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COMPUTER

S CIENCE I ILL

COMPUTER S CIENCE I ILL

WLADSTONFERREIRAFILHO

LasVegas

All rightsreserved.

EditedbyRaimondoPictet.

Publishedby CODEENERGYLLC

 hi@code.energy

 http //code.energy

 http //twitter.com/code_energy

 http //facebook.com/code.energy

 SJonesBlvd# LasVegasNV

Nopartofthispublicationmaybereproduced,storedinaretrieval systemortransmittedinanyformorbyanymeans, electronic,mechanical,photocopying,recordingorotherwise,withoutpermissionfromthepublisher,exceptforbrief quotationsembodiedinarticlesorreviews.

Whileeveryprecautionhasbeentakeninthepreparationof thisbook,thepublisherandtheauthorassumenoresponsibilityforerrorsoromissions,orfordamagesresultingfrom theuseoftheinformationcontainedherein.

Publisher’sCataloging-in-PublicationData FerreiraFilho,Wladston.

Computersciencedistilled:learntheartofsolvingcomputational problems/WladstonVianaFerreiraFilho.—1sted.

x,168p.:il.

ISBN978-0-9973160-0-1

eISBN978-0-9973160-1-8

1.Computeralgorithms.2.Computerprogramming.3.Computer science.4.Datastructures(Computerscience).I.Title. 004–dc22 2016909247

FirstEdition,February2017.

Friendsarethefamilywechooseforourselves.Thisbookis dedicated tomyfriendsRômulo,Léo,MotoandChris,who keptpushingmeto“finishthedamnbookalready”.

Iknowthattwo&twomakefour—andshouldbe glad toproveittooifIcould—thoughImustsayif byanysortofprocessIcouldconvert2&2into five itwouldgivememuchgreaterpleasure.

—LORDBYRON

1813lettertohisfuturewifeAnnabella. TheirdaughterAdaLovelacewasthefirstprogrammer.

Everybody inthiscountryshouldlearn toprogramacomputer,becauseit teachesyouhowtothink.

—STEVEJOBS

Ascomputerschangedtheworldwiththeirunprecedentedpower,a newscienceflourished: computerscience .Itshowedhowcomputers couldbeusedtosolveproblems.Itallowedustopushmachinesto theirfullpotential.Andweachievedcrazy,amazingthings.

Computerscienceiseverywhere,butit’sstilltaughtasboring theory.Manycodersneverevenstudyit!However,computerscienceiscrucialtoeffectiveprogramming.Somefriendsofminesimplycan’tfindagoodcodertohire.Computingpowerisabundant, butpeoplewhocanuseitarescarce.

Thisismyhumbleattempttohelptheworld,bypushing you tousecomputersefficiently.Thisbookpresentscomputerscience conceptsintheirplaindistilledforms.Iwillkeepacademicformalitiestoaminimum.Hopefully,computersciencewillsticktoyour mindandimproveyourcode.

I45hi4bookfo3me?

Ifyouwanttosmashproblemswithefficientsolutions,thisbook isforyou.Littleprogrammingexperienceisrequired.Ifyoualreadywroteafewlinesofcodeandrecognizebasicprogramming statementslike for and while ,you’llbeOK.Ifnot,onlineprogrammingcourses 1 covermorethanwhat’srequired.Youcando oneinaweek,forfree.Forthosewhostudiedcomputerscience, thisbookisanexcellentrecapforconsolidatingyourknowledge.

B65i4n’5com165e34ciencej645fo3academic4?

Thisbookisforeveryone.It’sabout computationalthinking .You’ll learntochangeproblemsintocomputablesystems.You’llusecomputationalthinkingoneverydayproblems.Prefetchingandcaching willstreamlineyourpacking.Parallelismwillspeedupyourcooking.Plus,yourcodewillbeawesome.

Maytheforcebewithyou, Wlad

1 http://code.energy/coding- courses.

C A Basics

Computer scienceisnotaboutmachines,in thesamewaythatastronomyisnotabout telescopes.Thereisanessentialunityof mathematicsandcomputerscience.

—EDSGERDIJKSTRA

COMPUTERSNEEDUS to breakdownproblemsintochunks theycancrunch.Todothis,weneedsomemath.Don’t panic,it’snotrocketscience—writinggoodcoderarely callsforcomplicatedequations.Thischapterisjustatoolboxfor problemsolving.You’lllearnto:

Model ideas into flowchartsandpseudocode, Knowrightfromwrongwith logic , Count stuff, Calculate probabilities safely.

With this,youwillhavewhatittakestotranslateyourideasinto computablesolutions.

. Idea4

Whenyou’reonacomplextask,keepyourbrainatthetopof itsgame:dumpallimportantstuffonpaper.Ourbrains’workingmemoryeasilyoverflowswithfactsandideas.Writingeverythingdownispartofmanyorganizingmethods.Thereareseveral waystodoit.We’llfirstseehowflowchartsareusedtorepresent processes.We’llthenlearnhowprogrammableprocessescanbe draftedinpseudocode.We’llalsotryandmodelasimpleproblem withmath.

Flo8cha354

WhenWikipediansdiscussedtheircollaborationprocess,theycreatedaflowchartthatwasupdatedasthedebateprogressed.Having apictureofwhatwasbeingproposedhelpedthediscussion:

Previous page state

Edit the page

Do you agree with the other person?

Discuss the issue with the other person

Was your edition modified by others?

New page state

you accept that modification?

Fig63e . Wikieditionprocess adaptedfromhttp//wikipedia.org.

Liketheeditingprocessabove,computercodeisessentiallyaprocess.Programmersoftenuseflowchartsforwritingdowncomputingprocesses.Whendoingso,youshouldfollowtheseguidelines 1 forotherstounderstandyourflowcharts:

•Writestatesandinstructionstepsinsiderectangles.

•Writedecisionsteps,wheretheprocessmaygodifferent ways,insidediamonds.

•Nevermixaninstructionstepwithadecisionstep.

•Connectsequentialstepswitharrows.

•Markthestartandendoftheprocess.

1 There’s evenanISOstandardspecifyingpreciselyhowsoftwaresystemsdiagramsshouldbedrawn,called UML :http://code.energy/UML.

Let’sseehowthisworksforfindingthebiggestofthreenumbers:

Fig63e . Findingthemaximumvaluebetweenthreevariables. P4e6docode

Justasflowcharts, pseudocode expressescomputationalprocesses. Pseudocodeishuman-friendlycodethatcannotbeunderstoodby amachine.Thefollowingexampleisthesameasfig.1.2.Takea minuteandtestitoutwithsomesamplevaluesof A , B ,and C : 2

function maximum A,B,C if A > B if A > C

max ← A else

max ← C else if B > C max ← B else

max ← C print max

2 Here, ← is theassignmentoperator: x← reads xissetto1 .

Noticehowthisexamplecompletelydisregardsthesyntacticrules ofprogramminglanguages?Whenyouwritepseudocode,youcan eventhrowinsomespokenlanguage!Justasyouuseflowcharts tocomposegeneralmindmaps,letyourcreativityflowfreewhen writingpseudocode(fig.1.3 ).

A model isasetofconceptsthatrepresentsaproblemanditscharacteristics.Itallowsustobetterreasonandoperatewiththeproblem.Creatingmodelsissoimportantit’staughtinschool.High schoolmathis(orshouldbe)aboutmodelingproblemsintonumbersandequations,andapplyingtoolsonthosetoreachasolution.

Mathematicallydescribedmodelshaveagreatadvantage:they canbeadaptedforcomputersusingwellestablishedmathtechniques.Ifyourmodelhasgraphs,usegraphtheory.Ifithasequations,usealgebra. Standontheshouldersofgiants whocreated thesetools.Itwilldothetrick.Let’sseethatinactioninatypical highschoolproblem:

LIVESTOCKFENCE

Yourfarmhastwotypesoflivestock. You have100unitsofbarbedwiretomakearectangular fencefortheanimals,withastraightdivisionforseparating them.Howdoyouframethefenceinordertomaximize thepasture’sarea?

Startingwithwhat’stobedetermined, w and l arethepasture’s dimensions; w × l ,thearea.Maximizingitmeansusingallthe barbedwire,sowerelate w and l with 100 :

Fig63e . PseudocodeinRealLife,courtesyofhttp//ctp .com.

Ma5hema5icalModel4

Pick w and l thatmaximizethearea A.

Plugging l from thesecondequation( l = 100 2w 3 ) intothefirst,

That’s aquadraticequation!Itsmaximumiseasilyfoundwiththe highschool quadraticformula .Quadraticequationsareimportant foryouasapressurecookingpotisvaluabletocooks.Theysave time.Quadraticequationshelpussolvemanyproblemsfaster.Remember,yourdutyistosolveproblems.Acookknowshistools, youshouldknowyours.Youneedmathematicalmodeling.And youwillneedlogic.

. Logic

Codersworkwithlogic somuch itmessestheirminds.Still,many codersdon’treallylearnlogicanduseitunknowingly.Bylearning formallogic,wecandeliberatelyuseittosolveproblems.

Wewillstartplayingaroundwithlogicalstatementsusingspecial operatorsandspecialalgebra.We’llthenlearntosolveproblems withtruthtablesandseehowcomputersrelyonlogic.

O1e3a5o34

Incommonmath,variablesandoperators( + , × , ,…)areused tomodelnumericalproblems.Inmathematicallogic,variablesand operatorsrepresentthevalidityofthings.Theydon’texpressnumbers,but True / False values.Forinstance,thevalidityoftheexpression“ ifthepooliswarm,I’llswim ”isbasedonthevalidityof twothings,whichcanbemappedto logicalvariables A and B :

A : Thepooliswarm.

B : Iswim.

They’reeither True or False . 3 A = True meansawarmpool; B = False meansnoswimming. B can’tbe half-true ,because Ican’thalfswim.Dependencybetweenvariablesisexpressed with → ,the conditionaloperator . A → B istheideathat A = True implies B = True :

A → B : Ifthepooliswarm,thenI’llswim.

Withmoreoperators,differentideascanbeexpressed.Tonegate ideas,weuse ! ,the negationoperator . ! A istheoppositeof A : ! A : Thepooliscold.

! B : Idon’tswim.

T C

Given A → B andIdidn’tswim,whatcanbe saidaboutthepool?Awarmpool forces theswimming,sowithout swimming,it’simpossibleforthepooltobewarm.Everyconditionalexpressionhasa contrapositive equivalent:

foranytwovariables A and B,

A → B isthesameas ! B → ! A.

3 Values canbeinbetweeninfuzzylogic,butitwon’tbecoveredinthisbook.

Anotherexample: ifyoucan’twritegoodcode,youhaven’treadthis book .Itscontrapositiveis ifyoureadthisbook,youcanwritegood code .Bothsentencessaythesameindifferentways. 4

T B

Becareful,saying “ifthepooliswarm,I’llswim” doesn’tmeanI’llonlyswiminwarmwater.Thestatementpromises nothingaboutcoldpools.Inotherwords, A → B doesn’tmean B → A .Toexpressbothconditionals,usethe biconditional :

A ↔ B : I’llswimifandonlyifthepooliswarm.

Here,thepoolbeingwarmisequivalenttomeswimming:knowing aboutthepoolmeansknowingifI’llswim andvice-versa .Again, bewareofthe inverseerror :neverpresume B → A follows from A → B .

A ,O ,E

O Theselogicaloperatorsarethemostfamous, asthey’reoftenexplicitlycoded. AND expressesallideasare True ; OR expressesanyideais True ; XOR expressesideasareofopposing truths.Imagineapartyservingvodkaandwine:

A : Youdrankwine.

B : You drankvodka.

A OR B : You drank.

A AND B : You drankmixingdrinks.

A XOR B : You drankwithoutmixing.

Makesureyouunderstandhowtheoperatorswe’veseensofar work. Thefollowingtablerecapsallpossiblecombinationsfortwo variables.Noticehow A → B isequivalentto ! A OR B ,and A XOR B isequivalentto ! (A ↔ B) .

4 And bytheway, they’reboth actually true.

Table . Logicaloperationsfor possiblevaluesof A and B .

AB ! A A → BA ↔ BA AND BA OR BA XOR B

BooleanAlgeb3a

As elementaryalgebrasimplifiesnumericalexpressions, boolean algebra 5 simplifieslogicalexpressions.

A Parenthesesareirrelevantforsequencesof AND or OR operations.Assequencesofsumsormultiplicationsinelementaryalgebra,theycanbecalculatedinanyorder.

A AND (B AND C )=(A AND B) AND C.

A OR (B OR C )=(A OR B) OR C.

D Inelementaryalgebrawefactormultiplicative termsfromsums: a × (b + c)=(a × b)+(a × c) .Likewise inlogic, AND ingafteran OR isequivalentto OR ingresultsof AND s, andviceversa:

A AND (B OR C )=(A AND B) OR (A AND C )

A OR (B AND C )=(A OR B) AND (A OR C )

D M ’ 4 6 Itcan’tbesummer and winteratonce,soit’seither not summer ornot winter.Andit’snotsummerandnotwinter ifandonlyif it’s not thecaseit’seithersummer or winter.Following thisreasoning, AND scanbetransformedinto OR sandviceversa:

5 After GeorgeBoole.His1854bookjoinedlogicandmath,startingallthis.

6 DeMorganwasfriendswithBoole.HetutoredtheyoungAdaLovelace,who becamethefirstprogrammeracenturybeforethefirstcomputerwasconstructed.

! (A AND B)= ! A OR! B, ! A AND! B = ! (A OR B).

Theserulestransformlogicalmodels,revealproperties,andsimplifyexpressions.Let’ssolveaproblem:

HOTSERVER Aservercrashesifit’soverheatingwhile the airconditioningisoff.Italsocrashesifit’soverheating anditschassiscoolerfails.Inwhichconditionsdoesthe serverwork?

Modelingitinlogicalvariables,theconditionsfortheserverto crashcanbestatedinasingleexpression:

A : Serveroverheats.

B : Airconditioningoff.

C : Chassiscoolerfails.

D : Servercrashes. (A AND B) OR (A AND C ) → D.

Usingdistributivity,wefactorizetheexpression: A AND (B OR C ) → D.

Theserverworkswhen( ! D ).Thecontrapositivereads:

! D → ! (A AND (B OR C )).

WeuseDeMorgan’sLawtoremoveparentheses:

! D → ! A OR! (B OR C ).

ApplyingDeMorgan’sLawagain,

! D → ! A OR ( ! B AND! C ).

Thisexpressiontellsusthatwhenevertheserverworks,either ! A (it’snotoverheating),or ! B AND! C (bothairconditioning and chassiscoolerareworking).

T365hTable4

Anotherwaytoanalyzelogicalmodelsischeckingwhathappens inallpossibleconfigurationsofitsvariables.A truthtable hasa columnforeachvariable.Rowsrepresentpossiblecombinations ofvariablestates.

Onevariablerequirestworows:inonethevariableisset True , intheother False .Toaddavariable,weduplicatetherows.We setthenewvariable True intheoriginalrows,and False inthe duplicatedrows(fig.1.5).Thetruthtablesizedoublesforeach addedvariable,soitcanonlybeconstructedforafewvariables. 7

Fig63e . Tableslistingtheconfigurationsof – logicalvariables. Let’sseehowatruthtablecanbeusedtoanalyzeaproblem.

FRAGILESYSTEM Wehavetocreateadatabasesystem with thefollowingrequirements:

I:Ifthedatabaseislocked,wecansavedata.

II:Adatabaselockonafullwritequeuecannothappen.

III:Eitherthewritequeueisfull,orthecacheisloaded.

IV:Ifthecacheisloaded,thedatabasecannotbelocked.

Isthispossible?Underwhichconditionswillitwork?

7 A truthtablefor30variableswouldhavemorethanabillionrows.

Firstwetransformeachrequirementintoalogicalexpression.This databasesystemcanbemodeledusingfourvariables:

A : Databaseislocked.

B : Abletosavedata.

C : Writequeueisfull.

D : Cacheisloaded.

I : A → B.

II : ! (A AND C ).

III : C OR D.

IV : D → ! A.

Wethencreateatruthtablewithallpossibleconfigurations.Extra columnsareaddedtochecktherequirements.

Table . Truthtableforexploringthevalidityoffourexpressions.

Allrequirementsaremetinstates9–11and13–15.Inthesestates, A = False , meaningthedatabasecan’teverbelocked.Notice thecachewillnotbeloadedonlyinstates10and14.

Totestwhatyou’velearned,solvetheZebraPuzzle. 8 It’sa famouslogicproblemwronglyattributedtoEinstein.Theysayonly 2%ofpeoplecansolveit,butIdoubtthat.Usingabigtruthtable andcorrectlysimplifyingandcombininglogicstatements,I’msure you’llcrackit.

Wheneveryou’redealingwiththingsthatassumeoneoftwo possibilities,remembertheycanbemodeledaslogicvariables.This way,it’seasytoderiveexpressions,simplifythem,anddrawconclusions.Let’snowseethemostimpressiveapplicationoflogic:the designofelectroniccomputers.

LogicinCom165ing

Groupsoflogicalvariablescanrepresentnumbersinbinaryform. 9 Logicoperationsonbinarydigitscanbecombinedtoperformgeneralcalculations. Logicgates performlogicoperationsonelectric current.Theyareusedinelectricalcircuitsthatcanperformcalculationsatveryhighspeeds.

Alogicgatereceivesvaluesthroughinputwires,performsits operation,andplacestheresultonitsoutputwire.Thereare AND gates, OR gates, XOR gates,andmore. True and False arerepresentedbyelectriccurrentswithhighorlowvoltage.Usinggates, complexlogicalexpressionscanbecomputednearinstantly.For example,thiselectricalcircuitsumstwonumbers:

Fig63e . Acircuittosum -bitnumbersgivenbypairsoflogical variables A1A0 and B1B0 intoa -bitnumber S2S1S0 .

8 http://code.energy/zebra- puzzle.

9 True =1, False =0 .Ifyouhavenoideawhy inbinaryrepresents thenumber5,checkAppendixIforanexplanationofnumbersystems.

Let’sseehowthiscircuitworks.Takeaminutetofollowtheoperationsperformedbythecircuittorealizehowthemagichappens:

Fig63e . Calculating 2 +3=5 inbinary, + = .

Totakeadvantageofthisfastformofcomputing,wetransform numericalproblemstotheirbinary/logicalform.Truthtableshelp modelandtestcircuits.Booleanalgebrasimplifiesexpressionsand thussimplifiescircuits.

Atfirst,gatesweremadewithbulky,inefficientandexpensive electricalvalves.Oncevalveswerereplacedwithtransistors,logic gatescouldbeproducedenmasse.Andwekeptdiscoveringways tomaketransistorssmallerandsmaller. 10 Theworkingprinciples ofthemodernCPUarestillbasedonbooleanalgebra.Amodern CPUisjustacircuitofmillionsofmicroscopicwiresandlogicgates thatmanipulateelectriccurrentsofinformation.

. Co6n5ing

It’simportanttocountthingscorrectly—you’llhavetodoitmany timeswhenworkingwithcomputationalproblems. 11 Themathin thissectionwillbemorecomplex,butdon’tbescared.Somepeople thinktheycan’tbegoodcodersbecausetheythinkthey’rebadat math.Well,Ifailedhighschoolmath ,yethereIam .The math thatmakesagoodcoderisnotwhat’srequiredintypicalmath examsfromschools.

10 In 2016,researcherscreatedworkingtransistorsona1nmscale.Forreference,agold atom is0.15nmwide.

11 Counting and Logic belongtoanimportantfieldtocomputersciencecalled DiscreteMathematics .

Outsideschool,formulasandstep-by-stepproceduresaren’t memorized.TheyarelookedupontheInternetwhenneeded.Calculationsmustn’tbeinpenandpaper.Whatagoodcoderrequires isintuition.Learningaboutcountingproblemswillstrengthenthat intuition.Let’snowgrindthroughabunchoftoolsstepbystep: multiplications,permutations,combinationsandsums.

M6l5i1lying

Ifaneventhappensin n differentways,andaanothereventhappensin m differentways,thenumberofdifferentwaysbothevents canhappenis n × m .Forexample:

CRACKINGTHECODE APINcodeiscomposedoftwo digits andaletter.IttakesonesecondtotryaPIN.Inthe worstcase,howmuchtimedoweneedtocrackaPIN?

Twodigitscanbechosenin100ways(00-99)andaletterin26 ways(A-Z).Therefore,thereare 100 × 26=2, 600 possiblePINs. Intheworstcase,wehavetotryeverysinglePINuntilwefindthe rightone.After2,600seconds(43minutes),we’llhavecrackedit.

TEAMBUILDING Thereare23candidateswhowant to joinyourteam.Foreachcandidate,youtossacoinand onlyhireifitshowsheads.Howmanyteamconfigurations arepossible?

Beforehiring,theonlypossibleteamconfigurationisyoualone. Eachcointossthendoublesthenumberofpossibleconfigurations. Thishastobedone23times,sowecompute2 tothepower23 : 2 × 2 ×···× 2 23 times =223 =8, 388, 608 teamconfigurations.

Notethatoneoftheseconfigurationsisstillyoualone.

Pe3m65a5ion4

Ifwehave n items,wecanorderthemin n factorial( n! )different ways.Thefactorialisexplosive,itgetstoenormousnumbersfor smallvaluesof n .Ifyouarenotfamiliar, n!= n × (n 1) × (n 2) ×···× 2 × 1.

It’seasytosee n! isthenumberofways n itemscanbeordered.In howmanywayscanyouchooseafirstitemamong n ?Afterthefirst itemwaschosen,inhowmanywayscanyouchooseasecondone? Afterwards,howmanyoptionsareleftforathird?Thinkaboutit, thenwe’llmoveontomoreexamples. 12

TRAVELINGSALESMAN

Yourtruckcompanydeliversto 15 cities.Youwanttoknowinwhatordertoservethese citiestominimizegasconsumption.Ifittakesamicrosecondtocalculatethelengthofoneroute,howlongdoesit taketocomputethelengthofallpossibleroutes?

Eachpermutationofthe15citiesisadifferentroute.Thefactorial isthenumberofdistinctpermutations,sothereare 15!=15 × 14 ×···× 1 ≈ 1.3 trillionroutes.Thatinmicrosecondsisroughly equivalentto15days.Ifinsteadyouhad20cities,itwouldtake 77thousandyears .

THEPRECIOUSTUNE

Amusicianisstudyingascale with 13differentnotes.Shewantsyoutorenderallpossiblemelodiesthatusesixnotesonly.Eachnoteshouldplay oncepermelody,andeachsix-notemelodyshouldplayfor onesecond.Howmuchaudioruntimeissheaskingfor?

Wewanttocountpermutationsofsixoutofthe13notes.Toignore permutationsofunusednotes,wemuststopdevelopingthefactorialafterthesixthfactor.Formally, n!/(n m)! isthenumberof possiblepermutationsof m outof n possibleitems.Inourcase:

12 By convention, 0!=1 .Wesaythere’sonewaytoorderzeroitems.

(13 6)! = 13 × 12 × 11 × 10 × 9 × 8 × 7!

= 13 × 12 × 11 × 10 × 9 × 8

6 factors

=1, 235, 520 melodies.

That’sover1.2millionone-secondmelodies—itwouldtake343 hourstolistentoeverything.Betterconvincethemusiciantofind theperfectmelodysomeotherway.

Pe3m65a5ion48i5hIden5icalI5em4

Thefactorial n! overcountsthenumberofwaystoorder n items ifsomeareidentical.Identicalitemsswappingtheirpositions shouldn’tcountasadifferentpermutation. Inasequenceof n itemsofwhich r areidentical,thereare r! waystoreorderidenticalitems.Thus, n! countseachdistinctpermutation r! times.Togetthenumberofdistinctpermutations,we needtodivide n! bythisovercountfactor.Forinstance,thenumber ofdistinctpermutationsoftheletters“ COD EE N E RGY ” is 10!/3! .

PLAYINGWITHDNA

AbiologistisstudyingaDNAsegment relatedtoageneticdisease.Thesegmentismade of23basepairs,where9mustbe A-T ,14mustbe G-C . ShewantstorunasimulationtaskoneverypossibleDNA segmenthavingthesenumbersofbasepairs.Howmany simulationtasksisshelookingat?

Firstwecalculateallpossiblepermutationsofthe23basepairs. Thenwedividetheresulttoaccountforthe9repeated A-T and the14repeated G-C basepairs: 23!/(9! × 14!)=817, 190 basepairpermutations.

Buttheproblemisn’tover.Consideringorientationofbasepairs:

A T C G A T

isn’tthesameas

Foreachsequenceof23basepairs,thereare 223 distinctorientationconfigurations.Therefore,thetotalis: 817, 190 × 223 ≈ 7 trillionsequences.

Andthat’sforatiny23basepairsequencewithaknowndistribution.ThesmallestreplicableDNAknownsofararefromtheminuscule Porcinecircovirus ,andithas1,800basepairs!DNAcodeand lifearetrulyamazingfromatechnologicalpointofview.It’scrazy: humanDNAhasabout3billionbasepairs,replicatedineachofthe 3trillioncellsofthehumanbody.

Combina5ion4

Pictureadeckof13cardscontainingall spades.Howmanyways can youdealsixcardstoyouropponent?We’veseen 13!/(13 6)! isthenumberofpermutationsofsixoutof13possibleitems. Sincetheorderofthesixcardsdoesn’tmatter,wemustdividethis by 6! toobtain: 13! 6!(13 6)! =1, 716 combinations.

Thebinomial ( n m) isthenumberofwaystoselect m itemsoutof asetof n items,regardlessoforder: ( n m) = n! m!(n m)! .

Thebinomialisread“ n choose m ”.

CHESSQUEENS Youhaveanemptychessboardand8 queens, whichcanbeplacedanywhereontheboard.In howmanydifferentwayscanthequeensbeplaced?

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Asphodelus microcarpus Viv.

Urginea maritima Bak.

batssal-'untssul (C. ),

Drog. Forsk. A.

batssal-el-hhánasch (Mar. )

bauāl (Nilt. F.)

bauual (Unt. A.)

beddān (C. )

Ornithogalum tenuifolium Guss.

Notobasis syriaca Cass.

Zygophyllum album L.

Mangifera indica L.

bēdd-el-ardd (Ros. )

bēdd-hommār (Sq.)

bēdd-el-'óschar (Nilt., ar. W.)

bedingān (Nilt. C. Al.)

bedingān-el-qūta (Nilt. C.)

behhēme (ar. W. )

békkem (Nilt. C.)

belbel (Al. )

S. A. S. A.

belbel (Al., lib. W. )

belbel (Unt. A.)

belbel (lib. W.)

Crepis bulbosa Tausch.

Panicum repens L.

Calotropis procera R.Br. (Frucht)

Solanum Melongena L.

Solanum aethiopicum L.

Odontospermum graveolens Sz.B.

Reseda luteola L.

Anabasis articulata Mq.T.

Haloxylon articulatum Bge.

Zygophyllum album L.

Zygophyllum coccineum L.

belbūsch (Al. )

Ehrenberg A.

!belessān (Nilt. Al.)

belhh. bélehh (Nilt. F. O.)

Bellevalia sessiliflora Kth.

Sambucus nigra L.

Phoenix dactylifera L. — (die frische Frucht).

bellahh-ma'īs (Lqs.)

bélle (Al.)

bellēch (O.)

benadūra (C.)

benéfschig. benéfssig (Nilt. C. Al.)

benefssig-frengi (C.)

beng. bing (Al., C. )

Drog.

Senecio aegyptius L.

Elaeagnus hortensis M.B.

Conyza Bovei D.C.

Lycopersicon esculentum Mill. (die Frucht).

Viola odorata L.

Duranta Plumieri Jacq.

Hyoscyamus albus L. (Kraut).

Fig.

beni-esch-schām (Brll. ) Imperata cylindrica P.B.

berberīss (C. )

Drog. Forsk.

Berberis vulgaris L. (Frucht)

berim-schan (Mens. ) Imperata cylindrica P.B.

berssīm (Nilt.)

berssīm-hhegāsi (Nilt.)

beschāft (Mens. Dam.)

beschmēleq (Nilt.)

bessílle (Nilt.)

bessūm (Nilt. )

bihēme (ar. W. )

bilēha ( )

bint-el-qóntssul (Al., C.)

Trifolium alexandrinum L.

Medicago sativa L.

Panicum colonum L.

Eriobotrya japonica Lindl.

Pisum sativum Alef.

Senecio aegyptius L.

Stipa tortilis Df.

Lippia nodiflora Rich.

Euphorbia pulcherrima W.

birdi (Nilt. , lib. W. Typha angustata Bor. Chamb.

A. A. Haut. A. S., W.

F.)

!bischenīn (Unt. A., O.)

bisr-el-kattūna (C. )

Drog.

Handl.

bisr-keschūth (C. )

Drog. Forsk.

bisr-maru (C. )

Drog. Forsk.

bissbāss (C. )

bissélla (Nilt.)

blīha. blēha (Nilt. )

Drog. Fig., Forsk. Forsk., S.

Nymphaea caerulea Sav.

Plantago ramosa Asch.

Cuscuta sp. (Same)

Origanum syriacum B.? (Same)

Myristica fragrans Hout. — (Frucht und Blüte).

Pisum sativum Alef.

Reseda luteola L.

bokkár (ar. W. ) Panicum turgidum D.

borrejtt (ar. W. ) Dipcadi serotinum Med.

Forsk. Forsk.

bortughal (C. )

Citrus Aurantium dulcis L. bortuqān (Nilt.)

boruaq (ar. W )

Asphodelus tenuifolius Cav

bottm (ar. W. ) Pistacia khinjuk Stocks.

bu-duēss (Dam., O. ) Imperata cylindrica P.B.

bu-lefen (Mar. ) Ifloga spicata Sz.B.

bunn (C., Nilt.)

Forsk. S., Forsk. M. A. A. S., Forsk.

Coffea arabica L. (Same)

burāq (ar. W. ) Asphodelus tenuifolius Cav.

burbēt (Nilt.)

Cyperus rotundus L.

burbēt (Unt. A. )

burdi (Nilt. , lib. W. F.)

burghl (ar. W. )

burkān (ar W. )

Cyperus laevigatus L.

Typha angustata Bor. Chamb.

Atriplex leucoclada B.

Centaurea Scoparia Sieb.

burrēd (el-Ar. ) Dipcadi serotinum Med. (= D. erythraeum Webb.).

!būss (Nilt.)

busīr (C. ) Verbascum Thapsus L. (Blüte)

Phragmites communis Trin.

buss-el-kelbe (ar. W. ) Zygophyllum decumbens D.

būss-fārisi (Unt. A. ) Arundo Donax L.

būss-haggni (Nilt. )

Drog. Fig. M. A. A. Forsk.

būss-giddaui (Ros. ) Saccharum biflorum F.

Arundo Donax L.

butssēl (Mar. ) Iris Sisyrinchium L.

butssēl (Ab. ) Muscari comosum Mill.

chafélla (ar. W. )

S., A. A. A. Abde., Kl.

!chafūr (Nilt.

Avena Wiestii St. A. S., W. M. M. A

butssēl (el-Ar. ) Urginea maritima Bak.

S., Forsk.,

Glossonema Boveanum Dcne. — (die Frucht)

A. M., Forsk., D., Ehrenberg, S.

ar. W. ) ⎱

chafūr (el-Ar. )

Avena fatua L.

Scleropoa memphitica Parl.

chalāf (C. ) Salix aegyptiaca L.

chalfi (C. )

A. Forsk. Forsk. A.

Eragrostis cynosuroides R. Sch.

chamīreh (el-Ar. ) Colchicum Ritchii R.Br.

chanīn (Matarieh in Unt. A. )

chantssaret-el-'arūss (ar. W. )

A. Forsk.

!chantsser-el-'arūss (ar. W. )

chanūf (Ob. A.)

charbaq-abjadd (C. )

Drog. Forsk.

charbaq-issuud (C. )

Panicum Crus-Galli L. var. Sieberiana A. Sf.

Astragalus trimestris L.

Astragalus Sieberi D.C.

Pulicaria crispa Cass.

Veratrum album L. (Wurzel)

Helleborus niger L. (Wurzelstock)

charchāfti (Nilt. )

chardal (Nilt.)

Drog. Forsk. D. Drog.

Ulmus campestris L.

Brassica lanceolata Lange.

Brassica nigra Koch.

chardal (C.), chardalabjadd (C. )

charfār (Unt. A. )

charīg (Mar.)

Sinapis alba L.

Phalaris minor Retz.

Vicia calcarata Df.

charna (ar. W. )

Ehrenberg M.

Salvia palaestina Bth.

charraq-el-bahhr (Ros.

Unt. A. )

Xanthium strumarium L.

chárrua (Faraskur )

charschūf (Nilt. Mar.) ⎰ ⎱

chárua'a (Ros. ), chárua (Nilt.)

charūb. charrūb (Nilt.)

chasāma (ar. W. )

Forsk., D. S. Forsk. M.

chaschāb (Ass.)

chaschab-el-ambieh (C. )

Drog. Fig.

chaschab-el-enbīja (C. ),

Drog. Forsk.

chaschab-el-qaddissīn (C. )

chaschchāsch (C.)

Hibiscus Trionum L.

Cynara Scolymus L.

Cynara Sibthorpiana B.

Ricinus communis L.

Ceratonia Siliqua L.

Reseda pruinosa D.

Acacia laeta R.Br.

Guaiacum officinale L. (Holz)

Guaiacum sanctum L. (Holz)

Papaver somniferum L.

chaschīr (ar. W. ) ⎰ ⎱

chasma (ar. W. )

chasraqūt (Ob. A. )

chas.s (Nilt. ), chass.ss (C. )

Drog. Forsk. M., Forsk. W. A. S. Forsk.

chássag (Al.)

chas.s-kelāb (Lqs.)

Echinops spinosus L.

Echin. galalensis Schf.

Salvia lanigera Poir.

Withania somnifera Dun.

Lactuca sativa L.

Medicago ciliaris W.

Scorzonera hispanica L.

chatmīje (Nilt. C.)

!chelle (Nilt. Al.)

chelle (Nilt.)

cherīt (ar. W )

Althaea ficifolia Cav.

Ammi Visnaga Lam.

Ammi majus L.

Salsola foetida D.

cheserān (ar. W. ) Amberboa tubuliflora Murb.

M. Forsk.

S., A.

!chēta (Mar. , Brll. ) Iris Sisyrinchium L.

chēta (Mar.)

chiār. chijār (Nilt.)

chiār-frengi (Al., C.)

Gladiolus segetum Gawl.

Cucumis Melo L. var. Dudaim Naud.

Cucumis sativus L.

chiār-schambar (Dam., C.) Cassia Fistula L.

chillāl. chillān. (el-Ar. ) Ammi Visnaga Lam.

chirschēf (el-Ar. )

chitēmi (Mens.)

A. A. A.

Gymnarrhena micrantha Df.

Reseda decursiva F.

Lippia nodiflora Rich.

chobbēsi (Nilt. Ob. A.) Rhynchosia Memnonia D.C.

chobbēs Malva parviflora L.

chōch (Nilt.)

chodar (ar. W )

Abde., S., Kl.

cholongan (C. )

chorēssah (Brll. )

Prunus Persica Sieb. Zucc.

Trichodesma africanum R.Br

Polypodium Calaguala Kz. (Wurzelstock)

Zygophyllum album L.

chosāme (C. Lavandula Spica Cav.

Drog. Fig. A. Drog.

)

chóscheruf (ar. W. , lib. W.)

chóscheruf (ar. W.)

choschjēn (C., Al. )

Forsk.

Carduncellus eriocephalus B.

Atractylis flava Df.

Helianthemum rosmarinifolium Prsl.

chrēss. chrēsi (Al. Abq. ) Salicornia fruticosa L.

chrēssi (ar. W. ) Zygophyllum album L.

!chrīt. chrījet. chrēt (ar. W )

Salsola foetida D.

chubb (Mahsama ) Carex divisa Huds.

chubbēse (Nilt., ar. W.) Malva parviflora L.

chudān (Mar.)

Zollikoferia nudicaulis B.

chu'ēteme (el-Ar. ) Convolvulus stachydifolius Chois.

chulengān (C. )

Drog. Forsk.

Alpinia Galanga S. W. (Wurzelstock)

churm-el-ibra (Brll. ) Koniga maritima R.Br

chusāme (ar. W ) Reseda pruinosa D.

dahami (ar. W. ) Echium longifolium D. Forsk.

dahāmi (ar. W., C.) ⎰ ⎱

Erodium bryoniifolium B.

Erodium laciniatum W.

dahasīr (Ssudan )

Koenig

dahmeh (ar. W. )

Indigofera oblongifolia F. (= Ind. paucifolia D.)

Erodium arborescens W.

dam'a (C. ) Coix lacryma L.

M. Forsk. S.

damm-el-achuēn (C. )

Drog., S., Fig.

damassēna (Ass. ) Ambrosia maritima L.

Dracaena Cinnabari Balf. (Harz)

danūn-el-ādirr (el-Ar. ) Orobanche cernua Lœfl.

A. Forsk.

darákrak (C. ) Trigonella hamosa L.

dār-filfil (C. ) Piper Chaba Hout.

datūra (Al. )

Drog. Forsk. A.

deb'áhh (ar. W. Mar. ) ⎰ ⎱

Datura Stramonium L.

Scorzonera alexandrina B.

defle (C. Al.)

A. Forsk. D.

Scorzon. Schweinfurthii B.

debscheh (Nilt. ) Scirpus maritimus L.

Nerium Oleander L.

dehorāg (Nilt. ) Vicia calcarata Df.

dekīss (Al. ) Plantago crypsoides B.

dēl-el-qutt.tt (Mahsama ) Imperata cylindrica P.B.

demssīss (Nilt. Lk. )

A. A. Forsk. S.

!derēri (ar. W. ) Aristida plumosa L. A.

Ambrosia maritima L.

derēre (Unt. A. ) Aristida lanata F.

Forsk. M.

derēsse (Ob. A. Lqs.)

Medicago hispida Urb.

ddáffera (ar. W. ), sáfara (Sinai)

Iphiona mucronata A. Sf. Iphiona scabra D.C.

ddummeiri (C. ) Cucumis Melo L. var. Chate Naud. — forma

ddurra. ddurra-béledi (Nilt.)

ddurra-'auuēge (Nilt.)

ddurra-schāmi (Nilt.)

ddurra schítaui (Nilt.)

ddurra-tssēfi (Nilt.)

dhalāthli. dhelādhli (ar. Chenolea arabica B. M., Forsk. Forsk.

Andropogon Sorghum Brot.

Andropogon Sorghum Brot. var. Schweinfurthianus Kcke.

Zea Mays L.

Andropogon Sorghum Brot. var. niloticus Kcke.

Andropogon Sorghum Brot. var. Schweinfurthianus Kcke.

W. )

dhamarān (ar. W. ) Traganum nudatum D.

!dhamarān (ar. W ) Salsola tetrandra F. (= Salsola tetragona D.)

dhanūn (ar. W. )

M., S.,

Orobanche cernua Lœfl.

W. W.

dhanūn (ar. W. )

dhanūn-el-dschin (Isth. )

dhaqn-el-bascha (Nilt. C.)

dhauanīn (ar. W. )

dhēl-el-fār (Nilt.)

dhēl-el-qott.tt (Ros. , Mens.)

M. S.,

dhēl-el-thāleb (Nilt.)

dhēl-thā'leb (Mar.)

Cistanche lutea Lk. Hoffm.

Albizia lebbeck Bth. — (die Blüte).

Orobanche cernua Lœfl.

Polypogon monspeliensis Df.

dhenāba. dhenebān (ar W. )

dhéneb-el-fār (Nilt.)

dhenebān (ar. W. , elAr. )

A. Forsk. M. A.

dhenebūn (ar. W. )

MteratAr.

dherag (Nilt.)

!dhinēb (Unt. A. Sq.)

Bromus rubens L.

Caylusea canescens St.H.

Polypogon monspeliensis Df.

Oligomeris subulata Webb.

Trigonella laciniata L.

Panicum Crus-Galli L.

dhirr (ar. W. )

Traganum nudatum D.

difl (C. Al.)

díffre (Lqs.)

diker-el-fūl (Nilt. )

Nerium Oleander L.

Panicum colonum L.

Orobanche crenata F.

dimma-Ajūb (C. ) Coix lacryma L.

dimssīss (Nilt.)

diqn-el-bedden (ar. W. )

dirr (Al.)

Conyza aegyptiaca Ait.

Centaurea eryngioides Lam.

Noaea mucronata A. Sf.

dirss-el-'agūs (Ros. ) Emex spinosa Camp.

dīss (Mens. ) Cyperus auricomus Sieb.

dīss (Nilt. ) Cyperus alopecuroides Rottb.

dobbāri (ar. W. ) Atractylis flava Df.

dochn (Ob. A. Ssudān) Pennisetum spicatum Kcke. (= P. americanum K.Schum.)

dochrēq (Ros. Dam. ) Vicia sativa L.

dohhrēg (Ros. ) Vicia calcarata Df.

ed-drēissi (ar. W. C. )

A. Forsk. M. A. A. S., W. Haut. A. A. Forsk.

Tribulus alatus D.

duchān (Nilt.)

duchān-béledi (Nilt.)

dulb (C.)

dul-ghētt (Al. )

dūm (Ob. A.)

dumssēn (Si.)

durrēhss (el-Ar. )

'echschīr (ar. W. )

edcher-mekki (C. )

egdīm (ar. W. )

Haut. Drog. Forsk. W.

endivia (Nilt., O.)

'éneb (Nilt. F. O.)

'eneb-ed-dīb (Nilt.)

'eneb-ed-dīb (Ass.)

'ennāb (Nilt. Al.)

Nicotiana tabacum L.

Nicotiana rustica L.

Platanus orientalis L.

Chrysanthemum Coronarium L.

Hyphaene thebaica Mart.

Chrozophora plicata A.Juss.

Hippocrepis bicontorta Loisl.

Echinops spinosus L.

Ech. galalensis Schf.

Andropogon Schoenanthus L. (Wurzelst.)

Andropogon Nardus L. (Wurzelst.)

Helianthemum Kahiricum D.

Cichorium Endivia L.

Vitis vinifera L. (die Beere).

Solanum nigrum L.

Solanum villosum Lam.

Cissus ibuensis D.C.

Ziziphus vulgaris Lam.

'ēn-qott.tt (ar. W. ) Anthemis retusa D.

entābe (Mens.)

Forsk. Forsk.

Alternanthera sessilis R.Br.

'erejām (ar. W. ) Bassia muricata L.

el-'ern (ar. W. )

Rhus oxyacantha Cav. erqēta (el-Ar. ) Helicophyllum crassipes Schott.

'erq-ikar (C. )

M. A. Drog. Forsk.

'erq-lingibār (Ros. Markt.)

'erq-edh-dháhab (C. )

Drog. Fig., Forsk.

Acorus calamus L.

Statice Limonium L.

Uragoga Ipecacuanha Baill. (Wurzel)

'erq-edh-dhahab (C. ) Piper Chaba Hout.

Drog. Forsk. Drog.

'erq-el-hhalaūi (C. )

Fig.

'erq-ssūss (Nilt., C., O.)

Gypsophila Struthium L. (Wurzel)

Glycyrrhiza glabra L. erssēl (ar. W. ) Bellevalia flexuosa B.

esbānach (Nilt. Al. C.)

'ēsch (Ssudan)

M. Forsk.

Spinacia glabra Mill. Spinacia oleracea Mill.

Andropogon Sorghum Brot.

'eschb (Al. ) Lotus creticus L.

'escheb (ar. W. Qossēr ) Lotononis dichotoma B.

'éscheb (Unt. A. ) Medicago ciliaris W.

'escheb-ed-dīb (ar. W. ) Linaria aegyptiaca Dum.Cours.

Abde. A. Forsk.

'ēsch-el-ghorāb (Nilt. ) Agaricaceae

'ēsch-er-rīf (Ssud. arab.) Zea Mays L.

'eschūb (ar. W. , el-Ar. )

S. Abde., Kl., S. A.

Cyperus conglomeratus Rottb.

'ēsch-u-gibne (Ab. ) Raphanus Raphanistrum L.

estachudess (C. )

A. Drog. Forsk.

Lavandula Stoechas L.

ethbáhh (ar. W. ) Scorzonera alexandrina B.

etírr (ar. W. )

'ettr (Nilt.)

faq.qūtss (Nilt.)

Glossonema Boveanum Dcne.

Pelargonium Radula Ait.

Cucumis Melo L. var. Chate Naud. (forma).

fatssūlia (C. ) Phaseolus vulgaris L.

felfel-ahhmar (Nilt.)

Capsicum annuum L.

el-fessīre (C. ) Bryonia alba L.

!figl (C. Nilt. O.)

W. Abde., S., Kl. Sickenberger Drog. Fig. A.

Raphanus sativus L.

figl-el-gemāl (Ros. ) Sisymbrium Irio L.

figl-el-gemāl (Al. )

Cakile maritima Scop.

figl-el-gemāl (Ros. ) Brassica Tournefortii Gouan.

fileījeh. fleīeh (Nilt.)

!filfel (C. Nilt.)

filfil-malti (Nilt. Al.)

fílfil-rūmi (C.)

foqqēsch (O. )

!fóstuq. fústuq (Nilt. C.)

Mentha Pulegium L.

Piper nigrum L. (Frucht).

Schinus molle L.

Capsicum annuum L.

Withania somnifera Dun.

Pistacia vera L. (Frucht).

frassijūn (Al. ) Marrubium Alysson L.

frēs (Nilt., C.)

fūa (Nilt.)

fuggēla (Al. )

fūl (Nilt. F. O.)

fūl-el-'arab (O. )

fūl-iblīss (Nilt. )

fuqqēla (el-Ar. )

Fragaria grandiflora Ehrh. (Frucht).

Rubia tinctorum L.

Cakile maritima Scop.

Vicia Faba L.

Vaccaria segetalis Gcke.

Vicia narbonensis L.

Brassica Tournefortii Gouan.

fuschfāsch. (el-Ar. ) Statice pruinosa L.

fustuq (Al. C.)

fustuq-scherqi (C. )

fuuet-etss-tssabbaghīn (C. )

A. A. Forsk., D. A. A. A. A. A. Drog. S. Drog. Forsk.

Pinus Pinea L. (Same).

Pistacia Lentiscus L. (Frucht).

Rubia tinctorum L.

ga'āde. dscha'āde (ar. W. )

!gachuān. gochuān (Mar. , Al. )

dsch

Teucrium leucocladum B.

Teucrium pilosum A. Sch.

Teucrium Polium L.

Chrysanthemum Coronarium L.

S. Forsk. M.

gaddīm (ar. W. )

Helianthemum Sancti-Antoni Schf.

gáhhui (C. ) Benzoin officinale Hayne (Harz).

galāb (C. )

Drog. Fig.

Drog. Fig., Forsk.

Exogonium purga Bth. (Wurzel).

garād (ar. W. ) Gymnocarpos decander F.

garaiai'īf (ar. W. )

M., Forsk. M

Phagnalon nitidum Fres.

Phagnalon Barbeyanum B.

garbā. dscharba (ar. W. ) Farsetia aegyptiaca T.

gargas (ar. W. ) Trigonella stellata F.

gármal (ar. W. C.)

garna (ar. W. )

garūnijeh (Nilt. C.)

gatba (ar. W. )

Forsk. Forsk. Forsk. F.

Zygophyllum simplex L.

Erodium malacoides W.

Pelargonium zonale W.

Tribulus pentandrus F.

gauuīn (ar. W. )

gehauān (Al. )

M. A.

Linaria aegyptiaca Dum.Cours.

Calendula officinalis L.

gehennamīe (Al. C.)

Bougainvillea spectabilis W.

gelauīl (Nilt. )

A., D.

gelauīn. gelauēn (Dam. Ros. )

gellauēn (Nilt. Ros.)

Sonchus glaucescens Jord.

A. Forsk. Haut., S.

Sonchus oleraceus L.

gēmde (ar. W ) Fagonia glutinosa D.

génem (ar. W , el-Ar. )

Plantago cylindrica F

genēme (ar. W. )

gera'īt (ar. W. )

A. M. M

geraū (Nilt. Lqs. C.)

!gergīr (Nilt.)

dscherdschīr (ar. W )

Plantago ovata F.

Phagnalon Barbeyanum B.

Phagnalon nitidum Fres.

Andropogon halepensis Brot.

Eruca sativa Lam.

Senecio coronopifolius Df. dscherdschīr-el-dschebel (el-Ar. )

gereibīeh (ar. W. ) Farsetia aegyptiaca T.

A. Haut. Drog. Fig.

gerfe (C. ) Cinnamomum zeylanicum Breyn. (Rinde).

gēri (Ob. A., Lq., Si.)

dscherrād (ar. W. )

M., Forsk.

„gerres“ (Nilt.)

géser (Nilt.), dschesar

Brassica nigra Koch.

Gymnocarpos decander F.

Triticum vulgare Lam. (das Ährchen).

Daucus Carota L. var. Boissieri Wittm.

(O. )

gettiāt (Nilt.)

Psoralea plicata D.

ghāb (Nilt. )

Phragmites communis Trin.

ghāb-rūmi (Dam. ) Arundo Donax L.

ghalga (ar. W. )

ghalqet-ed-dīb (C. )

ghannūm (ar. W. )

A., Ehrenberg, S. S., A. Haut. Forsk. M.

ghār (C.)

ghardaq. gharqad (ar. W. )

Daemia tomentosa Vatke.

Peganum Harmala L.

Globularia arabica J. Sp.

Laurus nobilis L.

Nitraria retusa Asch. gharghed (ar. W. )

S., Forsk. S., Forsk.

gharghār (C.)

Ulmus campestris L.

!ghassūl (Al. ) Salicornia fruticosa L.

ghassūl (Al.)

ghassūl-frengi (Mar. , Brll. )

Mesembryanthemum nodiflorum L.

Mesembryanthemum crystallinum L.

ghebēschi (ar. W. ) Salvia aegyptiaca L.

A. S. A. M.