mathematics of magic.doc

8/12/2019 Mathematics of Magic.doc

1/29

The Mathematics of Magic: The Gathering

A study in probability, statistics, strategy, and game theory

By Jon Prywes

May-June 1999

Here are some of Jon Prywes' Magic accomplishments:

He wrote a Magic magazine online called he !ibrary o" !eng, "rom 199# to 199$%

He wrote three articles "or &crye Magazine in 199' and 199$%

He started a Magic club at his high school in 199$, which ran through 1999%

He played in se(eral semi-competiti(e tournaments including the 1999 Junior &uper &eries )astern *i(isional%

He wrote numerous articles "or he Magic *o+o "eatured on this page%

He wrote a paper about the mathematical components o" Magic in 1999 also "eatured on this page%

He has done hundreds o" Magic eBay auctions%

He wor.ed at a day camp teaching Magic strategy to .ids in the summer o" 1999

1/09


2/29

Table of Contents

Introduction to Magic: The Gathering

A Mathematical ntroduction22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222223

A *escription o" the 4ame222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222225

A &ample 4ame222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222#

6hy )7perience 8ounts222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222$

TwoPerson Game Theory

6hat *oes t Mean222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222:

he Basic 8oncepts22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222229

wo-Person 4ame heory and Magic; he 4athering2222222222222222222222222222222222222222222222222222222222222222222211

Probability

Probability and Magic; he 4athering 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222215

&hu""ling and ersus &.ill 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222219

!ec"building

*ec.building and Magic; he 4athering 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222220?

*ec. Archetypes and *ec. &trategy 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222201

*ec. 8omparison and 6inning


3/29

Introduction to Magic: The Gathering

$ Mathematical Introduction

Magic; he 4athering is a game which some ta.e seriously, and others lightly2 Many understand it, e(en more do not2 &omeplay it nonstop and others ne(er see the point2 Personally, thin. it is a great game2 @nly those who actually play the gamecan truly understand how wonder"ul it is2 Most people who write it o"" as pointless do not see the mechanics o" the game that

ma.e it simple, yet comple72 n its "i(e years o" e7istence, the crowd that Magic has drawn can alone show how great thegame really is2 @n the wee.end o" May 1 and 0 tra(eled with some "riends to Pro our; ew or. in &ecaucus, ewJersey2 Hundreds o" people were gathered inside the Meadowlands )7position 8enter +ust to play the game they lo(e2 hePro our is no day in the par.% it is a great competition "or the mind2

Ce( 4urwitz, a +unior at &carsdale High &chool, along with mysel", had both Duali"ied "or the Junior &uper &eries e(ent2&i7ty-"our players age se(enteen and under Duali"y "or the J&& at challenge tournaments held during the winter2 6e hadplaytested "or this tournament a "airly lengthy amount, yet was still s.eptical2 As it turned out, went 1-3 and dropped out2Ce( went ?-3 and dropped2 A record o" at least "our wins was necessary to ma.e the top eight, in order to play at the J&&8hampionships where E0#?,??? in scholarship money is awarded2 6e tried, but what was it that .ept us "rom winning 6hatis it that .eeps e(ery Magic player "rom winning all the time 6hat .eeps e(en the most s.illed o" players at bay "romwinning e(ery single game t is a word called chance2

8hance is what ma.es Magic di""erent "rom many other classic games2 6hen one plays 8hess, he gets to see all the pieces2Both players ha(e eight pawns and eight other pieces, either white or blac.2 )(ery piece mo(ed is public in"ormation, andnothing is held secret2 8hess is ruled by complete s.ill2 othing random can possibly happen2 Magic, on the other hand, addsa new element2 ot only are each personFs playing materials o"ten di""erent or (aried the dec.s, but randomness e7ists inthis game2 @ne starts the game by randomizing his dec. o" si7ty or more cards, then drawing se(en2 He will de"initely notdraw the same se(en cards in e(ery opening handG >ariety is what ma.es this game di""erent than 8hess2 here are twoelements o" (ariety in(ol(ed; *ec. construction and random card draws2

*ec. construction is part o" what go(erns this element o" (ariety2 he card draws are simply a result o" the dec. construction,as well as any cards played during the game which may alter the contents o" what is remaining in the dec. "or e7ample,remo(ing cards "rom the dec.2 he *uelistsF 8on(ocation nternational *8 is the authority go(erning sanctionedtournaments2 Although many players do not participate in tournaments, the *8Fs dec.building guidelines pro(ide "or a "airgame2

&ome cards were printed be"ore the research and de(elopment team realized they were o(erpowered2 Hence, the cards arebanned "rom dec. construction2 &ome cards are only restricted, meaning only one is allowed in a dec.2 Any card that is notbanned or restricted is allowed up to "our copies per dec.2 A player has to thin., how many copies o" a card do want in mydec. @ne, two, three, or "ourI " he puts "our copies in, he is li.ely to draw one (ery soon2 " he only uses one, his chanceso" drawing it are slim2 Maybe ha(ing multiple copies is redundant and that is the reason "or using "ewer2 " he uses threecopies o" a card, then what will be his chances o" drawing one by turn si7 Probability po.es its nose up into the game now2

n 8hess, there is no probability, other than thin.ing about the number o" possible mo(es2 Howe(er, a playerFs play style ismore li.ely to go(ern how he plays the game o" 8hess than probability will2 =or all intents and purposes, there is noprobability in 8hess2 Probability goes hand in hand with chance and random e(ents2 t is the chance that an e(ent will occur,or not occur2 6ill your ad(ersary draw the card he needs to win the game this turn e7t turn 6ithin "i(e turns 8alculatingthe chances can help a player decide whether to play de"ensi(ely and anticipate the other person drawing their win card, or

play aggressi(ely and assume that the card will not be drawn by the other player2

&ome o" the other math in(ol(ed in the game is less ob(ious2 he chance (ersus s.illI Duestion is a (ery hot topic amonghigh-le(el players2 How much o" playing a gi(en dec. is based on s.ill, and how much on chance he ob(ious solutionwould be to ta.e a "airly ine7perienced player and gi(e him one dec., and gi(e an e7perienced player the other2 hee7perienced player will win with either dec. i" there is complete s.ill in(ol(ed2 " the dec. is straight"orward to the point thatthere are no decisions to be made, and it beats the other dec. almost automatically, then it is possible the e7pert will losegames2 n Magic, though, there are always at least some thought"ul decisions to ma.e2 he chance o" getting paired against asuperior dec. in a tournament does not mean an automatic loss2 " your s.ill is lower to that o" the superior dec. your chancesmay be slim, but i" your s.ill is higher than your chances become much greater2

3/09


4/29

&o then, what constitutes a superior dec. 6hat dec.building techniDues are necessary "or one to ma.e a dec. that winsmore o"ten than a dec. which is not as good How does one determine which card selections will guarantee a "lawless road to(ictory uestions li.e these are what e(ery player must try to answer when attempting to create the per"ectI dec.2 @"course, will aim to pro(e that the per"ectI dec. does not e7ist2 !aws o" game theory are an in(aluable aid in pro(ing this26hile game theory is a (ery conceptual science, its laws (ery well do apply to games such as Magic2 6hile .nowledge o"these laws will not ma.e one a Pro our Player, they can help one who is trying to understand the game ha(e an easier timema.ing decisions2 4ame theory is simply a series o" laws regulating how one goes about ma.ing decisions in a situation2 AgameI in game theory is not necessarily always a game as de"ined in common tal., but game theory certainly has many

game applications2

hrough the laws o" probability and game theory, along with statistical analysis and actual e7periments, will be wor.ing oncoming up with conclusions that can ta.e these laws and correlate them to the game o" Magic2 =irst and "oremost, will bee7plaining the basic concepts o" game, in order "or the reader who is unacDuainted with Magic to "amiliarize himsel" withthese terms2 n describing the game, will attempt not to reiterate the entire ruleboo., though will summarize the basics o"how the game is played2 Kpon completion o" these details, will begin my outline o" game theory and begin on relating themath and game theory to its Magic applications2 will pro(ide pro"iles o" many o" the people who consult with as part o"my pro+ect2 will be discussing a sur(ey conducted in order to determine how much math Magic players recognize as parto" the game2 his entire report will "ollow closely the outline gi(en in my pro+ect description2

$ !escription of the Game

he ob+ect o" Magic is simple2 ou start with 0? li"e points, and when they are reduced to ?, the game is o(er and the otherperson wins2 ou and another person both ha(e a dec. o" si7ty or more Magic cards, which are used in playing the game2he cards can be "rom a (ariety o" di""erent sets, with "or the most part no more than "our o" any one card2 he playersalternate in ta.ing turns2 )ach turn is a seDuence o" e(ents that in(ol(e drawing a card, putting cards into play, attac.ingIthe other player using cards that represent creatures and then discarding i" necessary2

he basic resource in Magic is called mana2 Mana comes "rom a Polynesian word meaning energy2 Mana let you bringcertain cards into play, and use abilities on cards already in play2 &ome cards reDuire more mana to use than others do2 hereare "i(e di""erent colors o" mana in the game, each representing a separate "orce2 he "orces represented are typical o" anad(enture gaming genre; 6hite mana represents the powers o" good2 Blue mana is "or the powers o" the mind2 Blac. mana is"or the powers o" e(il2


5/29

would be played on a creature2 t might ma.e the creature wea.er, or stronger2 &ome enchantments are not played on othercards, and ha(e a global e""ect on the game2 &orceries and instants cause a one-time e""ect on the game2 &orceries can only beplayed during your turn% instants can be played anytime2

At the beginning o" the game each player shu""les his dec.2 he players roll a die or "lip a coin to determine who chooseswho goes "irst, then each draws se(en cards2 he player who chooses to go "irst does not draw a card on his "irst turn2 heseDuence o" a turn is as "ollows;

Kntap phase; Kntap all cards you control2 his means to rotate them so they are all "acing upward and not rotated tapped2

Kp.eep phase; his is a maintenance phase2 &ome cards will ma.e you do an e""ect during this phase, such as pay mana to.eep the card in play, "or e7ample2

*raw phase; *raw a card2

Main phase; ou can do these things, in any order;

a2 Put one land into play

b2 *eclare one attac.

c2 Play spells2 ou can play spells be"ore or a"ter your attac., as well as both2 ote that all cards e7cept "or instants canonly be played on your turn during the main phase2

*iscard phase; *iscard down to se(en cards2

8leanup phase; Any e""ect that lasts until end o" turnI wears o"" now2 Any damage on a creature, which does not destroy it,wears o"" as well2

he attac. wor.s li.e this; ou choose any untapped creatures you control that you ha(e had in play at least one turn, and tapthem2 our opponent either bloc.s them or ta.es damage eDual to their power2 " he bloc.s, both creatures deal damage toeach other eDual to their power2 A creature has a pair o" numbers in the bottom right corner2 hese are its power andtoughness2 6hen a creature deals damage, it deals damage eDual to its power2 6hen it recei(es, the damage is applied to itstoughness2 " it ta.es damage eDual to or greater than its toughness it will be buried2 hat means it will be played in a pile

ne7t to your draw pile called your discard pile2 =or e7ample, a 4iant &pider has a power and toughness o" 0/52 t deals 0damage to a creature bloc.ing or bloc.ed by it, and i" unbloc.ed during an attac., deals 0 to the player it attac.ed2 " it ta.es5 damage during one turn, it will be placed in its controllerFs discard pile2

he game gets a lot more comple7 than this with the "ramewor. o" the rules, but these are the basics2 6hat the cards do isanother story2 here are a wide (ariety o" abilities in Magic, and this summary is +ust the "oundation2 o "ully understand thegame, one must see it being played2 @therwise the rules are hard to ma.e sense out o"2 t may seem daunting at "irst, but a"termany games, it becomes automatic2 he strategy is immense, and the game lea(es players with many decisions to ma.e2hese decisions are what ma.e the game the strategic tas. it is2

$ %ample Game

Perhaps, at this point, one who is un"amiliar with the game still may be con"used2 A sample game can at least demonstrate the

"low o" the game and gi(e a general idea2 Howe(er, to one who is already "amiliar with Magic, this will be super"luous2 willbe introducing new terms and concepts as the game progresses2 n this e7ample will be playing a dec. which is red and hasmany arti"acts2 My pro(erbial ad(ersary will be using a green dec. with some white cards2 his is based on a game playedonline using Apprentice, a program that allows the online play o" Magic through a graphical user inter"ace2

he li"e total score will be presented in the notation L/, where L is my score and the other personFs2 8ards will be"ootnoted in the notation; 8ard type, casting cost2 8ard te7t2 Power/oughness2 8olored mana symbols will be represented byB "or blac., K "or blue, 4 "or green, < "or red, and 6 "or white2 At the beginning o" the game, both players shu""le theirdec.s and draw se(en cards2 will win coin "lip and decide to go "irst2

#/09


6/29

urn 1 Me; &.ip "irst draw because o" play-draw rule whoe(er goes "irst s.ips his "irst draw phase to cancel out thead(antage o" getting to play "irst2 Play Mountain&(2 ap Mountain to play >oltaic ey&)(2 *eclare end o" turn2 # cards inhand2 0?/0?

urn 1 @pp; *raw a card2 Play =orest&*(2 ap =orest to play )l(ish !yrist&+(2 *eclare end o" turn2 ' cards in hand2 0?/0?

urn 0 Me; Kntap Mountain2 *raw a card2 Play Mountain2 ap two Mountains to play 4rim Monolith&,(2 Playing this cardin con+unction with the >oltaic ey allows "or enhanced mana production, as the ey can be used to untap the Monolith2

*eclare end o" turn2 5 cards in hand2 0?/0?

urn 0 @pp; Kntap =orest2 *raw a card2 Play Brushland&-(2 ap =orest and Brushland to play Priest o" itania&.(2 Attac.with )l(ish !yrist "or 1 damage2 *eclare end o" turn2 # cards in hand2 19/0?

urn 3 Me; Kntap2 Kntap all cards which donFt say otherwise, such as 4rim Monolith which says it does not untap2Howe(er, it can be untapped in other methods2 =or instance, it says you can pay 5 mana at any time to untap it2 he >oltaicey allows you to pay a mana and tap it to untap any arti"act in play, so that, too, can be used to untap the Monolith2Howe(er, it is not tapped right now so that is not a problem2 *raw a card2 Play Mountain2 ap 3 mountains to cast &tone


7/29

4rim Monolith; Arti"act, 02 ap "or 3 generic mana2 *oes not untap normally, and costs 5 to untap any other time2 4enericmana does not ha(e color% they may only be used to play cards or part o" cards not reDuiring colored mana2 ote that coloredmana is downwardly compatible% it may be used to play arti"acts and such2Brushland; !and2 ap "or one green or white mana and ta.e a point o" damage, or tap "or a generic mana2Priest o" itania; 8reature N )l", 42 ap "or a green mana "or e(ery )l" in play2 1/12&tone


8/29

t will always seem that Ce( has countermagic because when am hoping he does not, he will, simply because he .nowswhat to sa(e it "or2 " ha(e a .ey card in my dec., he will wait until play it be"ore using his countermagic2 How hedetermines what are the biggest threats is another comple7ity2 =irst, i" he .nows what cards his opponent has in his dec., hecan .now what to sa(e his countermagic "or2 He will ha(e other ways o" dealing with the smaller, lesser threats2 Being able todeal with an opposing situation and presenting an e""ecti(e opposition yoursel" is what lets you win the game2 Ce( is prettygood at this2

Both playing and dec.building s.ills both de(elop o(er time2 &ome players are better players than dec.builders, and (ice

(ersa2 &ome people ha(e di""erent playing styles2 All o" these "actors are what comprise the many aspects o" many di""erentMagic players2 )ach player has indi(idual characteristics2 6hile one may be able to build a great dec., he may not be able toplay it to its "ull capacity2 He may stumble and ma.e wrong decisions within the game2 his could mean he may counter thewrong spells, use remo(al spells such as &words to Plowshares&-(ine""iciently, or use the wrong mana in playing a spell2Ksing a &word on the "irst creature an opponent plays may be the wrong mo(e2 @r rather, it may be the right mo(e in a gi(ensituation2 A player needs to determine this2 Perhaps the creature played was a Birds o" Paradise&.(2 By using the &words onthis opposing card, the opponentFs mana production will be slowed down2

8on(ersely, the opponent might ha(e better targets "or the &words2 he player has to decide a "ew things2 6ill be able todeal with any other creatures i" waste this remo(al on the Birds 6ill the ad(antage gain by remo(ing the Birds now beworth my using this spell right now he player has to decide whether it is an e""ecti(e strategy based on the resources in hishand and dec. and the resources he thin.s his opponent has2 uestions such as these will help him come to a conclusionabout whether to play the card now or wait until later2 &imply put, to play oneFs cards right is the way to win2 Ma.ing the

right decision is o"ten easier said than done, though2 )(en the best o" players "ace tough decisions2 s there always a beststrategy in this game n answering a Duestion such as that, one can only turn to the logic o" game theory2

2ootnotes:!ightning Bolt; nstant,


9/29

Additionally, a game such as &tratego is classi"ied is a zero-sum game2 A zero-sum game is one where all payo""s add up tozero2 n this instance, a loss is simply a negati(e win, which when added together, combine to zero2 Po.er is also a zero-sumgame2 )(en when not played "or money or some sta.es, there be one clear winner in po.er2 6hen played with -players aloss can simply be accounted "or as a negati(e "raction o" a win i2e2 -1/0 i" there are three players, because two losses andone win add up to ?2 6hen played "or sta.es, the total money a player earns is eDual in magnitude and opposite in sign to themoney the other players lose2 All the money totaled will always eDual the same amount, and the total change in money wonbetween all the players combined will always add up to zero2 here"ore po.er is an -player zero-sum game o" imper"ectin"ormation2

A non-zero-sum game can be more o" a real-li"e application, where negotiation helps two people both succeed2


10/29

@b(iously, player AFs best option would then be to choose scissors2 8learly there is a best strategy in this game2 he beststrategy is to choose the item that would be beaten by the disclosed item that B did not guess in this case, scissors2 =i"typercent o" the time player A will win with this strategy, and "i"ty percent he will tie2 Had player B told player A no paperIinstead o" no roc.I, this strategy would lead to a tie instead o" a win as in this situation2 Knless he does not use this strategy,player A cannot lose2 ow e7amine this e7ample in game tree "orm;

By choosing scissors, player A can ma7imize his winnings in this game2 his is a (ery simple e7ample2 o add somecomple7ity, allow the assignment o" payo""s2 Perhaps player A would be recei(ing a higher payo"" i" he chose roc. instead o"scissors2 o compensate "or his lower chance o" winning, he would need to determine whether the payo""s were "air2 Be"oreadding payo""s, a game matri7 is a method that needs to be introduced2 A matri7 is similar to a grid in where all possibilitiescan be charted2 As used in the pre(ious e7ample, here is the game in matri7 "orm;

Player B no roc.I Player B - Paper Player B - &cissors

Player A -


11/29

Player A -


12/29

in his dec. to bloc. oncoming attac.ers with2 6rath o" 4od will destroy all o" JimFs creatures2 6ith three lands out, Jon canpossibly be holding a 6rath o" 4od in his hand, along with the land he needs to cast it2 here"ore, it may not be such a goodidea "or Jim to play any more creatures2 &eeing that Jon is at 10 li"e, attac.ing twice more and casting the &hoc. will let Jimwin2 " he plays any more creatures, Jim will still ha(e to attac. twice to deal su""icient damage, yet will put Jon on theo""ensi(e2

Howe(er, i" he waits too long, Jon may be able to ta.e control o" the board2 " Jon can play a &erra Angel, he will be able tobloc. one o" JimFs attac.ers2 here"ore ha(ing another creature to attac. with might be a good idea2 Mogg =anatic can be

sacri"iced to deal one damage, so playing it is ne(er too ris.y2 Being able to attac. with it, too, can be a bene"it, nonetheless2" he plays the Mountain, he will not gain a su""icient ad(antage, and may lea(e himsel" susceptible to a mid-gameArmageddon2 " he plays the &hoc. now, he may gi(e up the opportunity to use it on a creature Jon plays, or may +usteliminate the surprise (alue2 Howe(er, Jon may be pretty sure that Jim has one in his hand so that may not be an issue2 hesethree decisions ha(e many "actors a""ecting them, as now demonstrated2 Here is a general summary o" JimFs options andJonFs resulting li"e total;

here"ore, i" Jon plays a 6rath o" 4od ne7t turn he can ha(e a shot at winning the game2 " he does not, he will need one ne7t

turn, as ha(ing a bloc.er alone will not sa(e him2 &ince Jim has a &hoc. in his hand, he can use it to do 0 damage to Jon28ombined with the two Mogg =anatics, he can beat Jon simply by getting him to 5 li"e or less2 Howe(er, i" Jon plays the6rath, Jim will ha(e to sacri"ice his Moggs immediately2 Playing the &hoc. ends up not ma.ing a di""erence because todestroy the &erra Jim will need both the &hoc. and another creature or two2 here"ore the e""orts are better spent on reducingJimFs li"e total2 &ubseDuently, i" Jon is able to get out a &erra a"ter casting a 6rath, Jim cannot do anything about it e7cepthope to draw something good2

6ith the resources he currently has he cannot do anything2 Playing the Mountain, howe(er, does not seem worth it, since i"Jon casts Armageddon, Jim will need to hope to draw another Mountain as well as the additional cards he already needs2hin.ing ahead, Jim realizes the best plan is to play the Mogg =anatic2 &a(ing it will do no good since he already has enoughattac.ers to beat Jon regardless o" the &erra, pro(ided he does not play a 6rath2 By using this all-or-nothing strategy, Jim isassuming that Jon does not ha(e a 6rath to cast ne7t turn2 " he does, the game will be grim "or Jim2 " he has one the turna"ter, that is where the only di""erence comes in2

By sa(ing the Mogg, the ad(antage gained is that Jim will be able to play it a"ter Jon casts the 6rath2 Howe(er, since it is solate in the game, Jon will probably be ready to play his &erra the turn a"ter, or something else that can control the Mogg2Hence it is better i" he plays it and gets in an e7tra attac. with the Mogg2 Being able to put Jon down to 0 li"e, as opposed to 3li"e, will help2 " Jim draws another &hoc. he can win the game right then2 6ithout the e7tra damage, he would need twodirect damage&(cards2 Ksing an all-out aggressi(e strategy is there"ore the best thing Jim can do in hopes o" winning thisgame2

n order to demonstrate the concept o" a game matri7 in Magic; he 4athering, will use a di""erent e7ample2 ote that in thecase o" Magic, both players almost ne(er ma.e game decisions at the same time2 he decision ma.ing process o"tenalternates "rom one player to the other2 Howe(er, the topic o" dec. selection has se(eral game matri7 applications2 n Magic,

10/09


13/29

any gi(en .ind o" dec. has both strengths and wea.nesses2 *ec. A may win against dec. B most o" the time but lose to dec.8 +ust as o"ten2 Being as there are se(eral types o" strategies and sub-strategies within Magic, the choice o" dec. one usesmay o"ten be a great .ey in determining how he "ares at a tournament2 6hile play s.ill is important, dec. match-ups and luc.o" the draw are "actors o" the game, also2

&ome dec.s try to play slowly and ta.e control o" the game and then win, while others play aggressi(ely and try to winimmediately be"ore the opponent can control the game2 Jim has a "ast red dec. with "ast creatures and direct damage, a "astgreen dec. with many power"ul low casting-cost creatures, and a slow blue control dec.2 Jon has a "ast blac. creature dec., a

white control dec., and a red control dec.2 =or the sa.e o" discussion, let it be gi(en that Jim and Jon tested these dec.sagainst each other e7tensi(ely and "ound how o"ten their own dec.s won2 6in percentages are "rom JonFs point o" (iew2

JonFs 6in Q Jon


14/29

&hoc.; nstant,


15/29

beat you, "irst you ha(e to determine whether you played your own dec. correctly2 Perhaps you played your dec. incorrectlyand your ad(ersaries played "lawlessly2 " that was the case, then you were no match "or them2

Ksing the people ha(e a(ailable, will attempt to draw conclusions on sample dec. types about how much o" a chance onedec. has "or winning against another2 By then re(ersing who is playing each dec., new data ma.es itsel" present, since eachplayer plays a dec. di""erently2 he player who ma.es the best o(erall decisions is the one who will win the most with agi(en dec.2 By applying that theory, will be able to chart the statistics o" dec.s and players using this chance (ersus s.illImethod2

2ootnotes:sland; !and2 ap "or one blue mana2

%huffling and 7andomi8ation

&ince Magic is supposed to be played with a randomized dec. o" si7ty or more cards, the issue o" shu""ling presents itsel" atthe beginning o" the game, as well as some situations within the game that reDuire shu""ling2 6hat constitutes "ullyrandomizing a dec. he three most common shu""ling techniDues used by players are ri""le shu""les, pile shu""les, ando(erhand shu""les2 A ri""le shu""le is when the dec. is di(ided into two hal(es and ri""led together, alternating one card "romeach2 A pile shu""le is when the dec. is dealt out into a number o" piles and then grouped bac. up2 An o(erhand shu""le is +ustta.ing part o" the dec. and mo(ing it to another part o" the dec., such as ta.ing a "ew cards "rom the middle and mo(ingthem to the top or bottom, o(er and o(er2

@b(iously the o(erhand shu""le is the least e""ecti(e shu""le, as RclumpsR o" cards that is, small groups o" cards tend to staytogether more o"ten and not get randomized in this method2 he ri""le shu""le and pile shu""les are indeed the methods thatpro"essional players use2 &imply put, these methods randomize the dec. to a much higher degree2 Pile shu""les (ary amongplayers2 &ome li.e to use "our piles, while some use se(en2 ha(e seen higher-end players use se(en piles2 his is e""ecti(ebecause clumps are none7istent in this method2 Any group o" se(en cards becomes completely di(ided using this method28ombined with a "ew ri""le shu""les, this randomizes a dec. to a great degree2

How good is a ri""le shu""le, though o randomize a dec. o" #0 playing cards, it only ta.es $ times&(2 6ith '? cards, itshould there"ore ta.e appro7imately : shu""les2 6hen many players play a game o"ten see them o(erhand shu""le once ortwice, and "eign a ri""le shu""le2 hese players then draw land a"ter land and wonder whyG According to the same web site, itwould ta.e 0#?? o(erhand shu""les in order to get the same randomization as $ ri""le shu""lesG Again my analysis holds true;


16/29

A "actorial is calculated by multiplying all the whole numbers "rom 1 up to that number together2 t is written in the "ormRGR where is a positi(e number2 =or e7ample, 1 7 0 7 3 7 5 7 # S 10? S #G which is read as R# "actorialR2 Cero "actorial isde"ined as 12 6hile it does not immediately ma.e sense mathematically spea.ing, trying to "ind the number o" permutationsa(ailable "or zero items ma.es sense% there is only one way to arrange zero items2

A permutation represents the number o" di""erent ways one can arrange ob+ects2 " you ha(e # boo.s and want to see howmany di""erent ways you can stac. them, you would use a "actorial2 he "irst boo. would ha(e # possible positions% the ne7tone would ha(e 5 possible positions le"t, and so on2 here"ore there are 10? #G ways to arrange # boo.s2 " you only wanted

to arrange 3 o" the # boo.s, you would only need to compute # 7 5 7 3 which is eDual to '?2 hat is because you are onlyusing 3 positions2

A combination simply represents a number o" possible subsets, and is not concerned with the arrangement o" these items2 "you were to choose 3 out o" # boo.s, how many di""erent combinations are possible =or this, you would ta.e the number o"permutations, but then di(ide by the number o" repeated combinations2 ou do this by ta.ing the "actorial o" the number o"items2 =or this e7ample, you would "ind # T 5 T 3 and di(ide it by 1 T 0 T 32 here"ore '? di(ided by ' is 1?, and there are 1?di""erent ways to pic. 3 out o" # boo.s2

A hypergeometric distribution is something much more comple72 t is used to determine the probability o" certain sets o"occurrences when e7tracting elements without replacement2 hat de"inition certainly applies to drawing cards "rom a dec.,since you are ta.ing them out2 Hypergeometric distribution may seem li.e an un"amiliar phrase, but it is a concept that we areall "airly "amiliar with2 6hen drawing cards "rom a dec. without putting them bac., this concept applies2 his "ormula can be

used to determine how o"ten you draw certain cards "rom a dec. o" cards2

he "ormula synta7 is a bit comple7, though2


17/29

urn $ 3'2#:Q

urn : 3325'Q

urn 9 3?2##Q urn 1? 0$2:5Q

hose are the chances o" not drawing a !ightning Bolt2 @n turn 1, you will ha(e $ cards, and there is a '?2?#Q chance that

you ha(e not drawn one o" your 5 !ightning Bolts out o" '? cards2 By turn 1? this chance diminishes to 0$2:5Q2 !i.ewise,the chance o" drawing one or more by turn 1 is 3929#Q, and the chance increases to $021'Q by turn 1?2

Hypergeometric distribution has other use"ul applications within the game2 8alculating how many land cards to use in a dec.is the base o" dec.building, as one needs land in order to play his cards2 oo many land cards will cause you to draw notenough good cards late in the game, and too "ew will cause you to stall, gi(ing your opponent the ad(antage2 &ome dec.s willwant more land and be able to ta.e ad(antage o" it, and others will want "ewer because o" smaller mana reDuirements2 " youwant to draw "our land by turn "our o"ten, but not too o"ten, this "ormula is help"ul2

he best thing to do within the game would be to ha(e a $?-:?Q chance o" this happening, then use cards that allow you toloo. through your dec. to get more2


18/29

here are a "ew cards in Magic that ma.e you "lip a coin, but it is rarely an issue within the game2 A couple o" years agothere was a coin-"lip card that saw plenty o" tournament play called =renetic )"reet&(2 t utilized a game mechanic calledphasing2 6hen a card phases out it is remo(ed "rom the game2 t phases bac. in during its controllerFs ne7t untap phase2 hiscard was good because i" it was the target o" a card such as !ightning Bolt or &hoc., its controller could attempt to sa(e it byphasing it out2 t would ha(e a #? percent chance o" being lost as opposed to a 1?? percent chance o" being lost i" it lac.ed itsspecial ability2 @(er time, this ability would net its controller an e7treme ad(antage2 " the opponent had to use two cards inorder to remo(e a single card, that meant that the player who only lost one card had more resources le"t2 &uch card ad(antageis crucial to gaining an ad(antage in the game2

8ards li.e 6rath o" 4od, which can ma.e the opponent lose more than one card, are power"ul2 6hen playing more than onecreature against a dec. with 6rath o" 4od in it, you need to try and "igure out whether it is better to play more creatures andtry to de"eat the person "ast, or to anticipate a 6rath and play creatures a"ter2 =iguring the probability o" a 6rath is use"ul2Just .nowing some basic "igures, such as the chance o" the opponent drawing one a"ter #, ', $ turns is all a player should becom"ortable2 *oing hypergeometric distributions on the spot is de"initely not something a player needs2 8ommon senseusually dictates the right play2

" there is one card in a dec. that can gi(e the dec. a se(ere ad(antage against an opposing dec., ha(ing drawn that card atthe right point in the course o" a game also adds to randomness2 " the card is needed early, getting it within the "irst three or"our turns will o"ten be a deciding "actor2 his chance is around "i"ty percent, gi(en there are "our o" the card in a dec., asshown through hypergeometric distribution2

he largest random "actor in Magic is by "ar randomness o" the dec.2 he cards a player draws "rom his dec. are chosen atrandom2 here"ore what card a player is going to draw is an un.nown until he draws his ne7t card2 ot ha(ing this.nowledge limits a playerFs ability to thin. and plan ahead2 8ompare this to the popular game o" etris2 n etris, the ob+ectis to stac. "alling bloc.s so they arrange into lines without gaps2 By (iewing the ne7t piece, the player easily plans a strategy,and plans ahead in order to succeed2 n Magic, the player does not .now his ne7t card, and can only plan in the present2 hatis, unless the player has a card in play which allows him to (iew the ne7t card or cards in his dec.2 4enerally, being able toplan ahead is limited to the cards in oneFs hand2 nowing the chances o" what cards will be drawn later in is only s.etchy,because the player cannot determine e7actly when the cards will be drawn2

a.e the "ollowing situation2 ou are in a duel with Andrew and you are at : li"e2 Andrew is playing a "ast blac. dec. withlots o" small creatures that are slowly diminishing your li"e total2 ou are playing a blue and white control dec. and ha(ealtered your dec. between games this is called sideboarding&)( in order to help you de"eat AndrewFs dec.2 He has not put inany cards "rom his sideboard2 n order to guarantee your (ictory against Andrew, you ha(e put 3 copies o" the card !ight o"

*ay&*(in your dec.2

&ince you .now the entire contents o" AndrewFs dec., you .now that !ight o" *ay will stop him cold in his trac.s and let youdo whate(er you want, since his one method o" (ictory will be gone2 t is currently the end o" your se(enth turn2 ou played"irst this game2 ou still ha(e not drawn a !ight o" *ay, but ha(e all the land you need "or it; 5 slands and 3 Plains2Howe(er, i" you draw one o" your 5 copies o" ntuition&+(you will be able to search your dec. "or a !ight o" *ay and play itimmediately2 6hat are the chances o" you winning ne7t turn 6hat are your chances o" winning the turn a"ter 6hat do youdo in the meantime i" you do not draw a !ight o" *ay

&ince you ha(e ta.en $ turns and played "irst, and we will assume you ha(e used no dec. manipulation cards, you ha(e gonethrough 13 cards2 @" those 13 cards, none were ntuitions or !ight o" *ays2 &ince there are 5 ntuitions and 3 !ight o" *ays,you ha(e $ "a(orable outcomes out o" 5$ cards that can be drawn "rom ne7t turn2 =or the turn a"ter, you will ha(e $ out o" 5'i" you do not draw a .ey card ne7t turn2 hat does not mean a chance o" $/5', though2 Ksing the "ormula "rom the lastchapter, you would "ind HP4)@M*& 1, 1, $, 5$ because you are chec.ing "or 1 card, you are drawing 1 card, you ha(e$ "a(orable outcomes, and 5$ total outcomes possible2 our chance o" winning ne7t turn is 1529Q2 o "ind the chance o"drawing the card the turn a"ter i" it is not drawn ne7t turn, change the 5$ with 5' since there would be one other card drawnne7t turn which is not directly use"ul2

o "ind the total chance between the two ne7t turns, you would need to "ind the sum o" HP4)@M*& 1, 0, $, 5$ andHP4)@M*& 0, 0, $, 5$ since you are drawing 0 cards, and there can be either 1 or 0 o" your $ .ey cards drawn2 ourchance o" winning in two turns is 0$2:Q i" you do not win ne7t turn2 6hat is your chance o" winning in 3 turns "rom now i"you do not win within two turns

1:/09


19/29

By adding up the chances o" drawing 1, 0, or 3 copies o" your .ey cards by then, the answer comes to 392?Q2 @" course, thatis the chance in ad(ance2 " a"ter two turns you ha(e not drawn any o" the .ey cards, your chance is actually lower2 he sameapplies to the pre(ious e7ample "or winning in two turns, a"ter drawing the "irst card2 hen, the chance is similar to the "irste7ample, with one card being drawn2 " you do not win a"ter one turn, your chance o" winning on the second turn is 1#20Q2 "you do not win on the second turn, your chance o" winning on the third turn increases to 1#2#Q2 his is "ound by ta.ing thehypergeometric distribution o" 5' and 5# cards, respecti(ely, instead o" 5$2

" you "ind that these chances are too low, you will need to plan an alternate strategy2 &uch a strategy might include using

your a(ailable resources to stalling out Andrew instead o" sa(ing them to win the game immediately a"ter you get your loc.2=or e7ample, you might play a creature now and use it to bloc. one o" his in the meantime in order to sa(e you some damage2here is no clear method o" probability ma.ing the right decision2 nstead, a player needs to use probability to plan a strategy2A strategy will pro(ide decisions "or any gi(en situation2

2ootnotes:=renetic )"reet; 8reature, 1


20/29

he "ollowing statistics can be obtained "rom this matri7;

*ec. @(erall JonFs 4ames JimFs 4ames

Blue 09Q 50Q 1$Q

4reen $9Q :3Q $#Q

6hite '3Q :3Q 50Q


21/29

@"ten it is said that swarm beats control, control beats combo, and combo beats swarm2 hat is +ust a generalization, andsome dec.s per"orm better than others do2 *etermining which dec. is best to play in a tournament reDuires some simplethought2 here is no e7act way to determine which dec. to play unless you .now what other people are playing, and whatyour odds are o" beating them2 And still, you only get an appro7imation2

More importantly than choosing the dec. is the dec. construction aspect o" Magic2 &electing which cards to use "or a dec.means trying to ma7imize e""iciency and e""ecti(eness2 Ksing cards which are o(erall e""ecti(e means trying to "ind a balancebetween general use"ulness and situational use"ulness2 8ombined with proper management o" resources, creating a dec. that

wins is a comple7 tas.2 hat is o"ten why people rely on dec.s that ha(e already pro(en themsel(es2

6hich is more e""ecti(e though, designing your own rogue original dec. or using a stoc. "amiliar dec. Most peopleagree that each has its own inherent ad(antages2 &toc. dec.s ha(e already pro(en themsel(es, so one only needs to be able toplay the dec. well in order to succeed with it2 Howe(er, not .nowing the dec. well is a disad(antage2 6hen you build yourown dec. you .now why the cards are there since you put them there2 ot only that, but someone you "ace in a tournamentwill most li.ely .now the e7act contents o" your dec. i" you play a stoc. dec.2 " not that, he will .now e7actly how to beatit, which cards to deal with "irst, and the li.e2 " you play a dec. you made yoursel", the person will not .now i" you ha(esomething more threatening coming up2 He will ha(e to guess and ma.e tough +udgment calls2

Jamie 6a.e"ield is a Pro our player who has been renowned "or his success with rogue dec.s2 he dec. Jamie has been"amous "or is a green control dec. he calls &ecret =orceI2 Ksing large green creatures and utility, the dec. is able to wrec.most o" the popular dec.s2 6hen he "irst made his appearance with this dec., people did not .now how to deal with it since

they had ne(er seen it be"ore2 he result was that since his dec. was at least eDually e""ecti(e, but not .nown to the opponent,he won2 Jamie .new what was in his opponentsF dec.s2 Howe(er, that same in"ormation they did not .now about him2 hesurprise "actor won him many games, and Duali"ied him "or Pro our; ew or.2

Ksually, though, rogue dec.s ha(e limited success2 &till, the respect "actor is great "or those who can design new dec.s, playthem well, and win games2 t is easier "or someone to ta.e a dec. already in e7istence and play it to per"ection than "or thesame person to design a dec. and win +ust as much2 he reward in the latter case is that the person can truly call the dec. hisown2

!ec"building: !ec" $rchetypes and !ec" %trategy

6hile it is impossible to classi"y e(ery dec. into a concrete category, the (ast ma+ority o" tournament-le(el dec.s "all intoone o" three categories2 here are control dec.s, aggressi(e dec.s, and combo dec.s2 6hile it is not a de"inite roc.-paper-

scissors scenario, aggressi(e dec.s tend to beat control dec.s more o"ten than not2 Howe(er, it really depends on which .indso" dec.s they are2 8ontrol dec.s can o"ten win against combo dec.s but only i" there is a lot o" countermagic in the controldec.2 8ombo dec.s can o"ten win against aggressi(e dec.s simply because they can ignore the pressure and win immediatelyat a certain point2

he control dec. utilizes cards that can neutralize the opponentFs threats2 8ounterspells, creature elimination, and other threatneutralization cards "orm the bac.bone o" a control dec.2 *irect damage can be used de"ensi(ely to destroy creatures, but hasthe "le7ibility o" being used to damage the opponent i" necessary2 Blue-red dec.s were once a popular phenomenon based onthis theory2 8ounterspells could ta.e care o" any ma+or threats while direct damage either destroyed opposing creatures orwor.ed at "inishing o"" the opponent directly2 8ards that can tap creatures or pre(ent them "rom attac.ing are also use"ul in acontrol dec.2 hreat management is the critical idea here2 Being able to deal with opposing threats means pre(enting theopponent "rom winning2 By being able to pre(ent the opponent "rom winning, the control player allows himsel" to win2

How the control player wins is usually unimportant2 A single large creature is usually "air game, while cards such asMillstone&(are also popular in this .ind o" strategy, in order to run the opponent out o" cards2 6hile damage is the mostcommon method o" (ictory, "orcing the opponent to not be able to draw a card is an o"ten-o(erloo.ed method o" winning28ontrol dec.s can be any color, but o"ten use blue "or countermagic2 6hite has mass destruction cards such as Armageddonand 6rath o" 4od to destroy lands and creatures, red has direct damage to augment countermagic, and blac. has discard andmore e""icient creature remo(al2 4reen is not too popular as a control color but has cards such 6all o" Blossoms&)(that ma.e"or good de"ense2

Aggressi(e dec.s o"ten ta.e the "orm o" a creature swarm strategy2 " you can present the opponent with more threats than hecan deal with, you will be able to de"eat his control strategy2 8on(ersely, though, i" the control player can assert himsel"be"ore the swarm strategy wins, the control player can ta.e (ictory "or himsel"2 &ometimes the game is not o(er immediately

01/09


22/29

a"ter it seems the control player has his plan wor.ing, it can happen that the swarm player uses a snea.y tactic in order to winhis (ictory bac.2 Ksually a 6rath o" 4od will put a gaping hole in the creature-based dec.Fs plan by destroying all thecreatures he has already played2 Howe(er, once more creatures hit the table the control player will either ha(e to come upwith another 6rath or lots o" creature elimination2

&ometimes the aggressi(e dec. ta.es on other "orms2 A dec. relying mostly or e(en totally upon direct damage cards can bee""ecti(e2 n this case, creature elimination attempts are "utile2 A burnI dec. can cast lots o" direct damage spells li.e &hoc.,!ightning Bolt, and =ireblast&*(with one simple goal; o bring the opponent "rom 0? to ? li"e2 6ith plenty o" Mountains in

this all red dec., =ireblast ma.es "or a great "inisher2 n order to beat a burn dec., the control dec. has to either use a lot o"countermagic, or a lot o" li"e gain2

Another aggressi(e dec., usually a creature swarm dec., will ha(e a greater chance o" winning against a burn dec.2 Bringingout large amounts o" "ast creatures can o"ten o(erwhelm the burn dec., and "orce the burn player to be de"ensi(e2 By "orcingthe aggressi(e player to play de"ensi(ely, that puts a large hole in his plans and ruins his strategy2 &ometimes, howe(er, thecreature swarm dec. will get a slow start and the burn dec. will win2 Anything can happen in the world o" Magic2

he third ma+or type o" dec. is the combo dec.2 6hile many players "rown upon many .inds o" combo dec.s, this reigns tobe one o" the most e""ecti(e strategies in Magic2 @(er the course o" Magic history there ha(e been many "amous Magic dec.sbased around combos2 6hile will not go into detail about them, will e7plain the concept2 C(i Mowshowitz, a student at8olumbia Kni(ersity, designed a dec. in 199: that he dubbed urboC(i2 he dec. brought out a card called *ream Halls&+((ery "ast, which allowed both players to play spells practically "ree2 he dec. drew lots and lots o" cards up until the point

where it could either create a lot o" mana to win with a large direct damage spell, or ma.e the opponent run out o" cards26ith a large mana engine, anything is possible2

" C(i played against a creature swarm dec., he would +ust ignore the threat and win third or "ourth turn, be"ore his opponenthad any chance o" winning2 &ince the creature swarm dec. o""ered no threats, he was completely sa"e "rom its simplisticstrategy2 )(en still, the dec. had some countermagic in it in order to sa(e itsel" +ust in case2 Against a control dec., though,the dec. became a lot tougher to play2 " the opponent played his cards right he could easily win2

*ream Halls helped the countermagic user more than C(i2 he dec. was o.ay but not as good as other combo dec.s such asProsperous Bloom, named by two o" the main cards in it, Prosperity&,(and 8ada(erous Bloom&-(2 his dec. was (ery"amous, and brought Mi.e !ong (ictory at Pro our; Paris2 By generating a large amount o" mana with a card combinationthat +ust about wor.s itsel" together (ery easily, this dec. was power"ul in its day2 @" course, there were ways to beat it, byusing e""ecti(e enchantment remo(al and countermagic2 Mi.e .new how to play his dec. to per"ection and too. on the

competition his best2

6hile combo dec.s, creature dec.s, and control dec.s are the three main types o" dec.s which ha(e presented themsel(es,there are many dec.s which "all into more than one category, and e(en some which "all into none o" the categories2 Howe(er,it is sa"e to generalize that the (ast ma+ority o" dec.s are one o" these types2 ha(e played many dec.s and see that these.inds o" dec.s stand out about all others2 How does a gi(en dec. per"orm hat is the ne7t Duestion will attempt to answer2

2ootnotes:Millstone; Arti"act, 02 Pay 0 and tap to ta.e the top 0 cards "rom any playerFs dec. and put them in his discard pile26all o" Blossoms; 8reature - 6all, 142 6hen 6all o" Blossoms comes into play, draw a card2 ?/52 6alls cannot attac.2=ireblast; nstant, 5


23/29

the other dec.s, it is no good i" it cannot beat the most popular dec. consistently2 @" course, no one dec. is usually e(erdominant2

o date, only twice was there a dec. that truly dominated the tournament scene2 n o(ember-*ecember o" 199:, a"ter therelease o" the KrzaFs &aga standalone e7pansion, a completely o(erpowered combo dec. based around a card called olarianAcademy&(emerged2 his was by "ar the best dec., beating e(erything else hands down2 n the summer o" 199', a dec.based around ecropotence&)(ga(e the card popularity, and the combined elements o" total resource destruction made thisdec. dominant2 Mainly, a dominant situation only e7isted with these two dec.s2 @therwise, there ha(e been popular dec.s,

but ne(er a dominating dec. Duite the way these dec.s dominated2 hat being said, what is the best way to see how e""ecti(eyour dec. is against a "ield o" other dec.s

he "irst ob(ious step is to .now how to play your dec. as e""ecti(e as possible2 By playing the dec. against (arious otherdec.s, you need to learn e(ery dec.Fs strengths and wea.nesses2 @nce you .now that, you can capitalize on these areas andplay your dec. to ma7imum e""ecti(eness2 A"ter youF(e done all your playtesting, you need to "igure out the most populardec.s and ma.e sure your dec. and/or sideboard can handle them2 " you .eep losing to the most popular dec.s, you cannotpossibly win a tournament2

=or this purpose, played my red control dec. many games in order to demonstrate whether playing it in a tournament wouldbe a good idea2 came to the conclusion that it is a "airly good dec., though it can by no means handle the competition handsdown2 he players played against had di""erent dec.s2 Because o" this, the dec.Fs per"ormance is better analyzed by (iewingthe particular dec.s played against2 A"ter playing thirty games against (arious players with (arious dec.s, this is how "ared

against all the players, and the most common dec.s;

>incent Matty Mi.e


24/29

My o(erall win ratio was 1$ to 132 he "act that won #$Q o" the games played is nice to .now, but not too use"ul2 How per"ormed against the a(erage dec. is more use"ul2 By a(eraging how did against each dec. and putting it together, wouldha(e a rough estimate o" how would do at a tournament against an a(erage "ield2 he result is that win 53Q against thea(erage dec.2 his does not ta.e into account the sub-standard dec.s play against e(ery now and then, as would ne(ere7pect to play against anything li.e that in a tournament2 A player with little e7perience is not li.ely to pay the admission "ee"or a tournament, when he is more li.ely to +ust play "or "un2 hat being said, it is not a good idea to include those wins, asin"lating my win ratio does not help me as a player2 )(en still, a much larger number o" games are necessary to get a betterappro7imation "or my true win percentage2

his procedure assumes there is an e(en "ield, howe(er2 Perhaps would end up ha(ing a higher chance o" winning i" therewere more dec.s in the tournament that had a higher chance o" winning againstG 4i(en a "ield o" #?Q ron Maiden dec.s,0#Q blac. weenie swarm dec.s, 1#Q &ligh, and 1?Q *eath, would ha(e a higher chance o" winning2 @" course, *eath and&ligh are a lot more popular than Maiden and blac. swarm2 4i(en two tournament scenes with di""erent percentages o" thesedec.s, with my dec. being the only red control dec., will gi(e a sample situation2


25/29

!ec"building: Card 4fficiency and 7esource Management

6hen choosing cards "or a dec., the player needs to decide which cards will do the best +ob2 hat is to say, i" want to usewhite creature remo(al, which card is most e""ecti(e "or this purpose 6hich countermagic cards will "it into my dec. thebest without a mana problem 6hich creatures will wor. best in my dec. hese Duestions relate to the idea o" carde""iciency2 4etting the best possible card "or the lowest possible cost means the highest e""iciency2

n current Magic, many cards that are (ery e""icient are no longer printed because they were so power"ul2 &words to

Plowshares, which mentioned earlier, is one o" these cards2 =or one white mana, this card can remo(e any creature "rom thegame2 he li"e gain simply does not balance out this card2 8hances are, you may be playing a control dec. and you donFt carehow many li"e points your opponent has2 his card will simply .eep you ali(e and that is all that matters2 8ombined withcards li.e imetwister&(, this card is e(en more power"ul because you get bac. your &words and your opponentFs creaturesare hopelessly gone "or the game2 imetwister was another card which ceased being printing, albeit a lot sooner than the&words2

6hen choosing which creatures to use "or a dec., i" any, the best ones are ob(iously ones with a higher power and toughnessto casting cost ratio2 =or a completely aggressi(e dec., only sheer power is o" importance2 he almighty Ball !ightning&)(boasts a power o" si7 with a casting cost o" three2 =or "ast red dec.s, this has always been a popular card2 " the de"endingplayer cannot deal with it he will ta.e si7 damage, which is about one third o" his beginning li"e pointsG

8ombined with direct damage spells li.e !ightning Bolt which deals three damage "or one mana, and =ireblast which can

deal "our damage "or no mana at the slight cost o" losing your own land, "ast red dec.s were popular while those cards werein-print2 6inning the game is usually the result o" the resource sacri"ice "rom =ireblast2 he popularity o" the dec. wasbecause o" the supreme card e""iciency o" cards li.e these2 @" course, a"ter these cards were ta.en out o" print and replacedby newer sets, this dec. was no longer (iable2

Ksually creatures with a casting cost o" one will ha(e a power o" one2 &ometimes they will ha(e a power o" two, but at a cost2t may not always be able to be used to attac., or it may deal damage to its controller2 he purest breed o" aggressi(e dec.swill ta.e blazing speed at any cost2 &uch are "ast blac., red, and green dec.s2 Blue and white do not ha(e any e7tremelye""icient creatures, but they do ha(e creatures with good utility2 hat ma.es blue and white creature-based dec.s better as ade"ensi(e strategy2 Aggressi(e dec.s utilize low-cost creatures and damage spells2 *e"ensi(e dec.s utilize high-e""iciencythreat management remo(al spells2 *isenchant&*(is another prime e7ample o" a popular remo(al spell2 Just as &words toPlowshares is to creatures, *isenchant is "or arti"acts and enchantments2 Howe(er, creatures are usually a bigger threat inMagic, so *isenchant is a more balanced card2 hat is why it has been in the current tournament en(ironment since the

beginning o" the game2

Being able to ma7imize what resources you ha(e a(ailable is what ma.es your strategy more e""ecti(e2 By being able toremo(e more than one opposing card with only one o" yours, you gain card ad(antage2 8ard ad(antage is not the only .ind o"ad(antage, though2 By using cards which .eep the opponentFs resources una(ailable such as 6inter @rb&+(, you gain timead(antage2 his means that since your opponent cannot play his cards as usual, he is delayed and you gain more time to dowhate(er you ha(e to within your strategy2 his also means that the opponent will o"ten need more time to react to your otherthreats2

et another .ind o" time ad(antage lies in ta.ing multiple turns2 " you can ta.e more turns than the opponent can, you ha(emore time to ma7imize your strategyFs e""ecti(eness2 his is a ma+or resource ad(antage because it allows you to bring moreresources into play2 By ta.ing another turn you get to attac. again, play another land, and draw another card2 his is why thecard ime 6al.&,(was banned "rom tournament play2 =or two mana it let the caster ta.e another turn2 Howe(er, a more

balanced (ersion was later printed, called ime 6arp&-(2 =or "i(e mana this had the same e""ect2 By delaying when the playercan use it, it balanced out its ad(antage with a high cost2


26/29

2ootnotes:imetwister; &orcery, 0K2 )ach player shu""les his hand, dec., and discard pile together, and draws se(en cards2 Putimetwister into your new discard pile2Ball !ightning; 8reature, arious &peed *ec. 0#Q T 102#Q S 321Q>arious 8ontrol *ec. $#Q T 102#Q S 925Q

otal 6in Percentage ':2:Qhis ga(e me an e7tremely high chance o" winning2 nowing the metagame, that is, what dec.s other people were going tobe using, was an e7tremely e""ecti(e tactic2 6hile would lose to "ast dec.s, guaranteed mysel" a win against the slowerdec.s and especially the combo-reliant dec.s2

n round 1, played against a combo dec. called &piral BlueI and won2 n round 0, played against another &piral Bluedec. and lost, as the dec. was played by a pretty good player, and the only other match had played against this dec. was inround 12 n round 3, played against a "ast red dec. and lost2 hat was the small price paid "or playing the metagame2Howe(er, "aced two other combo dec.s in round 5 and # and won against them both2 he 3-0 record put me into the top :2 had won $#Q against the combo dec.s 3-1 so my estimate was (ery good2 n the top :, played against a control dec. andwon, and then two combo dec.s and won against both o" them2 he last round was, incidentally, the same opponent "rom

0'/09


27/29

round 02 his time around, got luc.ier perhaps2 6hate(er it was, had won because .new the chances and played themetagame correctly2

nowing which dec. to play is a de"inite game theory application2 he decision ma.ing process "ormed by game matricesand relati(e probabilities constitutes a calculation o" chance2 he statistics that used in order to "ormulate my chances o"winning against each dec. are stri.ingly similar to those used during my win-loss ratio calculation e7periment2 &uch atechniDue is (ery handy when trying to win a ma+or tournament2 6hile do not win them on a regular basis, do "airlyconsistently2 &tatistics and probability are only an a(erage2 hey are not a "i7ed (alue2 nstead, they ser(e as an

appro7imation2 nowing how to use them is what ma.es a player more in"ormed2

Math and the $Berage Player

By now ha(e clearly demonstrated how math is a big part o" Magic2 =rom probability, decision ma.ing, and strategicapplications, it is apparent that math "orms the bac.bone o" what ma.es the game so intellectual2 Howe(er, the Duestionarises to whether players actually identi"y and use the math components2 o "ind out about how much math is actually seenby players o" the game, conducted a poll on he Magic *o+o2 as.ed players to send me responses to the "ollowingDuestions;

6hat is your name and age

How long ha(e you been playing Magic

6hat is your bac.ground in math what courses ha(e you ta.en

o what e7tent to you thin. math is an important part o" strategy and game play

How, i" at all, does math play a role in your game play )7plain2

*o you thin. someone who is more .nowledgeable in math has a higher playing potential than someone who does not does6hy or why not

his sur(ey was posted on he Magic *o+o www2thedo+o2com on May 15th2 @(er the course o" May 15 through themorning o" June 9, recei(ed 1:0 responses2 hey came "rom players o" (arying age and e7perience2 Most recognized themathematical aspects% yet most agreed that math does not ma.e "or a better player2 8ritical thin.ing s.ills are the basis o"

being a better player2 onetheless most recognized that math does e7ist behind the scenes o" the game2 A"ter all,


28/29

Please note that L- years denotes greater than L, and less than or eDual to 2 here"ore 0-3 years means more than but notincluding 0 years, up to and including 3 years o" e7perience playing Magic2 t appears that the ma+ority o" players ha(e beenaround since when the ery !ittle &omewhat A great deal

8oncept recognition 92?Q 3'2$Q #520Q

Kse o" math 1:21Q 3123Q #?2'Q

mportance o" math 0?2#Q 3'21Q 5325Q

t is (ery apparent that Magic players recognize the mathematical concepts behind Magic, use these concepts in their gameplay, and see their relati(e importance2 Beginners may only see the subtleties such as li"e counting, damage dealing, andother addition/subtraction ideas such as creature combat2 More ad(anced players identi"y mana cur(es, casting cost analysis,land ratios, and timing issues2 )7perts see ad(anced probability, statistics, and metagame analysis2 Most players agree thatdecision ma.ing is superior to mathematical ability when it comes to playing the game2

n dec.building, mathematical s.ill comes into play more o"ten, though intuition o"ten reigns supreme2 6hile math is a bigpart o" Magic, it may not necessarily ma.e a better player2 *ecision-ma.ing does, though2 4ame theory, an economicsapplication, is more use"ul to the a(erage player then numerical mathematics2 *ec. creation is where the math comes intoplay2 4ame mechanics are "ull o" math2 he game play is not as "ull o" math2 6hile beginners only (iew math as li"e totals,e7perts see the big picture2 Math thin.ing s.ills are use"ul in the game either way2

Players use a "air amount o" math either at a conscious or subconscious le(el2 6hile it is an underlying part o" the game, itdoes not necessarily enhance oneFs playing s.ill to a large degree2 nowing math ne(er hurts, though2 n dec. construction,math is great2 n playing, "ast arithmetic is also a good thing2 !ogic and reasoning s.ills are also good things to ta.e with youonto the gaming "ield2 Play e7perience is more (aluable than pure mathematical e7perience, though2 All players seem toagree on these ideas2

Personal $pplications

will now attempt to answer my own Duestions a"ter doing my research and analysis2 My name is Jon Prywes, and am 1$2 ha(e been playing Magic "or "our and a hal" years2 ha(e ta.en e(erything up to B8 8alculus and will be ta.ing !inearAlgebra in the "all2 see math as a (ery important part o" Magic2 nowing how to create an e""icient dec. and play it wellreDuires a basic understanding o" probability and statistics2 8reating dec.s on your own time can utilize some ma+ormathematical ideas, while in the actual playing area, logic and reasoning are tested (ery intensely2

will base my strategy o"ten upon what anticipate my opponent doing2 &ometimes ha(e to "igure out the odds and base mystrategy around them2 6hile math is a big part o" my game, donFt thin. that it ma.es me too much o" a better player2 t maygi(e me an ad(antage against someone who does not .now as much math, but it is de"initely a part o" the game itsel"2

0:/09


29/29

Mathematical reasoning s.ills are (ery use"ul, and would not be able to play the game as well as do without them2 Ma.ingdecisions at lightning speed is something reDuired o" any tournament-le(el Magic player2 need to be able to do that a lot2@ne small mista.e has meant a match loss too many times2 Playing "lawlessly with reasonable speed is a .ey to success2Being able to acDuire this s.ill reDuires countless hours o" practice, sleep, and a proper diet2 he mathematics are de"initely inthere, though2

mathematics of magic.doc

Documents