ir ups`hc tarkct oir akkrckatc wcajtb. Xbc ihhs,biwcvcr rcgitc, arc fjcarjy hco`ech. Xb`s fcrta`ety io tbc ihhs suppirts tbc prcg`ug ti cxpcftchvajuc oir jittcr`cs.

@esuraefc ajsi rcprcscets a pis`t`vc ut`j`ty ti tbc `eh`v`huaj `e tbc oirg io a drcal-cvce coocft.[b`jc fajfujat`ek tbc pridad`j`t`cs io a eaturaj h`sastcr ir sigc itbcr cxikceius cvcet arc eit ash`sfrctc as tbc jittcry cxagpjc, tbc asyggctr`faj payiut strufturc oir sufb ae cvcet `s cquajjy hco`ech.Xb`s fcrta`ety `e payiuts suppirts tbc prcg`ug ti cxpcftch vajuc oir `esuraefc pij`f`cs.4

Xbc ahh`t`vc eaturc io fcrta`ety gay ajsi bcjp cxpja`e tbc pipujar`ty io hcr`vat`vcs, eitadjy tbcad`j`ty ti sat`soy tbc rijc io a jittcry aeh `esuraefc oir ae `evcstir dy hco`e`ek spcf`o`f payiuts tiuehcrjy`ek pis`t`ies.

o. Hcaj ir Ei Hcaj

^ist, vae hce Asscg, Dajtussce aeh Xbajcr (4>>2) aeajyzc tbc fietcstaet dcbav`ir `e tbc kagcsbiw Hcaj ir Ei Hcaj ti `jjustratc tbc patb-hcpcehcefc eaturc io r`sly hcf`s`ies uehcr uefcrta`ety.Xbc kagc `s as oijjiws=

4 FHV fietrafts wb`fb rcprcscet hcoaujt `esuraefc ie ae uehcrjy`ek hcdt idj`kat`ie `s tbc jarkcst spcfujat`vc garlct `e tbcwirjh. Xbc asyggctry io payiuts s rcspies`djc oir tb`s, rckarhjcss io tbc eckat`vc cvcet rcqu`rch oir paygcet.

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Xbc kagc sbiw wc usc `e tb`s stuhy, Hcaj ir Ei Hcaj, bas sufb hcs`radjc ocaturcs tbat`t ajgist appcars ti dc hcs`kech ti dc ae cfieig`fs cxpcr`gcet ratbcr tbae a X_ sbiw.Bcrc `s tbc csscefc io tbc kagc. A fietcstaet `s sbiwe 4? dr`cofascs wb`fb cafb fieta`ea b`hhce agiuet io giecy, raek`ek orig >.>1 ti 3,>>>,>>> (`e tbc Hutfb ch`t`ie). Xbcfietcstaet p`fls iec io tbc dr`cofascs aeh tbce iwes `ts ueleiwe fietcets. Ecxt, sbcscjcfts ? io tbc itbcr 43 dr`cofascs ti ipce. Cafb ipcech dr`cofasc rcvcajs iec io tbc 4?pr`zcs tbat arc

eit`e bcr iwe dr`cofasc. Xbc fietcstaet `s tbce prcscetch a dael ioocr

tbc ippirtue`ty ti wajl away w`tb a surc agiuet io giecy aeh aslch tbc s`gpjcqucst`ie= Hcaj ir Ei Hcaj9 @o sbc says Ei Hcaj, sbc bas ti ipce o`vc gircdr`cofascs, oijjiwch dy a ecw dael ioocr. Xbc kagc fiet`eucs `e tb`s oasb`ie uet`j tbcfietcstaet c`tbcr affcpts a dael ioocr, ir rcmcfts ajj ioocrs aeh rcfc`vcs tbc fietcets io bcriwe dr`cofasc. Xbc dael ioocrs hcpceh ie tbc vajuc io tbc ueipcech dr`cofascs< `o, oircxagpjc, tbc fietcstaet ipces b`kb-vajuc dr`cofascs, tbc dael ioocr oajjs.

Xb`s kagc sbiw sccgs wcjj-su`tch oir aeajyz`ek r`sly fbi`fc. Xbc stalcs arc vcry b`kbaeh w`hc-raek`ek= fietcstaets fae ki bigc as gujt`g`jj`iea`rcs ir praft`fajjy cgpty-baehch. Sej`lc itbcr kagc sbiws, Hcaj ir Ei Hcaj `evijvcs iejy s`gpjc stip-kihcf`s`ies (Hcaj ir Ei Hcaj) tbat rcqu`rc g`e`gaj sl`jj, leiwjchkc ir stratcky, aehtbc pridad`j`ty h`str`dut`ie `s s`gpjc aeh leiwe w`tb ecar-fcrta`ety (tbc dael ioocrs arcb`kbjy prch`ftadjc, as h`sfussch jatcr). O`eajjy, tbc kagc sbiw `evijvcs gujt`pjc kagcriuehs, aeh fiescqucetjy sccgs part`fujarjy `etcrcst`ek oir aeajyz`ek patb-hcpcehcefc,ir tbc rijc io carj`cr iutfigcs. ^ist ct aj. (4>>2).

_`ijc aeh Eawrifl` (4>11a) priv`hc ae cxpaehch aeajys`s io tbc Hcaj ir Ei Hcaj hataprcscetch dy ^ist ct aj. (4>>2), suppirt`ek Ajja`s' fcrta`ety cqu`vajcefc idscrvat`ies.

Xb`s h`sfiuet ti cxpcftch vajuc w`tb`e a sijcjy pis`t`vc ut`j`ty (ei jisscs `evijvch) io?2.42% ($31,4>0 fcrta`e fiesugpt`ie h`v`hch dy tbc cxpcftch vajuc $;3,>>>) Gael`waeh ]cjhcs (1001) `hcet`oy `s `e tbc rcajg io tbc ;?.6% dael ioocr affcptch `e tbc Hutfbvcrs`ie io Hcaj ir Ei Hcaj, aeh fjcarjy eit ioo dy ae irhcr io gake`tuhc. _`cwchh`oocrcetjy, tbc 8.2% fcrta`e fiesugpt`ie jcvcj Gael`w aeh ]cjhcs (1001) `hcet`oy($1,4>0 prcg`ug ti tbc fcrta`e g`e`gug h`v`hch dy tbc cxpcftch vajuc h`oocrcefc origtbc fcrta`e g`e`gug io $43,>>>) `s ajsi fies`stcet w`tb tbc pcrfcetakcs priv`hch dy ^istct aj. (4>>2)< wbcrc tbc`r patb hcpcehcet gihcj `gpj`ch fcrta`ety ficoo`f`cet oir tbcecutraj kriup io Hutfb fietcstaets `gpj`cs `eh`oocrcefc at 6.4% io cxpcftch vajuc oirjarkc pr`zcs.

@@@. ^iss`djc Cxpjaeat`ies oir tbc Vt. ^ctcrsdurk ^arahix

a. Fiefav`tyJikar`tbg`f St`j`ty Oueft`ie

Dcreiujj`'s sijut`ie= a fiefavc ut`j`ty oueft`ie aeh tbus a h`g`e`sb`ek gark`eaj ut`j`ty io giecy,`s eit a praft`faj sijut`ie ti tbc Vt. ^ctcrsdurk ^arahix.6 _`ijc aeh Eawrifl` (4>11a, 4>16) `hcet`oy a

6 Garliw`tz (1034) eitcs, Xi avi`h tbc oagius Vt. ^ctcrsdurk ^arahix, ir `ts kcecraj`zat`ie dy Fragcr, @ assugc tbat tbc ut`j` ty oueft`ie `s diuehchorig adivc. Oir aeajikius rcasies @ assugc `t ti dc diuehch orig dcjiw.

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twioijh fiefavc-fievcx ut`j`ty oir ka`es dasch ie Garliw`tz Vtifbast`f Hig`eaefc rcvcrsc-V ut`j`tyfurvcs (Jcvy aeh Jcvy (4>>4)) girc aptjy eit`fch w`tb pb`jaetbripy aeh itbcr `etaek`djc ut`j`t`csassif`atch w`tb giecy. Xbc h`oocrcefc `e ut`j`ty dctwcce $1 d`jj`ie aeh $1>> d`jj`ie `e wcajtb `s eit aoraft`ie io a ut`j as Dcreiujj` wiujh fietceh `e b`s Vt. ^ctcrsdurk ^arahix sijut`ie. Xbc r`sl avcrscjikar`tbg`f ut`j`ty oueft`ie bas sigc hcs`radjc quaj`t`cs sufb as prcvcet`ek ajj `e kagdjcs wbce $>tcrg`eaj wcajtb `s a rcgitc piss`d`j`ty. Biwcvcr, kcecraj`zch fiefav`ty as a gcaes io hco`e`ek jiss-avcrs`ie `s a djuet fiestruft. Xbc hcs`radjc hiwes`hc pritcft`ies arc cas`jy rcft`o`ch `e ajtcreat`vcut`j`ty oueft`ies w`tb tbc fiefavc jiss scft`ie bav`ek a stccpcr furvc tbae tbc firrcspieh`ek fievcx`tyio ka`es.

d. Gujt`pjc ^r`zc Jittcr`cs

Xbc h`sfrctc jittcry ihhs aeh `esuraefc pij`f`cs payioos `s jafl`ek `e tbc Vt. ^ctcrsdurk ^arahix,rcehcr`ek tbc `eo`e`tc cxpcftch vajuc gcae`ekjcss.

A stat`st`faj oaft `s tbat `o yiu pay $4>, tbc gist j`lcjy iutfigc `s tbat a bcah w`jj figcup qu`fljy aeh yiujj jisc giecy ect io tbc pr`fc yiu pa`h (w`tb pridad`j`ty.3+.43+.143+.>?437.06;3< yiu bavc iejy a ;% fbaefc io w`ee`ek, tbiukb io fiursc `oyiu w`e, yiu g`kbt w`e d`k). Fagcrcr (4>>3).

Spie ourtbcr `espcft`ie, tbc Vt. ^ctcrsdurk ^arahix appcars ti dc a gujt`pjc pr`zc jittcry wb`fbGarliw`tz eitcs (1034) aeh ioocrs a priio aka`est (4>1>)< wbcrcdy cxpcftch ut`j`ty gax`g`zcrs wiujheit w`jj`ekjy cekakc. Appceh`x A ioocrs ae cxpaehch cxpjaeat`ie rckarh`ek akcet assugpt`ies.

f. Ivcrwc`kbt`ek io Vgajj ^ridad`j`t`cs

Xbc o`cjh io dcbav`iraj o`eaefc bas figc `eti cx`stcefc `e irhcr ti cxpja`e tbcsc aeigaj`cs w`tbcxpcftch ut`j`ty tbciry. Fugujat`vc prispcft tbciry as prcscetch dy Labecgae aeh Xvcrsly (1004)sukkcsts tbc ivcrwc`kbt`ek io sgajj pridad`j`t`cs as a rcasie oir tbcsc idscrvch dcbav`irs...

a eiej`ecar traesoirgat`ie io tbc pridad`j`ty sfajc, wb`fb ivcrwc`kbts sgajjpridad`j`t`cs aeh uehcrwc`kbts gihcratc aeh b`kb pridad`j`t`cs. @e ae `gpirtaet jatcrhcvcjipgcet (Qu`kk`e, 1024< Vfbgc`hjcr, 1020< Paar`, 102;< [cygarl, 1021) bavcahvaefch a ecw rcprcscetat`ie, fajjch tbc rael-hcpcehcet ir tbc fugujat`vc oueft`ieaj,tbat traesoirgs fugujat vc ratbcr tbae `eh`v`huaj pridad`j`t cs. Xb`s art`fjc prcscets aecw vcrs`ie io prispcft tbciry tbat `efirpiratcs tbc fugujat`vc oueft`ieaj aeh cxtcehstbc tbciry ti uefcrta`e as wcjj as r`sly prispcfts w`tb aey eugdcr io iutfigcs.

Vbiujhe't tbc ivcrwc`kbt`ek io sgajj pridad`j`t`cs, sufb as jittcry part`f`pat`ie ir `esur`ek ae assct,jcah ti jarkcr sugs ioocrch ti pjay tbc jittcry ir Vt. ^ctcrsdurk ^arahix9

Xbcrc `s a >.>>>>>>314% fbaefc io w`ee`ek tbc avcrakc piwcrdajj maflpit io $181 g`jj`ie. 8,3 Xbcrc `sa >.>>>>>>;83% fbaefc io w`ee`ek $168 g`jj`ie (44;) pjay`ek tbc Vt. ^ctcrsdurk ^arahix. Xbcrc `s a

8 bttp=//www.piwcrdajj.fig/piwcrdajj/pdYpr`zcs.asp3 bttp=//www.piwcrdajj.fig/pdYfietaft.asp

http://www.powerball.com/powerball/pb_prizes.asphttp://www.powerball.com/powerball/pb_prizes.asphttp://www.powerball.com/pb_contact.asphttp://www.powerball.com/pb_contact.asphttp://www.powerball.com/pb_contact.asphttp://www.powerball.com/pb_contact.asphttp://www.powerball.com/powerball/pb_prizes.asp

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83% krcatcr fbaefc ti w`e a s`g`jar agiuet w`tb tbc Vt. ^ctcrsdurk ^arahix.

@e tbc cvajuat`ie io uefcrta`ety, tbcrc arc twi eaturaj diuehar`cs fcrta`ety aeh`gpiss`d`j`ty tbat firrcspieh ti tbc cehpi`ets io tbc fcrta`ety sfajc. H`g`e`sb`eksces`t`v`ty ceta`js tbat tbc `gpaft io a k`vce fbaekc `e pridad`j`ty h`g`e`sbcs w`tb `tsh`staefc orig tbc diuehary. Oir cxagpjc, ae `efrcasc io .1 `e tbc pridad`j`ty iow`ee`ek a k`vce pr`zc bas girc `gpaft wbce `t fbaekcs tbc pridad`j`ty io w`ee`ek orig.0 ti 1.> ir orig > ti .1, tbae wbce `t fbaekcs tbc pridad`j`ty io w`ee`ek orig .6 ti .8ir orig .? ti .;. H`g`e`sb`ek sces`t`v`ty, tbcrcoirc, k`vcs r`sc ti a wc`kbt`ek oueft`ietbat `s fiefavc ecar > aeh fievcx ecar 1. Oir uefcrta`e prispcfts, tb`s pr`ef`pjc y`cjhssudahh`t`v`ty oir vcry uej`lcjy cvcets aeh supcrahh`t`v`ty ecar fcrta`ety. Biwcvcr, tbc

oueft ie `s eit wcjj-dcbavch ecar tbc cehpiets, aeh vcry sgajj pridad j` t cs fae dcc tbcr krcatjy ivcrwckbtch ir eckjcftch ajtikctbcr. Labecgae aeh Xvcrsly (1004)?

Xbc aoircgcet`iech ihhs wiujh fcrta`ejy dc fies`hcrch `gpiss`djc, tbc jiwcrcehpi`et gcet`iech.Vi c`tbcr `eh`v`huajs wbi ioocr tbc $4> ti pjay tbc Vt. ^ctcrsdurk ^arahix arc ivcrwc`kbt`ek tbc 83%krcatcr ihhs ir arc eckjcft`ek tbcg ajtikctbcr (ioocr`ek s`gujtaecius `evcrsc cxpjaeat`ies sccgs ti dcae coocft`vc bchkc aka`est tbc ueleiwe). ^crbaps tbc `eh`v`huaj's agd`ku`ty avcrs`ie `s pjay`ek aprig`ecet part `e tb`s hcf`s`ie io uefcrta`ety. _`ijc aeh Eawrifl` (4>11d) `jjustratc tbat tbiscqucst`iech dy Ajja`s cxb`d`t a jiss-avcrs`ie 60 t`gcs tbc fcrta`e ajtcreat`vc. Xbcrc `s a fcrta`ety wbcepjay`ek tbc Vt. ^ctcrsdurk ^arahix tbat tbc maflpit, biwcvcr wc`kbtch, `s sijcjy tbc fietcstaet's.Gujt`pjc w`eecrs eckatcs tbat fcrta`ety w`tb tbc jittcry, tbus sigcwbat jcssce`ek tbc vaujtch fcrta`estatus as w`tb `ts pridad`j`t`cs. Xb`s prcg`ug oir s`g`jar stat`st`faj kagdjcs `s fies`stcet w`tb tbcagd`ku`ty avcrs`ie eitch w`tb Ajja`s' sudmcfts aeh figpjctcjy `efies`stcet w`tb tbc o`eh`eks ir`k`eajjyprcscetch `e prispcft tbciry.

@s tbc ivcrwc`kbt`ek io sgajj pridad`j`t`cs rcocrrch ti dy Labecgae aeh Xvcrsly rcjcvaet `e tb`s`estaefc9 Eit j`lcjy. ^ridjcg 2 prcscetch `e prispcft tbciry asls `o tbc rcspiehcet wiujh prcocr a>.1% fbaefc io ?>>> cjsc eitb`ek< ir a >.4% fbaefc io 6>>> cjsc eitb`ek.

@e ^ridjcg 2, tbcrc `s apiss`d`j`tyio w`ee`ek, ajtbiukb tbc pridad`j`t`cs io w`ee`ek arcg`e`sfujc (.>>4 aeh .>>1) `e ditb prispcfts. @e tb`s s`tuat`ie wbcrc w`ee`ek `s piss`djcdut eit pridadjc, gist pcipjc fbiisc tbc prispcft tbat ioocrs tbc jarkcr ka`e. V`g`jarrcsujts bavc dcce rcpirtch dy GafFr`ggie aeh Jarsie (10;0).Labecgae aeh Xvcrsly(10;0)

Xb`s hics eit dihc wcjj oir tbc adivc rcocrcefc io sudahh`t`v`ty oir vcry uej`lcjy cvcets (sufb as tbcjittcry ir V^^), aeh wiujh j`lcjy dc hcocehch w`tb cehpi`ets arc eckjcftch `e tb`s `estaefc. Xb`sv`ijat`ie io tbc v`ijat`ie io cxpcftch ut`j`ty tbciry arkucs oir tbc fies`hcrat`ie tbat, pcrbaps, prispcfttbciry `e eit a suoo`f`cet autieigius cxpjaeat`ie io hcf`s`ies uehcr uefcrta`ety.

h. Vug ivcr B`stir`cs

Iec kjar`ek qucst`ie rcga`es, wby iejy $4> oir ae `eo`e`tc cxpcftch vajuc, wby eit $1>>,$1>>>, ctf9 Vug ivcr b`stir`cs aeh tbc `ead`j ty ti v`cw tbc kagc as a wbijc gay prcvcet tbc

?Cgpbas`s iurs.

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`eh`v`huaj orig dc`ek adjc ti pripcrjy figputc ae cxpcftch vajuc oir tbc Vt. ^ctcrsdurk ^arahix aehitbcr gujt`pjc pr`zc jittcr cs. Xbc gyip`f v`cw io cafb iutfigc gay dc at oaujt, h`g`e`sb`ek orig tbcahh`t`vc eaturc io tbc iutfigcs.

A str`l`ek oaft adiut tbc jiss-avcrs`ie `e iur sav`eks cxpcr`gcet `s tbat cafb sudmcftgahc 40 pcr`ih`f fiesugpt`ie hcf`s`ies, ivcr ; j`oct`gcs. Vi tbc`r care`eks wcrchctcrg`ech dy 4>6 h`oocrcet p`cfcgcaj hcf`s`ies. Xbcy fiujh cas`jy aooirh ti adsirdpi`et jisscs ie gaey hcf`s`ies aeh st`jj ceh up abcah. Xbc oaft tbat tbcy arc rcjuftaet tihi si (muhk`ek orig O`kurc 1) gcaes tbcy arc eit iejy jiss-avcrsc (ie ae ard`traryut`j`ty sfajc)tbcy arc gyip`fajjy jiss-avcrsc ti cafb io adiut 4>> scparatc pcr`ihs.Fagcrcr (4>>3).

_`ijc aeh Eawrifl` (4>11a) eirgat`vcjy prcscet a sug ivcr b`stir`cs patb-hcpcehcet ut`j`ty gihcjwbcrcdy jifaj `evcstgcet ut`j`t`cs arc akkrckatch `eti a titaj wcajtb oueft`ie. Xb`s suppirts tbcgyip`f o`eh`eks io Fagcrcr as wcjj as tbc idscrvat`ie dy Garliw`tz (4>1>) @t hics eit v`ijatc tbccxpcftch-ut`j`ty bypitbcs`s ti assugc tbat ae akcet's ut`j`ty oueft`ie fbaekcs w`tb t`gc.

Ourtbcrgirc, `o tbc akcet `s gyip`f ti tbc cxtcet `jjustratch, tbce tbcy j`lcjy bavc pjafch asudmcft`vc o`e`tc pis`t`vc dcefbgarl ie tbc kagdjc at tbc t`gc io hcf`h`ek ti pjay tbc Vt. ^ctcrsdurkkagdjc. Xb`s sudmcft`vc cxpcftch vajuc gay oaftir `eti tbc`r ioocr`ek io a spcf`o`f agiuet aspistujatch `e tbc ecxt scft`ie.

c. Dcbav`iraj

@o wc wirl uehcr tbc assugpt`ie tbat ae `eh`v`huaj gust rcafb ae cqu`j`dr`ug `e irhcr ticekakc `e a wakcr, tbce tb`s ax`ig wiujh dc rcprcscetch dy cquat`ie 1,

() (1)wbcrc `s tbc gax`gug paygcet oir a kagdjcx, () `s tbc cxpcftch vajuc io tbc kagdjc, aeh`s tbc ivcrajj r`sl avcrs`ie ficoo`f`cet. Biwcvcr, r`sl `s eit sijcjy jiss avcrs ie aeh Cjjsdcrk (10?1)`hcet`o`cs agd`ku`ty as ae autieigius var`adjc pcr iur `etrihuftiry quitc. Cjjsdcrk (10?1) tbce ioocrsa hcf`s`ie rujc figpcesat`ek agd`ku`ty,

\. (1 )W() (4)wbcrc `s tbc hckrcc io fieo`hcefc, `s tbc cst`gatch pridad`j`ty vcftir, tbc pridad`j`ty vcftir`e > firrcspieh`ek ti g`exoir aft`iex aeh (R) `s tbc vcftir io payioos oir aft`iex. (>, >,R aeh arc ajj sudmcft`vc hata ti dc `eocrrch dy ae idscrvcr ir suppj`ch dy tbc `eh`v`huaj, hcpceh`ek iewbctbcr tbc fr`tcr`ie `s dc`ek usch hcsfr`pt`vcjy ir oir fievce`cet hcf`s`ie-gal`ek).

Ss`ek Cjjsdcrks hco`e`t`ie io a s`tuat`ie io b`kb agd`ku`ty,

Jct us `gak`ec a s`tuat`ie `e wb`fb si gaey io tbc pridad`j`ty muhkgcets ae `eh`v`huajfae dr`ek ti dcar upie a part`fujar pridjcg arc c`tbcr vakuc ir uesurc tbat b`s

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fieo`hcefc `e a part`fujar ass`kegcet io pridad`j`t`cs, as ippisch ti sigc itbcr io a sctio rcasieadjc h`str`dut`ies, `s vcry jiw. [c gay hco`ec tb`s as a s`tuat`ie io b`kbagd`ku`ty.

aeh oaftir`ek tbc sudmcft`vc eaturc io agd`ku`ty wc fae rchco`ec as

(+ ) (6)wbcrc `s tbc jiss-avcrs`ie paragctcr, `s tbc hckrcc io fieo`hcefc, aeh `s tbc var`aefc io tbccst`gatch pridad`j`ty vcftir.

[c fae eiw hcgiestratc tbc coocfts io agd`ku`ty ie tbc cqu`j`dr`ug ae `eh`v`huaj gust afb`cvc `eirhcr ti part`f`patc `e tbc kagdjc.

(

)

()Xbce tbc coocfts io tbc var`aefc io tbc cst`gatch pridad`j`ty vcftir ( ),

() ()

Ajtcreat`vcjy, wc fae scc tbc `sijatch coocfts io jiss-avcrs`ie ie tbc cqu`j`dr`ug cquat`ie=

() ()

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@_. Fiefjus`ies

[bce fatckir`z`ek parahixcs aeh puzzjcs `eti paygcet vcrsus cxpcftch vajuc ir affcptchpaygcet vcrsus cxpcftch vajuc, wc eitc tbc fies`stcefy io agd`ku`ty avcrs`ie. @eh`v`huajs arcfies`stcetjy rat`ieaj ie tbc`r asyggctr`faj payiut strufturcs as a figpcesat`ie oir var`iusuefcrta`et`cs. Cjjsdcrk (10?1) ajsi eitcs tbc rat`ieaj dcbav`ir io sufb aft`ies,

@t wiujh sccg `efaut`ius ti rujc pcrcgptir`jy tbat tbc pcipjc `e qucst`ie sbiujh eitajjiw tbc`r pcrfcpt`ie io agd`ku`ty, tbc`r uecasc w`tb tbc`r dcst cst`gatcs io pridad`j`ty,ti `eojucefc tbc`r hcf`s`ie= ir ti asscrt tbat tbc gaeecr `e wb`fb tbcy rcspieh ti `t `saka`est tbc`r jiek-rue `etcrcst aeh tbat tbcy wiujh dc `e sigc scesc dcttcr ioo `o tbcysbiujh ki aka`est tbc`r hccp ocjt prcocrcefcs. @o tbc`r rat`ieajc oir tbc`r hcf`s`ie dcbav`ir`s eit ue`qucjy figpcjj`ek (aeh rcfcet h`sfuss`ies w`tb X. Vfbcjj`ek bavc ra`schqucst`ies `e gy g`eh adiut `t), ec`tbcr, `t sccgs ti gc, arc tbc fiuetcrarkugcets.@ehcch, `t sccgs iut io tbc qucst`ie ti suggar`jy muhkc tbc`r dcbav`ir as `rrat`ieaj= @ ag`efjuhch agiek tbcg.

Ioocr`ek a H`sfiuet ti Cxpcftch _ajuc

Cjjsdcrk ^arahix (Jiw )Vt. ^ctcrsdurk ^arahix (Jiw , Sehco`ech )

Affcpt`ek a H`sfiuet ti Cxpcftch _ajucAjja`s' ^arahix (Fcrta`e Ajtcreat`vc)

Cqu`ty ^rcg`ug ^uzzjc (Fcrta`e Ajtcreat`vc)Hcaj ir Ei Hcaj (Fcrta`e Ajtcreat vc)

Ioocr`ek a ^rcg`ug ti Cxpcftch _ajuc

Jittcr`cs (B`kb )@esuraefc (B`kb )

Aecfhitajjy, wc arc w`tecss`ek tbcsc parahixcs dc`ek gae`ocstch `eti taek`djc furrcet idscrvat`ies.Xbc furrcet bius`ek fr`s`s bas cemi`ech uefcrta`ety `eti tbc bigciwecr's cxpcftch vajuc fajfujat`ies.Xbisc bigciwecrs furrcetjy uehcrwatcr arc pay`ek a prcg`ug ti tbc furrcet vajuc io tbc`r bigc.Xb`s `s `efies`stcet w`tb tbc eitch asyggctry `eh`v`huajs hcgaeh oir var`ius uefcrta`et`cs.

Xbc fiupjc spilc w`tb o`eaef`aj ahv`scrs aeh fies`hcrch a stratck`f hcoaujt.

"@t's eit adiut eit bav`ek giecy," says Vbcjdy, a purfbas`ek gaeakcr oir a sbicfigpaey. "@t's adiut eit tbriw`ek giecy away."

@e tbc ceh, tbcy iptch oir a sbirt sajc, ae akrccgcet w`tb a jcehcr ti scjj tbc biusc oirjcss tbae wbat `s iwch.;

; bttp=//www.usatihay.fig/giecy/cfieigy/bius`ek/4>1>->6-43-uehcrwatcr43YVXYE.btg

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"Vtratck`f hcoaujt fae dc a o`eaef`ajjy sipb`st`fatch tb`ek ti hi," sa`h Garl Ojcg`ek,fb`co cfieig`st oir FircJik`f, tbc o`eaef`aj aeajyt`fs figpaey. "Aeh `t galcs scesctbat girc o`eaef`ajjy savvy pcipjc hi `t. Xbcy gay trcat tbc`r girtkakcs j`lc tbcywiujh tbc`r `evcstgcet pirtoij`is -- `e a o`eaef`ajjy rutbjcss gaeecr."2

Affirh`ek ti b`stir`faj parahix aeh puzzjc aeajys`s, wbat `s rcqu`rch ti pay a prcg`ug ti cxpcftch

vajuc (`efrcasch vcrsus a stat`f ()) 9 A hcfrcasch v`a fcrta`ety `e tbc pridad`j`t`cs (ae`efrcasch jittcr`cs aeh `esuraefc) io a b`kbjy asyggctr`faj payiut - io wb`fb tbc furrcet bius`ekgarlct oa`js ti ioocr.

2 bttp=//giecy.fee.fig/4>11/>?/>;/rcajYcstatc/wajlYawayYgirtkakc/`ehcx.btg

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Tcocrcefcs

Djafl, O. (1021) ^r`vatc figgue`fat`ie w`tb T.Gcbra (4>>?).

Fagcrcr, F. (4>>3). Xbrcc Fbccrs ^byfbijik`faj, Xbcirct`faj, Cgp`r`faj oir Jiss Avcrs`ie.Miureaj io Garlct`ek Tcscarfb, 84 (4). 140166.

Fbiw, F.F., aeh Var`e, T.L. (4>>1). Figpar`t`vc @keiraefc aeh tbc Cjjsdcrk arahix. Xbc Miureajio T`sl aeh Sefcrta`ety, 44=4, 140-160.

Fragcr, K. (1;42). bttp=//ce.w`l`pch`a.irk/w`l`/Kadr`cjYFragcr

Cjjsdcrk, H. (10?1). T`sl, Agd`ku`ty, aeh tbc Vavakc Ax`igs. Quartcrjy Miureaj io Cfieig`fs, ;3(8), ?86-??0.

Bafl`ek, @. (102>). Vtraekc Cxpcftat`ies. ^b`jisipby io Vf`cefc8;, 3?4-3?;.

Labecgae, H., aeh Xvcrsly, A., (1004). Ahvaefcs `e ^rispcft Xbciry= Fugujat`vc Tcprcscetat`ie ioSefcrta`ety. Miureaj io T`sl aeh Sefcrta`ety, 3=40;646.

Jcvy G., aeh Jcvy B. (4>>4). ^rispcft Xbciry= Gufb Ahi Adiut Eitb`ek9 Gaeakcgcet Vf`cefc,v82(1>), 1668-1680.

GafFr`ggie, L. T., aeh Jarssie, V., (10;0). St`j`ty Xbciry= Ax`igs vcrsus ^arahixcs. @e G.Ajja`s aeh I. Bakce (chs.),Cxpcftch St`j`ty Bypitbcscs aeh tbc Ajja`s ^arahix, Distie, Tc`hcj.

Gael`w, E. Krckiry, aeh ]cjhcs, V.^. (1001). Xbc Fiesugpt`ie io Vtiflbijhcrs aehEiestiflbijhcrs.Miureaj io O`eaef`aj Cfieig`fs, 40, 0;-114.

Garliw`tz, B. (1034). Xbc St`j ty io [cajtb.Miureaj io ^ij`t`faj Cfieigy, ?>, 131-132.

Garliw`tz, B. (4>1>). ^irtoij`iXbciry as @ scc `t Vt`jj.Aeeuaj Tcv`cw io O`eaef`aj Cfieig`fs,_4, 1-81.

Gcbra, T. (4>>?). "Xbc Cqu`ty ^rcg`ug ^uzzjc= A Tcv`cw", Oiuehat`ies aeh Xrcehs `e O`eaefc, _ij.4, Ei. 1, 1-21.

Gcbra, T., aeh C.F. ^rcsfitt. (1023). Xbc Cqu`ty ^rcg`ug= A ^uzzjc. Miureaj io GiectaryCfieig`fs, vij. 13, ei. 4, (Garfb)=1831?1.

^ist, X., _ae hce Asscg, G., Dajtussce, K., aeh Xbajcr, T. (4>>2). Hcaj ir Ei Hcaj9 Hcf`s`ie Gal`ekuehcr T`sl `e a Jarkc-^ayioo Kagc Vbiw.Agcr`fae Cfieig`f Tcv`cw, Garfb 4>>2, (02=1), 62-;1.

Qu`kk`e, M. (1024). A Xbciry io Aet`f`patch St`j`ty. Miureaj io Cfieig`f Dcbav`ir aehIrkae`zat`ie, 6, 646-686.

Vfbgc`hjcr, H. (1020). Vudmcft`vc ^ridad`j`ty aeh Cxpcftch St`j`ty w`tbiut Ahh`t`v`ty.Cfieigctr`fa, 3;, 3;1-32;.

http://en.wikipedia.org/wiki/Gabriel_Cramerhttp://en.wikipedia.org/wiki/Gabriel_Cramer

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_`ijc, Orch, aeh Eawrifl` Hav`h, (4>11a). Xbc St`j`ty io [cajtb `e ae Sppcr aeh Jiwcr ^art`ajGigcet Oadr`f.Miureaj io @evcst`ek, Vuggcr 4>11, _ij. 4>, Ei. 4, 32-23.

_`ijc, Orch, aeh Eawrifl` Hav`h, (4>11d). Xbc Cqu`ty ^rcg`ug ^uzzjc= Hco`e`ek a Tidust T`s lAvcrs`ie Ficoo`f`cet. [irl`ek ^apcr.

_`ijc, Orch, aeh Eawrifl` Hav`h, (4>16). Ae Aeajys`s io Bctcrikcecius St`j`ty Dcefbgarls `e A]cri Tcture Cev`riegcet. Oirtbfig`ek,@etcreat`ieaj Tcv`cw io O`eaef`aj Aeajys`s.

[cygarl, M.A. (1021). Kcecraj`zch K`e` @ecquaj`ty @eh`fcs. Gatbcgat faj Vif`aj Vf`cefcs, 1, 8>0-86>.

Paar`, G.C. (102;). Xbc Huaj Xbciry io Fbi`fc Sehcr T sl. Cfieigctr`fa, 33, 03-113.

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Appceh`x AGujt`pjc ^r`zc Jittcr`cs

Orig Garliw`tz (1034),

Cvce eiw wc arc awarc io iec fjass io figgiejy idscrvch pbceigcea wb`fb sccgs`efies`stcet w`tb tbc bypitbcs`s `etrihufch `e tb`s papcr, as wcjj as tbc bypitbcs`s wb`fbtb`s iec was `etcehch ti supcrschc. Xbc cx`stcefc io gujt`pjc jittcry pr`zcs gayfietrah`ft tbc tbciry prcscetch. @o wc arc oirfch ti fiefchc tbat tbc `eh`v`huaj (jittcry-t`flct) duycr prcocrs, say, a oa`r gujt`pjc pr`zc jittcry ti say ajj itbcr oa`r jittcr`cs, tbcegy bypitbcs`s faeeit cxpja`e tb`s oaft. Eir fae aey itbcr bypitbcs`s fies`hcrch `e tb`spapcr cxpja`e a prcocrcefc oir h`oocrcet s`zch jittcry pr`zcs. Eir fae aey bypitbcs`swb`fb assugcs tbat pcipjc gax`g`zc cxpcftch ut`j`ty. Cvce eiw wc gust scclbypitbcscs wb`fb cxpja`e wbat iur furrcet bypitbcscs cxpja`e, avi`h tbc fietrah`ft`iesw`tb idscrvat`ie ti wb`fb tbcy arc sudmcft, aeh pcrbaps cxpja`e st`jj itbcr pbceigcea.

Orig Garliw`tz (4>1>), iec io twi priios aka`est gujt`pjc-pr`zc jittcr`cs.

Xbcircg 1

Vuppisc tbat ae akcet fae hcs`ke `ts iwe kagdjc dy fbiis`ek pridad`j`t`cs

, , , etbat tbc kagdjc w`jj bavc hijjar iutfigcs , , , e. B`s fbi`fc `ssudmcft ti twi fiestra`ets= (a) tbc gust dc pridad`j`t`cs, tbcrcoirc

1 (1?a)

7

> 1 , , (1?d)aeh (d) tbc kagc bas a k`vce cxpcftch jiss ir ka`e, `.c.,

(1;)7

@e part`fujar, `o >, tbce tbc jittcry `s rcqu`rch ti dc a oa`r kagc. Hco`ec a piss`djciutfigc io tbc jittcry as a w`tb 5 > @o tbc hcf`s`ie galcr gax`g`zcs cxpcftchut`j`ty

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(12)7 oir sigc k`vce ut`j`ty vcftir, , , etbce tbc akcet w`jj ecvcr prcocr a kagdjc w`tbtbrcc ir girc piss`djc iutfigcs ti ajj kagdjcs w`tb iejy iec ir twi piss`djc iutfigcs.

^riio

Xbc gax`g`zat`ie io tbc j`ecar oueft`ie, Cquat`ie 12, sudmcft ti tbc twi j`ecar

fiestra`ets, Cquat`ies 1?a aeh 1;, `e eieeckat`vc var`adjcs, Fiestra`ets 1?d. Xb`s `s a

4 e j`ecar prikragg`ek pridjcg. Haetz`ks (10?6) s`gpjcx ajkir`tbg `s dasch ie a

tbcircg fiefcre`ek aey j`ecar prikrag ti tbc coocft tbat `o ae ipt`gug sijut`iecx`sts tbce iec cx`sts w`tb > oir var`adjcs. (Iec ir girc io tbc gdas`svar`adjcs io tbc sijut`ie gay ajsi dc zcri aff`hcetajjy.) Oir iur 4 j`ecar prikrag,tbc oaft tbat tbc fiestra`et sct `s fjisch aeh diuehch `gpj`cs tbat ae ipt`gug sijut`ie

cx`sts< tbcrcoirc, ae ipt`gug sijut`ie cx`sts w`tb ajj dut iec ir twi io tbc cquaj tizcri. QCH

Garliw`tz's priio hics `ehcch eitc agd`ku`ty avcrs`ie. @o wc jiil at cquat`ie 1?a, b`s assugpt`ie `s

tbat tbc sug io tbc pridad`j`t`cs orig 1 , , cquaj 1. Xbat `s eit piss`djc uehcr tbc Vt. ^ctcrsdurkkagdjc. Xbc fugujat`vc pridad`j ty appriafbcs 1 dut ecvcr rcafbcs `t uehcr tbc `eo`e`tcjy `tcratchkagc. Xadjc 1 `jjustratcs tbc fugujat`vc pridad`j`ty oir tbc Vt. ^ctcrsdurk kagdjc w`tb pridad`j`ty >.3eoir e76>.

7/29/2019 Puzzles and Paradoxes

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1; | ^ a k c

Xadjc 1. Fugujat`vc pridad`j`t`cs oir tbc Vt. ^ctcrsdurk arahix

Xb`s fiestra`et tbat tbc sug io pridad`j`t`cs cquaj 1 ahhrcsscs `eh`v`huaj's h`stastc oir agd`ku`ty. XbcVt. ^ctcrsdurk ^arahix aeh itbcr j`lc kagdjcs wiujh oa`j Garliw`tz's akcet's rcqu`rcgcet aeh gayioocr sigc `es`kbt as ti wby ei sudstaet`aj agiuet `s ioocrch ti pjay sufb a kagc.

E 7 Fugujat`vc ridad`jty

1 >.3>>>>>>>>>>>>>>>>>>>

4 >.;3>>>>>>>>>>>>>>>>>>

6 >.2;3>>>>>>>>>>>>>>>>>

8 >.06;3>>>>>>>>>>>>>>>>

3 >.0?2;3>>>>>>>>>>>>>>>

? >.0286;3>>>>>>>>>>>>>>

; >.00412;3>>>>>>>>>>>>>2 >.00?>06;3>>>>>>>>>>>>

0 >.002>8?2;3>>>>>>>>>>>

1> >.000>4686;3>>>>>>>>>>

11 >.000311;12;3>>>>>>>>>

14 >.000;332306;3>>>>>>>>

16 >.0002;;040?2;3>>>>>>>

18 >.0000620?8286;3>>>>>>

13 >.0000?08248412;3>>>>>

1? >.000028;8141>062>>>>>

1; >.0000046;>?>38?0>>>>>

12 >.00000?1236>4;68>>>>>

10 >.000002>04?316?;>>>>>

4> >.000000>8?643?28>>>>>

41 >.0000003461?4284>>>>>

44 >.000000;?1321841>>>>>

46 >.00000022>;0>;1>>>>>>48 >.00000008>603633>>>>>

43 >.0000000;>10;?;2>>>>>

4? >.000000023>02260>>>>>

4; >.000000004380810>>>>>

42 >.00000000?4;8;1>>>>>>

40 >.00000000216;633>>>>>

6> >.000000000>?2?;;>>>>>