, so the next 5 terms are 31, 4, 37, 12, and The constant is π. Ptolemy used a 360-gon to approximate π as

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1 . The umber chose was, guessg as e tes dgt ad as e oes dgt each ears hal a pot.. The best way to do s s rough brute orce o a computer. The rst two are 6867 ad 68. The rd s The eve term s e prme whle e odd term s φ (), so e et terms are,, 7,, ad.. The costat s π. Ptolemy used a 6-go to appromate π as.6666 Vtruvus appromated π as.. Fboacc appromated π as Brahmagupta appromated π as.6... whch was e worst. Let a e restaurat I try becomes my avorte (meag t s better a each o e prevous oes) a oerwse. So e epected umber o avortes I have s E [ a ] E[ a ] Pr[ a ] ad Pr[ a ], so e epected umber o avortes I have s whch goes as l( ) + γ as. 6. Bo cubes eed to have ad o em; oerwse ad are t possble. Furermore, sce o oe de wll have each dgt to, bo cubes eed to have to be able to make each o to. Ths leaves 6 sdes let, but ere are 7 dgts remag ( rough ). However, sce 6 ad are ever bo used at e same tme, ad a 6 ca tur to a by rotatg t, t s sucet to just clude oe ad rotate t depedg o wheer you eed a 6 or a. There are may solutos at s pot, oe o whch s: rst de:,,,,, ; secod de:,,, 6, 7, 8. Ay o ese solutos wll oly have prme aces, e two s, e oe, e oe, ad e oe Ths ca be oud by eer wrtg a program or o e teret. The mllo prme s Ths problem ca be solved relatvely easly by brute orcg t w a computer or dg a table o φ () ole.. Trvally, ( ). The, k k k + k + d d ( ) ( ) d k + + k+ d d k+ ( ) k ( ) d. The rst ew umbers spoke are,, 6,,,, 8. lm Page o

2 . We should look at s data as just e umber o lps at laded o heads ad tals raer a dvdual trals. Obvously, ere were heads, sce each tral eded oe. The umber o tals s * 7 + * + * + *6 + * + * 8. So out o 8 lps, e 8 8 probablty o gettg heads s ( ) p ( p), whch s mamzed whe t s dervatve equals, so p ( p) 8 p ( ) ( ) ( ) p ( p ) 8p p.7 8. Tral ad error alog w plottg e data ad peelg away hgher order terms shows at e data ts ( ) + + +, so ( 7) 7. Ths s a smple Caesar Sht o spaces w a catch: e space s treated as a 7 letter. The two quotes decode to God does ot care about our maematcal dcultes. He tegrates emprcally. ad I do't beleve maematcs. whch are bo commoly attrbuted to Albert Este.. The Mu Alpha Theta Natoal Coveto shrts appromated π as.6, whch s.6 o rom e covetoal value o.6 6. The trck here s to pck a arbtrary pot (, ) o e curve y ad relect t about e le y +. Let s say at (, ) s gog to relect to, ). We ( y kow at bo pots all o e le y ( ) +, whch s e le perpedcular to e le o relecto at goes rough e pot we are relectg. Gve s ad e act at e dstace rom (, ) to y + equals hal e dstace rom (, ) to (, y) ad lots o messy algebra gves e equato o e curve: 6 76 y y 8y y + y 7y The dgts ca be mapped to letters usg e mappg o e telephoe buttos. Each o ose combatos o letters ca oly map to a ew deret Eglsh words, ad takg cotet to cosderato, t become clear at e setece s Fd e sum o ety ad e whch s. 8. Let s call e radus o e polygo (e dstace rom e ceter o e polyomal to a verte, whch s also e radus o e crcumscrbed crcle) r. The polygo ca be broke dow to detcal sosceles tragles w two π sdes o leg r w a agle o betwee em, so e area o each tragle π π r s( ) r s( ) s, so e area o e polygo s. The area o e π r s( ) s( π crcumscrbed crcle s π r, so e rato s ). πr π Page o

3 . Let y lm, l y lm l( ) lm l( ) lm l( ) whch s e Rema sum or l( ) l y e. Two umber are relatvely prme ad oly ey do t share ay prme actors, so or ay prme, e probablty at two radomly chose tegers share a actor o p s p, so e probablty at ey do t s. Sce we wat s to be true or all prmes, p we wat to multply over em, so e probablty o two umbers beg relatvely prme 6 s ( ). p ζ () π prmes ( +) ( ) + +. t, t, so t + t s. 6,,7,6 *7 * *, so all o e odd actors o 6,,7,6 just e actors o *7 ** at have a sum o 7 * * * * * log log log log. Graphg y log ad y log shows at ey tersect oly oce, ad specto shows at at tersecto comes at. The shortest perod leg s e smallest value o such at s a p p teger, so p ( ), so mod p. Sce p ad are relatvely prme, p mod p, by Fermat s Lttle Theorem, so e smallest perod ca be o larger a p. p 7 s a eample where e shortest perod equals p, so t s obtaable.. The cty ckames ad respectve real ames e correct order are as ollows: The Cty o Fve Flags (Pesacola), The Cty Beautul (Orlado), The Bg Guava (Tampa), Cty o Palms (Ft. Myers), Vece o Amerca (Ft. Lauderdale), The Magc Cty (Mam) 6. A al s a obscure ut o leg measure at s appromately.7 meters ad a metrc secod was e ut o tme suggested e metrc tme movemet at s equal to appromate.6 secods. 78m al.6 sec c * *. 8 sec.7m metrc sec 7. Frst, ote at e sce e crcle-square combato shares a as o symmetry w bo e square ad e tragle, e ceters o mass o e tragle, square ad combato wll all le o at le as well. Now, e square has area 6 ad Page o

4 e tragle has area. The dstace betwee e square s ceter o mass ad e shared edge o e square ad e tragle s. The dstace betwee e tragle s ceter o mass ad e shared edge s, so e dstace betwee e 6 + ceter o masses s. The rato o e dstace rom e ceter o mass o e system to e square s ceter ad e system s ceter to e tragle s ceter s :6, so e dstace rom e square s * The term descrbed s Eucld s Orchard ad e epsode s Brutus., 77, 66, 76, 8787, 887,, each have 7 prme actors, so ere are 8.. Fdg s w a computer s pretty straght orward (alough qute slow).. Let be e evet at I select. Let y be e evet at I select rom e rst dstrbuto ad y be e evet at I select e secod dstrbuto. Pr[ y] s e probablty o selectg a umber stadard devato rom e mea, whch s.86. Smlarly, Pr[ y] s e probablty o selectg a umber hal a stadard devato above e mea whch s.88. Pr[ ].* Pr[ y] +.7* Pr[ y]. 66 Pr[ y].7* Pr[ y] ad by Bayes Rule, Pr[ y ] Pr[ ].66. The tersecto o e two crcles ca be reduced to two quarter crcles o radus cotaed a square o sde leg. Ths meas at e tersecto s e area whch s bo quarter crcles, so t s e sum o e areas o e quarter crcles mus π e area o e square, whch s. 8. C ( F ), so we wat to solve F + ( F ) F F 6. ad 7 F ( F ) F F 7. whch correspod to. C ad 7. C respectvely.. There are 6 sttches o a regulato baseball. P( heads > ves) P( ves ) * P( heads > ) + P( ves ) * P( heads > ) * P( ves ) * P( heads > ) + P( ves ) * P( heads > ). P( heads < ves) P( heads ) * P( ves > ) + P( heads ) * P( ves > ) * P ( heads ) * P( ves > ) + P( heads ) * P( ves > ).6 Page o

5 so ( + ) + ( + ) ( + ) l 7. ( ) + + ( + )( + ) ( +) 8. Sce e tragular umber s, we eed oe o or + to be a square ad e oer to be twce a square so er product s a square. There s o ( +) oer way or to be a square, sce ad + are relatvely prme. Lookg at a table o prmes shows at e smallest square tragular umber you ca costruct s 7.. By puttg restrctos o partcular letters: K must by or, Y must be eve, E O mod, e ollowg solutos are oud: , + 886, , , + 788, , , , , US Patet 76 regards cryptography, ad e seed questo s e 6-bt seed D87 EB87F CDDA A6A76 FD8B7. b s used to dcate at ollowg s ad s represet a bary umber, so. Cosder each colum to be a elemet o e sequece, each row, rom e bottom up, s a bary bt, where s are shaded ad s are ot. Oce seeg s, decodg e rst ew should gve t away. Oer clues would be at e sequece s strctly creasg ad at e rst umber s ad e rest are odd. Page o