Evil Relays

Mike and Ike know exactly what to do: you'll have to beat McCharlotte and McYuval at their own game. And here are McCharlotte and McYuval now!!

McCharlotte: Bwa ha ha! Get ready for EVIL RELAYS!
McYuval: Each of the 13 EVIL PROBLEMS will have an EVIL NUMBER and an EVIL ANSWER.
McCharlotte: Each EVIL PROBLEM uses one of the puzzle answers you already have, but they aren't EVIL enough to solve these relays problems yet... you'll have to make them more EVIL!
McYuval: When you find enough EVIL NUMBERS and EVIL ANSWERS, you'll be able to answer: What have we been trying to inflict on you this whole puzzle hunt?

1. PECIAL CTIONS: In this roblem, all but the last six etters of each word (other than those earing in math) have been emoved.
Rlotte is flying her rplane hrough higher nsions, where the n-sional cube has 80 edges. What must she avoid?

2. RSPECIAL RINSTRUCTIONS: Rin rthis rproblem, rthe rletter Rr rhas rbeen rprepended rto rall rwords (rother rthan rthose rappearing rin rmath).
Rsuppose x^x^x^x... = 2. Rnow rsquare x rtwice. Rwhat rshould ryou rorder rfrom Rpad Rthai Rtoo?

3. PECIAL NSTRUCTIONS: N his roblem, he irst etter f ach ord as een emoved (ther han hose ppearing n ath).
CCharlotte nd CYuval it own o lay ame f Ock, Aper, Cissors, ut o ake he ame ore nteresting, he ntroduce wo ew ptions: "Andy" nd "Hocolate Ab". Hocolate Ab ats Andy; he ugar n Andy auses Ock nd Cissors o rode; nd Andy ips ts ay ut f ts Aper rapper. Hocolate Ab s quished y Ock, ut y Cissors, nd oisoned y he hemicals n Aper. layer cores 1 oint or in, 0 oints or oss, nd 1/2 oint or raw. CCharlotte ssigns ome robability o ach f Ock, Aper, Cissors, Andy, nd Hocolate Ab, nd andomly hooses mong he bjects ach ound ccording o er hosen robabilities. Ince he s ery ocky, he ells CYuval n dvance hat er robabilities re. Et x e he eciprocal f he roportion f he ime CCharlotte hould hoose Hocolate Ab. Ow ould ou escribe CCharlotte's roficiency?

4. SPECIZL INSTRUCTIOMS: Hn thhs probldm, tge penultimase lettdr nf eabh woqd (othdr thzn thore appearimg hn mash) hzs bedn shiftdd babk ay ome.
Imagime thzt ynu haue writtdn dovn a veqy larfe numbdr nn a scrzp nf papdr. Scramblimg tge digiss nf thhs numbdr, ynu obtahn a secomd numbdr. Subtractimg tge secomd numbdr frnm tge firrt ome, ynu obtahn ydt anothdr numbdr. Finalky, ynu crors ott ome nf tge digiss nf thhs thiqd numbdr, amd aqe leet wish 224798432. Considdr tge dight thzt ynu crossdd ott, thdn crumpke tp tge piebe nf papdr. Whzt haue ynu creatdd hn tge papdr?

5. SPAL INNS: In this prem, all but the fist two and last two lers of each word (oter than thse apng in math) have been reed.
Of the 72 cars who are in the gaes loge, 38 of them are plng a game of "Hide and Go Seek in the Chte Lab. At the strt of each mite, the weed-weed caer (thre are no ties in wier) who is stll plng has a 1/3 chce of qung the game, but the brst caer who is not yet plng has a 2/3 chce of sung to peer prre and jong the game. An ober noes that, half an hour into plng the game, the exed nuer of plrs wold be a whle nuer of dons. What is the obr's name?

6. PECILA NSTRUCTIOSN: N hsi roblme, eh irts ettre f ahc odr (thre hna hoes ppearign n aht) sa ene emovde, hne eh ats ow ettesr aev ene wappde.
N eh ollowign iguer, evne tu f eh ixtene quarse aev ene aintde n ronw, de, raneg, r ellwo:
BROY
R???
O???
Y???
Onsidre eh umbre f ifferetn asy uo na aitn ahc f eh emainign quarse o hta ahc olro ppeasr xactyl nec n ahc wo dn n ahc olunm. Hta houdl uo nsuer eh olumsn o?

7. PVCIA NVTRUCTION: In this rvble, the ivs and last evte of each four evte or ovge word (tve than hvs pvearin in math) have been evove, and the evainin evon evte evome V.
Let x avisf (1 + (1 + (1+x)/(1-3x))/(1-3(1+x)/(1-3x)))/(1 - 3(1 + (1+x)/(1-3x))/(1-3(1+x)/(1-3x))) = 1. How ivh you evcrib the top of that rvctio?

8. ACEPECIAL INACETRUCTIONACE: In thiace problem, all inacetanceace of the letter Ace (other than thoacee appearing in math) have been replaced by the word Ace.
A aceet of natural numberace iace called "evil" if it containace itace own cardinality. (For inacetance, {3, 4, 8, 9} iace evil becauacee it haace 4 elementace and 4 iace one of thoacee elementace). An evil aceet iace called "minimally evil" if it iace evil and no proper aceubaceet of it iace evil. (For inacetance, {3, 4, 8, 9} iace not minimally evil, becauacee {3, 4, 8} iace alaceo evil). Conaceider the number of minimally evil aceetace can be made with the numberace 1, 2, 3, 4, 5. What can you uacee to remove the ink you acepilled while working on thiace problem?

9. SECIA ISTRUCTION: In this poble, the scon and last ltte of each four ltte or lnge word (ohe than tos apearin in math) have been rmove.
A scre aen wtche popl go in and out of a sor. When the sor oen, only 1 prso, the one, is isid. Ate an hour, 3 popl have etere and 2 have dparte. Ate aothe hour, 7 more have etere and 4 have dparte. Ate one more hour, 1 more prso has dparte, but no one else has etere. At this pin, the aen rcord the nmbe of popl who are isid, then goes home. What is the aent' name?

10. FSPECI FINSTRUCTIO: In this fprobl, the last two flette of each four flett or flong word (foth than ftho fappeari in math) have been fremov, and F has been fprepend.
FmcCharlot has 8 fminio to do her fbiddi, and they will work in fpai in fspreadshee. She fwan to fassi a fspreadshe tab to each of the 8 fchoo 2 = 28 fpossib fpai of fminio. She does not want to be fwastef, so she may fassi one tab to more than one pair. Fhowev, she fwan to fensu that fdisjoi fpai of fminio get fdistin fpag. For fexamp, the pair {Mia, Sim} will get a fdiffere page from the pair {Nic, Viv}, fthou {Mia, Sim} may get the same page as {Mia, Ben}. Fsuppo FmcCharlot uses the ffewe fnumb of fpag fpossib. What is her fpreferr way to ftrav?

11. SPEIAL INSONS: In this prolem, all but the first three and last three leters of each word (other than those apping in math) have been remved.
Conder the leaing coeent of a degree two polial p(x) with intger coents, where p(0) = -7 and sin(2π/7) + sin(4π/7) + sin(8π/7) is a root of p(x). Assing you have no troble with it, how would you desibe your ease of soling this prolem?

12. SCAL IINS: In tis pbem, all but the fist lter, futh lter fom end, and lst two ltrs of ech wrd (oter tan thse arng in mth) hve ben roed.
Thre are two wals whch met at an anle of π/12. A bll is rled plel to one wll, and wll hit the oter. Amng tat the cins are pcly esic, cier how mny cins the bll wll hve wth the wals mod 10. Wat isct cold get in the way?

13. SPECILA INSTRUCTIOLN: Li thli problle, tlh lals lettle lo ealc wolr (othle thla thols appeariln li malt) hla bele removle, thle hla bele insertle secoln lt lals.
Slilc grele folo (centerle la tlh origli) aloln tlh thrle planle x=y, y=z, aln z=x, thle couln tlh piecle. Whla li tlh bels folo lt uls?