Question 1 Use the appropriate procedure in ComboProject1.txt to find the number of ways a King can travel on a 3 by 50 chessboard, return to the starting square and visit each of the 150 squares exaxtly once. Answer 1 The number of ways can be caluated by using the generating function I used maple for the following procedures KingToursGF(3, 50, t) 2 / 4 3 2 \ 2 t \3 t + 4 t + 2 t - 2/ ----------------------------- 4 3 2 6 t + 8 t + 15 t + 4 t - 1 f:=% 2 / 4 3 2 \ 2 t \3 t + 4 t + 2 t - 2/ f := ----------------------------- 4 3 2 6 t + 8 t + 15 t + 4 t - 1 taylor(f, t=0, 151): coeff(%, t^150) 7545888799163887583998489994476586235043351535115695573088910425917583779127904018211697778376609418770213095232667916486 Question 2 Use the appropriate procedures in ComboProject2.txt to find the number of ways of walking from [0,0] to [3000,3000] using, as atomic steps {[1,0],[0,1],[1,1],[1,2],[2,1]}, all the while staying in the region x ≥ y Don't do it directly! Use Findrec followed by SeqFromRec. Answer 2 I used it and found the massive number below 95169091361304497661016734197678134770349822930308569276916914467565347871540683291808752282824707570651985872820372363152563131551499362536825304711559361666563103386330829999710178259606336618359856705840905896307823039694186915923739801088785552537207794661641719063082723104319582945468736368770654686751378287041101572685985713084393957598627298986861938927166053927103431921104126027579703091359302890708953724730599070838409654723308763002613165079306548111632722909218562889533100675786353470112016022196477554880490589499289089162582632715079250461763612854611196880347954440249628671174622179674903282050871247325146410790010719334253540172397663997792921830518354359522669329363082614972729965686428855219511297199827548476675004412199720715365099792501311123143368483881397876775712439421054798831717981977936622337062303624546576898187959341646575490702549296915200857634048143923818701102393144599106994024665053648913638391157955194662231996921003344586526221812096107389514125990119451880185515845934698037055104486896573666844857573880742852639391722910231692794433520126339340352663869698430014805776170895308465157999383126154136534892638175457579256315002808693778321645459807540062485196275478961352224073937381392596094278147716000344452101216213718218846995422403246683326563202994807095282839244324106363086847480478463730345215985413064515655006280716371594297868314185041474747026816679035293127055797403755491278529530018110422188400807126145601136460544996907253108026710738133221241693706980842171873604900269856459541793234702167937163679886476596787643003424261664266808615823066346661804622908642236679528725542671747404314012249795021732170691008575336656050986306648593100164229972842611760785390415331373519508784472911981975537222743422312589751576257948909815276363791495173342977855094118355700016556226919615428250897733703209967582718607928464069524222159738340806195626241390924709235357374015016254839959209084054394569432471432576684522506998779788376200443126782132255047582891261666245361634285199523372050333572627938881441014457739772722775254541205569963083330946306737155223437987340685548656285567503838100825537658312256841838804244651258118811651131924827871017999511708744102921789247462817980894438730298698541980861879749020341945718568693083738317799328488585909649259726166330681988951268751122991503360075439622584181083778123619416313964846558075294830227299727374990124401764015434769152111199549890678059731046462531157554362228562028418125254044367001279506736510680059995930940034 Question 3 Use the appropriate procedures in ComboProject3.txt to find the number of ways of walking from [0,0,0] to [3000,3000,3000] using, as atomic steps {[1,0,0],[0,1,0],[0,0,1],[1,1,1]}, all the while staying in the region x ≥ y ≥ z Answer 3 The answer is 1055961597172638192205714533827955182069330710185887123715638338751912356623868934358954829265893265497184019516455497946501358614406592180916562995571435822638011864699948413518735379729837425167828197932773091617214603560704076763403058008870590210530506834889187670236069663000563403909291518139471141193933668569097230791592378327765453206880338367141465884774689273332974613327790670133864770839786907616252716172612810392322684357026318977543761631185892041752872792707472523339315464878092600122776510487117547611700101814824458512731561188677556925181977017823995965449343853533549328389007002003812727585993220294254740294866113923151479672429490481644088528411109634826771309376893827003349243068183476873533935594331988585675337762038394870883132602159173883922347999944078821728163370937052619155590568391069063317592509955598788483466418234408407388202299036233322379192615075021698393212868787733569268229184628289565589549741584574327892919908881010751517221722381012834556556824848577221227684137830816101020327876526681725444453682864636555747673248741474708131366459673977930419195224272687150280798146593067544755578201466060570093254935347120068461604351415848203621522648676607169207523056123186434484713066312780810175397901692245443312244602960976968454897460424230265560584944905249327347768975517831568652679595607920084575082584347314212810007985240690964634053969181170611215275387792197501045392327155456594027530404389193373166529959026406585893677752993907919001817631111686530612365744087948657981705511840245825626621270255357615436919191022104201504251202188647227781935143967146969540653262038857149021960306229530963054026491708200678862903006329234012195611754127295080517138419193838585920213543419469950079348946755215094694137847178186210286204470627994554250221170678469119109643939053546724329890548255475005106738694859949241621137720195040487766275618591781040165985869235563695957345264145252201054929162788252799455694036338904769031986390960043941243280986491172713827805499916660157385659508981509849147715973547457229029295526584351410281813731593170470817309110789876049747009019009949757200409333866914797127029381219525171081649367682900079983068290393154404118264018185364905606965589045505005572066630122121150375280811289992817027578011638093603433244372443437925332947876282657417807723726415290604403918202072098770116006990175350820492314867796387685661273702215556922453698175420527665425961486623775069041082819614980223231440570232620689816525157996858630797825675874156474646732065652602821773250214911095523839301872173514572945534777185829446594458412939559790823335335988943249612096741437113159733496107309438749517155816113611466181606413786593079760286476932502122540382520346014562754440684858961033148757521944516188781560550967342223679376726927253300939809903760523174681426976332580145397706760699651938934492314536611971407185558853691943983900622101911869433034988901671398749303046154807811343347079647262775389547959397251809798861843882195477113660504281821150366169039324164617598327207275003383926645719618807107118775499837007692943421107557066566151632494226460059669482223627000755320102291265532871104516620408021503469564960744840527318501733587855812003819890890023893498130260498951286347694871337136866334557342356711006118238958427510975755322669497692071461010165591733674071285408880586240632956831541705216908849696753484260512346231146506771015879263648500135313751979294236700008885800509693120930841769817301480352739004267277612475636269175159040483895116762728473969354479429135527425822803355504559236868842150825399906121046596018792183036698610028246005663710973350043476200695248674446173296357513573521498183102888985070332615743303290931992949087204064596760980551855974772314932770996526784911225488608910212901539625649853695492476183650987182998517921557222998098782725632700696246594120460326038060483755479352197136229776118789296545919836632814452748621216052104394615335775039904853067848468754958862041044060581067912446471855014833235066207766370613368257308071954943653479180111042281881188887660434936081241799850788470218012237849400298660425782403148718872653901005280466525095291577169601084339235634955604898374949141660578599870423680499360451073008559080164788699736386609233499025062855391709957273487073386537359165683384170380942428413622358059424469862102618894310733782269292492850029973291 Question 5 Use the appropriate procedures in ComboProject5.txt to find the number of 3 by 101 TicTacToe boards that ended in a tie (i.e. had 151 X's and 150 O'x and no three consectutive Xs, or O's horizontally, vertically, or diagonally. Answer 5 taylor(taylor(f, x=0, 200), t=0, 202): coeff(%, t^101) 76 75 74 73 56 x + 416218 x + 181132044 x + 17258609756 x 72 71 70 + 550527276492 x + 7182166538698 x + 42657412905204 x 69 68 + 122177173180284 x + 173058258270516 x 67 66 65 + 121177055746484 x + 40893176034048 x + 6294003943450 x 64 63 62 61 + 400601494604 x + 8883189024 x + 49505836 x + 31714 x We then have 514416748274428 ways to get the desired results Question 6 Use the appropriate procedures in ComboProject6.txt to approxinate the average degree of an induced graph if you take 1000 random choices of 257 vetrices in Bn(9). Of coure it changses from run to run, but it should be close Question 7 Use the appropriate procedures in ComboProject7.txt to find the coefficient of x2000 y2000 in the bi-variate Taylor series of 1/(1-4*x-5*y+11*x*y) Answer 7 Takes a long time. DiagSeq2(1/(1-4*x-5*y+11*x*y), x, y, 2000)[2000]