Let and . Prove that

Let and is a subset of such that if then Find the smallest of

Find all sequences of four consecutive integers such that twice their product is perfect square minus nine.

If , then prove that the following inequality holds:

Find all solutions for the functional equation , in which :
Note: the solution is actually quite obvious - , but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.

Find all functions > that satisfy , for all € .

This cipher is a lot like the Playfair Cipher as you might have guessed from it's name. We won't spend much time discussing this cipher, I will just explain it so that you know what it is. Instead of having just 1 key array, you need to have 2.

This cipher really isn't much harder to break than the playfair cipher and I am too lazy to explain it so if you really want to know what it is just click this link.

In the previous post, I made several assumptions about the key array that just happened to be correct, often, you will make several assumptions and find that when decrypted the plaintext looks completely wrong. If this happens go back to the last assumption you made and change it slightly. If that does not work, go back to a previous assumption. It can take a while but in the end, you will probably get the correct plaintext. This method works for cryptography in general. You can also look for words in the incorrectly decrypted ciphertext to give you a clue as to what the mistake in the key array is. We used this when we found that the last word in our incorrectly decrypted ciphertext was "AMERLCA". One of the most important skills you need to have when trying to decrpyt any ciphertext is patience.

We now can say that "QL" is "TH", "LM" is "HE" and "CD" is "RE" and "CQ" is "AT".

When trying to find the key array, there are a few things you need to keep in mind.

1. Remember that though the keyword is a word, any repeated letters in it will be deleted, so you should be looking to form a group of letters at the first row of the array that looks like it could be a word, but is not a word exactly.

2. When the person who wrote the message made the key array, after he chose the key word, he put the rest of the letters in alphabetical order to the end of the array, this means that "V W X Y Z" is probably the last line of the array and that the letter G will definitely go somewhere in the first 3 rows of the array. If, when making the array, you see a string of consecutive letters, like " F G H I" you are probably on the right track.

3. You want to assume that the digraphs you know form a rectangle with the letters at the corners. Let us say that you knew that the letters QL represented the letters TH,

draw a rectangle like this:

H L

Q T

We have no idea of the number of columns between Q and T, nor the number of rows between H and Q. but we do know that H and Q and L and T are on the same column and Q and T and H and L are on the same row. I drew the rectangle with H in the top left corner because it is alphabetically first. (Not e that all 4 could have been on the same row or column but that is not likely). We also know that LM and HE form a rectangle. We already have L and H in our grid so we just have to add L and M.

M E
H L
Q T

M and E technically could have gone under H and L but since E comes before H, above seems likely.

Next one, CD is RE. Since D comes directly before E in the alphabet, I'm gonna put D directly before E in our key array, once I've put D, C and R will follow:

RC
M DE
H L
Q T

Note that we could also have realized that C is on the same column as T using this next pair: CQ is AT. We have already put C, Q, and T in our array, so we just need to put the A t o complete the rectangle.

ARC
MDE
H L
Q T

Our first string of consecutive letters is "DE" so it seems that the key word ends at M and the rest of the letters are filled in alphabetically. F is not a common letter so we can assume temporarily that F comes after the E. We can also say that G is after F. And we can write VWXYZ in the last row.

ARC
MDEF
GH L
Q T
VWXYZ

We have completed enough of the ciphertext to just fill in the remaining letters in alphabetical order:

BARCI
KMDEF
GHNLO
PQSTU
VWXYZ

Looking at the key word: BARCIKM, it would look a lot more like a word if K and I were switched:

BARCK
IMDEF
GHNLO
PQSTU
VWXYZ

We could try decrypting the ciphertext now and find that it does not look very readable:

myfkevniwcdllydosxstaoifernxidayhthksnlkythdtaskknhfreusbvat
kfnehfrthdtrustyiuhaznknstiwedmlfeoynifthdsacrledxesnobfnkyi
uraodnstirslshaokpresldeosbupmhfbmlsservdxnxixiubfatlioaswns
nasthdhnodrisltyaodcixiperatliibdhaspmiwosmbiugmiutxshlstrao
slldiohfrtyfiuramerlcafphaznoiwtakdoshnuresldeoslaniathshdwi
mfpmavnkexdospikdodurlogrlsloglddesiforisperdlyafeshesldevnw
atersifoeadnrnxnznrysifhxnoshdiashlstakeoamldvsgathdrlogcniu
dvaofmaghobgtirmsatxshesemimdotsamerdxahascarxrlediooitslmpn
ybndausefhshesclevnirvlslioifthisehodggmfheldnbutbndausewesh
nueipndhaveremalodfnalshoyntithdpdeaepfhiurfirknarersafetrue
tiiurfiufehofgicumdotssidlhasbexdosidlmustknwlshshlshnodrald
iofhamerdxaosthatwearehoshempdstfhcrlslslsoiwxweevenfeerstix
idiubfatliohsatwaragalfptafarxreacdgogodtwirkifulisnoreafeha
trediurecioimylsbadfyweakeoddacifpeudodnfhbvexedaodprxrespio
slbllhtyioshnuartifsimnkutansiiurcinsnctlvkfalenretimakehamf
rhildnsafeprnuarnxhdoaldiohfraodwahnhimepmavnkexdonistlibspm
edbuslodsxsesxpmutxnrediurhdanshcarelstiicistnyiurschiinsfal
ntiximaoyafeearhdaybrlogsfurshernzpddodnshatshewaysweusedoer
gxstreogshdoiuralzersarlesaodtmbeaxnoiurpnaonxsheseareshrdfe
dxatirsifcrlslsxsublndtxtidataaodvtaldstdxpeessmeasurabnnkut
oisnsxsprihfuodpsasapxplogfhcitopddodnacrisxsiurnafeaoagghog
fkarshatamerdxasdecnhordslodvltabneafeshatshdoexsgdoeratliom
ustniwerdlsxslgmtstidasdsaytiyiushatshndhaevsnogeswefadnarer
eanthdyareserliusafeshnraremaoythdywhlloitknmexnashlyirloapm
irtspaoiftlmnkutkoiwshlsamerdxathdywhlnbemnxioshlsdaywnhathd
rbndausewehaznrhisdohiuniverfkaruodlyifourpiseivercitolhctao
dbdpscirdiosdgvdaywecimetipricnalmaododtithduntxtybvdrvaores
aofnanseprimlsessherndrlmloaldiosafewibfiutdigmasthatfirfart
ixiniogmavestraoglediurpindldxsweremaloayiuogoatliobuthoshew
irdvfhscrlpturnxhdldmehascimnxisnxaspdnddgfdlshsdgogsthdldme
hascimnxireafxelrmiurefeurhobgplrltxtirhixiseiurkntxxnbmlsti
ryticarxryhfrwardthatprexdiugbletxshatoibnrddeapasseditoring
doeratliosigdoeratlioshdgigflvdoprimlsnxhataevnareeuanannare
frexeafeanfdeserznachaornxipursunxhdlroyevnmeasurefhhapxplod
sxslfbeafxelrmhogshdbveasoessihfuroaldioweuoderstafeshatbvea
soesslpfnzeraglvdodlmustkneabfediurliubfnrhapfnzerkndoioeifs
hirtcutsirsetxtnholnirsnsxslshasoitbexdoshnuatheirshkfaloshd
arxnfnirshisewhiprkfersnlsurnxiwirkirsexekionyshnusnasuresfh
rlrhesaofnamerathdrlshaskndosherlsktakersthddiersthdmakersfh
shhobgsimednsnbratedbutmireiftdomeoafewimeoibscurehoshrdbeab
irwhihavndarrledusupshnsiobvtghnluathsiwardvprisunrltyaofnre
edimhfrusthdypackedupthdlrfkwxwibedfypissesxsliosafetravnsed
acrisxsidnaoslfpearchifaodwnlekfirusshnrtihledhosweatshipsao
dvnxtnedshewesxnfeuredshnsashifthdwhlpaoluniwedthdhamfearthe
irusshnrhftghsaodbdpedhopnaceslhkeciorirdafehntxtysburgoirma
feyafechzshaosdrsafeagaloshdsemeoafewimefptrtggledaodvacrlel
dndafewirkedldevnthdlrhafeswererawsishatwemlgmtnlveakntxxnbe
lenxhdysawamerdxaxasblbphnrthaoshdsumihfurhodpvlduanamblldio
gbreatershaoannthddpfxfkreoresfhblrthirweanthirfacldioshlsls
shrdiubfnrwecioshouetidayweremaloshdmistprisunriuspiwerfneoa
ldioioearthiurwirkersareoisnsxspriductlvnxhaowhdosdgscrlslsb
nhaoiurmhodvardoinesslovdoldzniuvbixidsafeservdxesoisnsxpfex
ededshaoshnrwerenastweekirnastmioshibeastyeariurcapaxdtyrema
hosufelmholshdlkutiurtlmeifstafehogpatfhpritndldogoarxriwlos
erestsaoluutldogfhoyopsnasaotdndlslifpshatldmehasxsurenypasx
sedvtarthogsidaywemustplckiursnsznsuulustiursenvesfhfafekngh
oagahoshewirkifremakhogamerlca

If you stare carefully at it you will find that it is semi-readable meaning that we probably just have 2 letters switched around in the key array. We can find which two letters by looking at the ciphertext and seeing what letters are getting distorted. The last 7 letters are amerlca which looks an awful lot like America, but the i is replaced with an l. So the problem is with the letter I.

'I' was one of the last letters we put in along with O, P and K. We could try switching I with each of these letters and seeing what kind of ciphertext results. We would find that when we switched I with O we got something readable.

BARCK
OMDEF
GHNLI
PQSTU
VWXYZ

When decrypted it produces:

myfelxlowcitizensxstandheretodayhumbledbythetaskbeforeusgrat
efulforthetrustyouhavebestowedmindfulofthesacrificesbornebyo
urancestorsithankpresidentbushforhisservicetoxournationaswel
lasthegenerosityandcoxoperationhehasshownthroughoutxthistran
sitionfortyfouramericanshavenowtakenthepresidentialoaththewo
rdshavebexenspokenduringrisingtidesofprosperityandthestilxlw
atersofpeaceyeteverysooftentheoathistakenamidstgatheringclou
dsandragingstormsatxthesemomentsamericahascarxriedonnotsimpl
ybecauseoftheskilxlorvisionofthoseinhighofficebutbecauseweth
epeoplehaveremainedfaithfultotheidealsofourforbearersandtrue
toourfoundingdocumentssoithasbexensoitmustbewiththisgenerati
onofamericansthatweareinthemidstofcrisisisnowxwelxlunderstox
odournationisatwaragainstafarxreachingnetworkofviolenceandha
tredoureconomyisbadlyweakenedaconseuenceofgrexedandirxrespon
sibilityonthepartofsomebutalsoourcollectivefailuretomakehard
choicesandpreparethenationforanewagehomeshavebexenlostiobssh
edbusinesxsesxshutteredourhealthcareistoocostlyourschoolsfai
ltoxomanyandeachdaybringsfurtherevidencethatthewaysweuseener
gystrengthenouradversariesandthreatenourplanetthesearetheind
icatorsofcrisisxsubiectxtodataandstatisticslessmeasurablebut
nolesxsprofoundisasapxpingofconfidenceacrosxsourlandanagging
fearthatamericasdeclineisinevitableandthatthenextgenerationm
ustloweritsxsightstodayisaytoyouthatthechalxlengeswefacearer
ealtheyareseriousandtheyaremanytheywillnotbemeteasilyorinash
ortspanoftimebutknowthisamericatheywillbemetonthisdaywegathe
rbecausewehavechosenhopeoverfearunityofpurposeoverconflictan
dxdiscordonthisdaywecometoproclaimanendtothepetxtygrievances
andfalsepromisestherecriminationsandwornoutdogmasthatforfart
oxolonghavestrangledourpoliticsweremainayoungnationbutinthew
ordsofscripturethetimehascometosetasidechildishthingsthetime
hascometoreafxfirmourenduringspiritxtochoxoseourbetxterhisto
rytocarxryforwardthatpreciousgiftxthatnobleideapassedonfromg
enerationtogenerationthegodgivenpromisethatalxlareeualallare
frexeandalldeserveachancetopursuetheirfulxlmeasureofhapxpine
sxsinreafxfirmingthegreatnessofournationweunderstandthatgrea
tnessisneveragivenitmustbeearnedouriourneyhasneverbeenoneofs
hortcutsorsetxtlingforlesxsithasnotbexenthepathforthefainthe
artedforthosewhopreferleisuretoworkorsexekonlythepleasuresof
richesandfameratherithasbeentherisktakersthedoersthemakersof
thingssomecelebratedbutmoreoftenmenandwomenobscureintheirlab
orwhohavecarriedusupthelongruggedpathtowardsprosperityandfre
edomforustheypackeduptheirfewxworldlypossesxsionsandtraveled
acrosxsoceansinsearchofanewlifeforustheytoiledinsweatshopsan
dsettledthewestenduredthelashofthewhipandplowedthehardearthf
orustheyfoughtandxdiedinplaceslikeconcordandgetxtysburgnorma
ndyandkhzshantiesandagainthesemenandwomenstruggledandsacrifi
cedandworkedtilxltheirhandswererawsothatwemightliveabetxterl
ifetheysawamericaxasbigxgerthanthesumofourindividualambition
sgreaterthanallthedifxferencesofbirthorwealthorfactionthisis
theiourneywecontinuetodayweremainthemostprosperouspowerfulna
tiononearthourworkersarenolesxsproductivethanwhenthiscrisisb
eganourmindsarenolessinventiveourgoxodsandservicesnolesxsnex
ededthantheywerelastweekorlastmonthorlastyearourcapacityrema
insundiminishedbutourtimeofstandingpatofprotectingnarxrowint
erestsandputtingoffunpleasantdecisionsthattimehasxsurelypasx
sedstartingtodaywemustpickourselvesupdustourselvesoffandbegi
nagaintheworkofremakingamerica

If you can't read it now, I can't help you.

Okay, so, first I want you to copy the entire plaintext from the previous post. Then I want you to click this link. Paste the ciphertext into the big empty box under the words "Enter your ciphertext or comparison text here:" and click Submit. You should see a page with 3 tables. Look at the one whose cells are mostly green and whose title is "Most common digraphs". It will then list the frequency of the digraphs in the ciphertext and tell you which one appears the most. I have never actually told you which digraphs appear the most often in the English language, but we can figure out which ones do based on the monograph frequencies. The most common word in the English language is "THE" so we would expect the digraphs "TH" and "HE" to be very high up on the list. We could also say that "TH" would probably appear more often than "HE" because it also appears in many other words. You may notice that the table also gives us the frequency of the reversed digraph. This is extremely useful when trying to decrypt a ciphertext encrypted with playfair because if you examine the algorithm playfair uses for encryption you will realize that if "QP" gets encrypted as "KL", "PQ" must get encrypted as "LK". In the table, the darker the shade of green, the more appearances of the digraph. In the English language, there is one very common digraph whose reverse digraph is almost just as equally as common. This pair of letters is "ER" (or "RE"). If we look at the table the pair of digraph and reverse digraphs that are both very high in frequency are the pairs "CD" and "DC". Also, the highest frequency is the digraph "QL" and the second highest is "LM". We can now safely assume that "QL" is "TH", "LM" is "HE" and "CD" is "RE" and "DC" is "ER". Our goal is to find the correct key array, then we can decrypt the entire ciphertext. We can probably say that the last row of our 5 by 5 key table is "V W X Y Z" since these letters will probably not appear in our keyword. Now, we need to identify more digraphs before beginning to construct our key table. It is very likely that somewhere in the passage the phrase "THAT THE" will appear. If it does and is spaced like this: "TH AT TH E*" (* represents an unknown character) then we we know that in the ciphertext will be "QL ** QL **". We can look in our ciphertext for two QL's separated by 2 characters, and then assume that those two characters are AT. It turns out that "QL ** QL" appears 3 times in our ciphertext and in every single one of the times the ** was "CQ". We now can say that "QL" is "TH", "LM" is "HE" and "CD" is "RE" and "CQ" is "AT". We are now done with the first step, we have figured out what a sufficient number of digraphs are. Step 2, which will be the next post, is figuring out what the key array is based on this.

ew of iy ge ya ls nx fl xr tu kh mi dc ly me cw nq oa tl or cy lm qc ur co om cd pt ib cq fo tn om cs lm sc pt yc fp qm yo co tu mv fe dh if nz ge eu lm qr kc nd lr dt og ki oc ve sk kh el tu db xs ql kh bu cd xs ef lu kp qi om ai sx td bx lr ly dv fp ki cq gd hk qx lt hc tu lm lo lf bd xs yc kh er dv gv dc cq gd gi ml rq qi mv lu ia fp hi fp sy ql sx sc kh xs sl fg om cs ze fp cr df ds kr iu qm yo gf yq rb fl ql ot cd xs ef lu hr ge cq lq lm vm di qi bw oc dy fl tq fb fl fs ds gh ds xs gh sl ef pd ou bd tq dc ls wc if ql dt sl iy hy cq dc pd ou mc el cl yl yo cx pd mo yl lu lm mb ql sx qc cf hk dh ix pl cq lm ds gh et fp ix kh id bh lg ip pe ad qr sy ql dt fd md fl ut mh dc lr mq rq kr dr ds fe fg gf ut hd tg vc le kq td mo ql dt rn iy ge bx sx gd gf eu gm td lg il hi mo dn el kp pc le kq td ym ql ot fm tg ml bw dc fd rh lf eo rh ql nz ty ep lm si mc it mo fp kd db co rc dc qr if sc tf pe fp kd fp if lg io eb qf fl ut pd ls qm pr dy fl pd ls fq tu co xh ql ql sx lo lf cr sl fg mo mh dc lr kh tu qm qy mc cd lg ql fd si tu mo kc sx sx sx gf xy ym iy nt if dc tu dv me fp ki cq gd gl qr qy rc bh rh iu qc mk dr cd rk il gh lf qy db bf oz gd tl lk mc if qm sc fe fp cd be gf ew sx ar ei zx mc cf lf mr be iu ft fl el mo ib dy fe kh is dr cd tq fg xs rg nl yc fg ql ot rc pe du md oc pu ch pd fp ck eg tl ey gx fo rh nt cd pe hm cf qm di al dg el qr if sb ot rc ly lm hk sl fg om cr lf am lo gm df qi bw oc dy fl ge tu gd rp qi fe kp xs lf xr td xr qi pu yl cd em sk lm ch ql kr cd sx pe eb dp yt ve sk tr gm eg ud rh ty dv md kh wc if mc al mr vc ds gh ud sk ql dc oy si fl el ql cq ql my cw qx ft td fl dc lv tu cd gh ql fl fp cr ox dc qr ds dt kh es ia mc yl gf sk tg kh ly ql dt mc cd ql dl if lr cq db pd ek ds xs xr tp rg le sy pe mr qc kh ix qc sl tu lr ti dt qd mc tp cr cg oc pu gf tl xr tq bd om zu is qr qr sv sg gh mo be un si fl el rk bd xr pd sk hc if kh bh hl gh of rc ql cq mh dc lr rq ef et lg dl xs lf xg qc cg mc if ql cq ql fl dy pl fl dc cq gd hf pt yt mv dc ls xr xs hi ut pe mr xl qr cy ev fp ql cq ql le qm iy tl gh dt ym mk el rc dc mc ty lm wc cd td ds fp qr if ql lc rc fd kh cy lm zx ln ng ep co df yl rq ln ve ds hk qi db ut qb gf eu hd oc pu fu mv ql sx mh dc lr cq lm zx ln gc fd ly fg ql sx mr zx ol cq lm ca le kq td ym qm yo al dp fl gm to gb dc of rc zu ls ve ou sk vg td gb dc be un nl ey kh ir is tr db em lu il xi cw ym be df pe sb eb hc hd kh fl es ep lm to sy yc ib ld wb lk dt kh eo ch td sb md sx dt ql dc le ds dh hk sl fg qr if vm ki fp se gp hm tu qm ue db mk cs dv eg fg hi bw dt sc kh hn fe fp bs eg ls lr qx dc fd rh hk ve zu hg cq gd gk pu lg ql my db ix mo tr ds qu sk ly lm sl df qm tr md ly dp ly rq si le il ie sx lq il gh tu lm sl df qm tr md ly db mc dz dn ad fp cd if sk lg ip sg ds sy pe al dv dp fm sk co sy yl ai sx pe cx pe kr dr cx om ax rc es qm uq cd rl fp pi nd sy ql cq gf cg dl ef bq rq td em un bd oh fl dc cq gd lu gp fl dc cq gd lu lm pg oi gx fl sb md sx ly qm qc iy hc cd ft ch ch hc cd dk dy mc if ch ie dt dc yo rk qm lk ly gv sk tp ly lm sd nz iy he mc tp cd mo qm sv sg lf xr xs ik mc dz dn ad lg lp lm ib mc ul dt pd om sk hk sl fg ym zu ef dx qc if ql cq ib mc ul dt xs ui oy dc bh gx fl ls fq tu co mc ki fe fp ds fp ki lc qm ui oy dc co fl fg fm du gm cs kt ut db td sy yt lg no db tl xr xs ql rq gf pc dy fl ql ot cq nm db ql fo rh lu lm rc yl eo db ql dp my gm sb fo dc tl sx sk ly mv db bf dx dy fc fg tc ql ot tl rq sk dt mo ds al dt kh eo mh dc cq lm ds ql rq co fl ql dc sx cu rb dc tu lm em dc tu lm hm cf dx mo ql lg ip pd df el tl ac cq fe kp qe db fm eu fl df hk if vm df gf rp kt cd lg ql dl ci ra db aq mg bw le rc ds fe pt pq ql lt fg ib pn lo os cq lq mv rc ix sb dp to ds yc kh eo cd fe md om ks tu lm vt rk cf fs qu lm sd of xy vm ci ei vt dp td xr xs fg qr if sc bw lt fe rk bd xr pd el kh xs iu mc ck gm mk lf yh nd fo db pt ql lc pe ln fe lg qx mc ut gm qt kh ix ly yt fe ql my dt yl if sk fe ql lt rq gm eu lm aq gs kh os ge ym es lm qm di mc cs nm db pt ql lc om pn lq kh ir is fe lg tg rk dt nl cf be lk db mr if lo sy yc pr sk hg db hm if wc if an xu qm lu ld qr if bh rh lu lm td df hk if vm df iu sc pn hn fe kh ix rk ds dn el mr if vm cb fe sl iy ty lm sd qm if qx dc dc ma pd ql cq ym dh hi yt gx mc co sy yl ci nd ly lm xt ma mh dc lr rw rq rg iv lo cs qm lu lm tp dm om sk lg is xg fs ch mh rg sl fg pi cd cq dc ql kh ch ty lm is dz of cd lk dt mo rg cs gm ax mc ty gm kd rk sl fg ql sx sx ql dl fp ki lc ym be lu lg tf pe mr zx dc fd rh lu lm dm tu sb dp to bd pt vg ym kd tn hk sl fg fg mc cs gm sk vm cb dc qr cd gf tl xr tq bd fs ey gx ly qm hz lm lu il tr ds xs pr ol kh fp ad lg ix rc fl eg dt xs gz fl sl yo fp bi dv me qr if td bx lr dt gf tl xr ui dy fe fe ql kh ql lc ym cd hc tu ym fc db hc tu dm lu gm ci rq yc mc bd sk kr qb rl yc cd hm lg tp if hd lg sx lm or pu fp cs hd fm du qc if lg pv cq mo sb ep le sl gh hk dr bd xh lu dc dt ut kh os pu sl gh mo nz gu tl rq kh se le sx gd iu ql cq sl df qm xr tp cd tc qb xr td ix qc cs lg lp me cw ym fq tu sg kb fp dx lt yo tp so pt pe sk td gy dt mo mk if co hl hk hb lg ql my db bf dk fd rb lg hb df ds kr

Find all triplets of positive integers such that .