### Magic The Gathering
Magic: The Gathering is a trading card gam...

## Playing Magic
The full rules of Magic are extensive - you can f...

Rengo Kriegspiel is a "blindfolded", team variant of Go. There are ...

Zero-sum games are 2-player games where the sum of the payoffs of t...

Example of a Turing Machine built by a player in Minecraft
![](htt...

As the authors state, there are a number of real world games that s...

Caulombe and Lynch were able to reason about those video games by s...

Unlike games like Chess and Go, researchers still haven't been able...

Rice's Theorem states that all nontrivial semantic properties of pr...

![](https://i.imgur.com/UmfRdn6.jpg)

### Rogozhin Turing Machines
When Alan Turing came up with the ide...

Cards mentioned by the authors:
![](https://i.imgur.com/KktkX2L.jp...

#### What is a Creature Token?
Creature tokens are effectively mar...

![](https://i.imgur.com/TkQmL50.jpg)

Rogozhin's (2,18) Turing Machine is a Turing machine with 2 states ...

![](https://i.imgur.com/KXbMR8g.jpg)
![](https://i.imgur.com/QmIDX...

![](https://i.imgur.com/j7hi0HL.png)
![](https://i.imgur.com/bKSaD...

![](https://i.imgur.com/JzoBkQm.jpg)
![](https://i.imgur.com/KOgOp...

![](https://i.imgur.com/cz2y2uY.jpg)
![](https://i.imgur.com/NmRaC...

![](https://i.imgur.com/x7qX86o.jpg)
![](https://i.imgur.com/dITxN...

arXiv:1904.09828v2 [cs.AI] 23 Apr 2019

Magic: The Gathering is Turing Complete

Alex Churchill

Indepe ndent Researcher

Cambridge, United Kingdom

alex.churchill@cantab.net

Stella Biderman

Georgia Institute of Technolo gy

Atlanta, United States of America

stellabiderman@gatech.edu

Austin Herrick

University of Pennsylvania

Philadelphia, United States of America

aherrick@wharton.upenn.edu

Abstract—Magic: The Gathering is a popular and famously

complicated trading card game about magical combat. In this

paper we show that optimal play in real-world Magic is at

least as hard as the Halting Problem, solvin g a problem that

has been open for a decade [1], [10]. To d o this, we present a

methodology for embeddin g an arbitrary Turing machine into a

game of Magic such that the ﬁrst player is guaranteed to win the

game if and only if the Turing machine halts. Our result applies

to how real Magic is played, can be achieved using stand ard-

size tournament-legal decks, and does not rely on stochasticity

or hi dden information. Our result is also highly unusual in that

all moves of both players are forced in the construction . This

shows that even recognising who wi ll win a game in which n ei ther

player has a non-trivial decision to make for the rest of the game

is undecidable. We conclude with a discussion of the implications

for a un iﬁed computational theory of games and remarks about

the playability of such a board in a tournament setting.

I. INTRODUCTION

Magic: The Gathering (also known as Magic) is a popular

trading card game owned by Wizards of the Coast. Formally, it

is a two-player zero-sum stochastic card game with imperfect

informa tion, putting it in the same category as games like

poker and hearts. Unlike those games, players desig n their

own custom decks ou t of a card-pool of over 20,000 cards.

Magic’s multifaceted strategy has made it a p opular topic in

artiﬁcial intelligence researc h.

In this paper, we examine Magic: The Gathering from the

point of view of algorithmic game theory, looking at the

computational complexity of evaluating who will win a game.

As most games have ﬁnite limits o n their complexity (such

as the size of a game board) most research in algorithm ic

game theory of real-world games has primarily looked at

generalisations of commo nly played games rather than the

real-world versions of the games. A few real-world games have

been found to have non-trivial complexity, inc luding Dots-and-

Boxes, Jenga and Tetris [8]. We believe that no real-world

game is known to be h arder than NP previous to this work.

Even when looking at generalised games, very few examples

of undecid a ble games are known. On an abstract level, the

Team Computation Game [ 9] shows that some games can be

undecid able, if they are a particular kind of team game with

imperfect information. The authors also present an equivalent

construction in their Constraint Logic fra mewo rk that was used

by Coulombe and Lynch (2018) [7] to show th a t some video

games, including Super Smash Bros Melee and Mario Kart,

have u ndecidable gen eralisations. Constraint Logic is a highly

successful and highly ﬂexible framework for modelling games

as computations.

The core of this paper is the c onstruction presente d in

Section IV: a universal Turing machin e embedded into a game

of Magic: The Gathering. As we can arrange for the victor

of the game to be determined by the halting behaviour of

the Turing machine, this construction establishes th e following

theorem:

Theorem 1: Determining the outcome of a game of Magic:

The Gathering in which all remain ing moves are forced is

undecid able.

A. Previous Work

Prior to this work, no u ndecidable real games were known

to exist. Demaine an d Hearn (2009) [10] note that almost every

real-world game is trivially decidable, as they produce game

trees with only computable paths. They further note that Rengo

Kriegspiel

1

is “a game humans play that is not obvio usly

decidable; we are not aware of any other such game.” It

is conjectured by Auger and Teytaud (2012) [1] that Rengo

Kriegspiel is in fact undecidable, and it is posed as an open

problem to demonstrate a ny real game that is undecida ble.

The approach of embedding a Turing machine inside a

game directly is generally not considered to be feasible for

real-world games [10]. Althoug h some open-world sandbox

games such as Minecraft and Dwarf Fortress can support

the construc tion of Turing machines, those machines have no

strategic relevance and those games are deliberately design e d

to su pport large-scale simulation. In contrast, leading formal

theory of strategic game s claims tha t the unbounded memory

required to simulate a Turing machine entirely in a game

would be a violation of the very natur e of a game [9].

The computational complexity of Magic: Th e Gathe rin g in

has been studied previously by several authors. Our work is

inspired by [4], in which it was show n that four-player Magic

can simulate a Turing machine under certain assumptions

about player behaviour. In that work, Churchill conjectures

that these limitations can be removed and preliminary work

along those lines is discussed in [5]. The com putational

complexity of checkin g the legality of a particular decision

in Magic (blocking) is investigated in [3] and is found to

be coNP-complete. There have also been a number of papers

1

Rengo Kriegspiel is a combination of two variations on Go: Rengo, in

which two players play on a team alternating turns, and Shadow Go, in which

players are only able to see their own moves.

investigating algorithmic and artiﬁcial intelligence approaches

to playin g Magic, including Ward and Cowling (2009) [15],

Cowling et al. (2012) [6], and Esche (2018) [11]. Esche (2018)

brieﬂy considers the theoretical computational complexity of

Magic and states an op en problem that has a positive answer

only if Magic e nd-games are decidable.

B. Our Contribution

This paper complete s the project started by Churchill [4]

and continued by Churchill et a l. [5] of embed ding a u niversal

Turing machine in Magic: The Gathering such that determin-

ing the outcome of the game is eq uivalent to determining the

halting of the Tur ing machine. This is the ﬁrst result showing

that there exists a real-world game for which deter mining

the winning strategy is non-computable, answering an open

question of Demaine and Hearn [10] and Auger and Teytaud

[1] in the positive. This result, combined with Rice’s Theorem

[13], also answers an open problem from Esche [11] in the

negative by showing that the equivalen c e of two strategies for

playing Magic is undecidable.

This result raises important f oundational questions about

the nature of a game itself. As we have alr eady discussed, the

leading formal theory of games holds that this construc tion

is unreasonable, if not impossible, and so a reconsideration

of those assumptions is called for. In section V-A we discuss

additional foundational assumptions of Constraint Logic that

Magic: The Gathering violates, and present our interpreta tion

of the implications for a uniﬁed theory of ga mes.

C. Overview

The paper is structured as follows. In Section I I we provide

backgr ound information on this work , including previous work

on Magic Turing machines. In Section III we present a sketch

of the construction and its key pieces. In Section IV we provide

the full construction of a universal Tu ring machine embedded

in a two-player game of Magic. In Section V we discu ss the

game-theoretic and real-world implications of our result.

II. PRELIMINARIES

One initial challenge with Magic: The Gathering is the

encodin g of infor mation. Som e cards ask players to choose

a number. Although rules for how to specify a number are not

discussed in the Compreh e nsive Rules [16], convention is that

players are allowed to specify n umbers in any way that both

players can agree to. For example, you are allowed to choose

the number 2

100

or ⌈log 177⌉. This presents an issue brough t

to our attention by Fortanely [12]. Consid e r the following

situation: both playe rs control Lich, Transcendence, an d

Laboratory Maniac. O ne player then casts Menacing Ogre.

The net effect of this play is the “Who Can Name the Bigger

Number” game – whoever picks the biggest number wins on

the spot. This makes identifying the next board state non-

computable [2], so we require that any numbers speciﬁed by

a playe r mu st be expressed in stand ard binary notation.

We believe that with this r e stric tion Magic: The Gathering is

transition-computable, meaning that the function that maps a

board state and a move to the next board state is computable

2

.

However, it is unclear how to prove this beyond exhaustive

analysis of th e over 20,000 card s in the game. We leave that

question open for future work:

Conjecture 1: The function that takes a board state and

a legal move and returns the next b oard state in Magic: The

Gathering is c omputable.

In this conjecture we say “a legal move” because it is also not

obvious that checking to see if a move is legal is computable.

Chatterjee and Ibsen-Jensen [3] show that checking the legality

of a particular kind of game action is coNP-complete, but the

question ha s not be e n othe rwise considered. Ag ain, we leave

this for future work:

Conjecture 2: There does not exist an algorithm for

checking the legality of a move in Magic: The Gathering .

A. Previous Magic Turing Machines

In [4], the author presents a Magic: The Gathering end-

game that embeds a universal Turing machine. However, this

work ha s a major issue: it’s not qu ite deterministic. At several

points in the simulatio n, players have the ability to stop the

computation at any time by opting to decline to use effects that

say “may.” For example, Kazuul Warlord read s “Whenever

Kazuul Warlord or another Ally enters the battleﬁeld under

your control, you may p ut a +1/ + 1 counter on each Ally

you con trol.” Declining to u se this ability will interfe re with

the Turing machine, either causing it to stop or cau sin g it

to perform a different calculation from the one intende d.

The construction as given in Churchill [4] works under the

assumption that all players that are given the option to do

something actually do it, but as the a uthor notes it fails w ithout

this assumption. Attempts to correc t this issue are discussed

in Churchill et al. [5].

In this work, we solve this problem by reformulating the

construction to exclusively use cards with mandatory effects.

We also substan tially simplify the most co mplicated aspect

of the construction, the recording of th e tape, and reduce the

construction from one involving four players to one involving

two, and whic h only places constraints on one player’s d eck,

matching the format in which Magic is most commonly played

in the real world (two-player duels). Like the previous work,

we will embed Rog ozhin’s (2, 18) universal Turing machine

[14].

III. AN OVERVIEW OF THE CONSTRUCTION

In this section we give a big picture view of the Turing

machine, with full d etails deferred to the next section. The

two players in the game are named Alice and Bob.

To construct a Turing machine in Magic: The Gathering

requires thr ee main elements: the tape which encodes the

computation, the c ontroller which determines what action to

take next ba sed on the current state and the last read cell, and

the read/write head which interacts with the tape under the

control of the controller.

2

We avoid the term “computable game” which is more commonly used to

mean that the game has a computable winning strategy.

2

A. The Tape

As the rules of Magic: The Gathering do not contain a ny

concept of geometry or adjacency, encoding the tape itself

is tr ic ky. Our solution is to have many creature tokens with

carefully controlled power and toughness, with each token’s

power and toughness r epresenting the distance fr om the head

of the Tur ing machine. The tape to the left of the Turing

machine’s current read head position is represented by a

series of creature to kens which all have the game colour

green, while th e tape to the right is represented by white

tokens. Our distan c e-counting starts at 2, so there is one 2/2

token representing th e space currently under the head of the

Turing machine; a green 3/3 token represents the tape space

immediately to the left of the Turing head, a gr een 4/4 is

the space to the left of that, and so on. Rogozh in’s universal

Turing mac hine starts with the read head in the middle of the

tape [14].

To represent the symbols on the tape, we use creature types.

We choose 18 creature types from the list of creature types in

Magic to correspond to the 18 symbols in Rogozhin’s (2, 18)

UTM. We can choose th e se creature types to begin with suc-

cessive letters of the alphabet: Aetherborn, Basilisk, Cephalid,

Demon, Elf, Faerie, Giant, Harpy, I llusion, Juggernaut, Kavu,

Leviathan, Myr, Noggle, Orc, Pegasus, Rhino, and Sliver. For

example, a green 5/5 A etherborn token represents that the 1

st

symbol is w ritten on the 3

rd

cell to the left of the head, and a

white 10/10 Sliver represents that the 18

th

symbol is written

on the 9

th

cell to the right of the head. These tokens are all

controlled by Bob, except the most recently c reated token (the

space the Tu ring head has just left) which is controlled by

Alice.

B. The Controller

Control instructions in a Turing machine are repr esented by

a table of conditio nal statements of the form “if the machine

is in state s, and the last re ad cell is symbol k, then do such-

and-such.” Many Magic cards h ave triggered abilities which

can function like conditional statements. The two we shall use

are Rotlung Reanimator (“Whenever Rotlu ng Reanimator or

another Cleric dies, create a 2/2 black Zombie creature token”)

and Xathrid Necromancer (“Whenever Xathrid Necromancer

or another Human creature you c ontrol dies, create a tapped

2/2 black Zombie creature token”). We will use both, and the

difference between tapped and unta pped c reature tokens will

contribute to the design of the Tu ring m achine.

Each Rotlung Reanimator

3

needs to trig ger from a dif-

ferent state being read – that is, a different creature type

dying – and needs to encode a different result. Fortunately,

Magic includes cards that can be used to ed it the text of other

cards. The card Artiﬁcial Evolution is uniquely powerful for

our purposes, as it reads “Change the text of target spell or

permane nt by replacing all instanc e s of one creature type with

another. The new creature type can’t be Wall. (This effect

3

For now we will speak about Rotlung Reanimator for simplicity. Some of

these will in fact be Xathrid Necromancers as explained in the next section.

lasts indeﬁnitely.)” So we crea te a large number of copies

of Rotlung Reanimator and edit e ach one. A similar card

Glamerdye can be used to modify the colour word s with in

card text.

Thus, we edit a Rotlung Reanimator by casting two c opies

of Artiﬁcial Evolution replacing ‘Cleric’ with ‘Aethe rborn’

and ‘Zombie’ with ‘Sliver’ and one copy of Glamerdye to

replace ‘blac k’ with ‘white’, so that this Rotlung Reani-

mator now reads “Whenever Rotlung Rean imator or another

Aetherborn dies, create a 2/2 white Sliver creature token ”

4

.

This Rotlung Reanimator now encodes the ﬁrst rule of the

q

1

program of the (2, 18) UTM: “When reading symbol 1

in state A, write symbol 18 and move lef t.” The Aetherborn

creature token represents symbol 1, the Sliver crea ture token

represents symbol 18, and the fact that the token is white leads

to processing that will cause th e head to move left.

We similarly have seventeen more Rotlung Reanimators

encodin g the rest of the q

1

program from [14]. Between them

they say:

1) Whenever an Aetherborn dies, create a 2 /2 white Sliver.

2) Whenever a Basilisk dies, create a 2/2 green Elf.

Whenever a . . . dies, create a 2/2 . . .

18) Whenever a Sliver dies, create a 2/2 green Cephalid.

See Table II for the full encoding of the program.

C. The Read /Write Head

The operation “read the current cell of the tape” is rep-

resented in-game by forcing Alice to c a st Infest to give all

creatures

−

2/

−

2. This causes the unique token with 2 tou ghness

to die. It had a colour (green or white) which is irreleva nt,

and a creature type which corresponds to the symbol written

on that cell. That creature type is noticed by a Rotlung

Reanimator, which has a triggered ability that is used to carr y

out the logic encoded in the head of the Turing machine . It

produces a new 2/2 token, co ntaining the information written

to the cell that was just read.

The Turing machine then moves either left or right and

modiﬁes the tokens to keep the tape in order by ad ding

+1/ + 1 counters to all tokens on one side of the head and

−

1/

−

1 counters to all tokens on the other sid e. M oving left o r

right will be accomplishe d by casting ﬁrst Cleansing Beam

and then So ul Snuffers.

D. Adding a Second State

Everything described so far outlines the operation of one

state of the Turing machine. However, our Turing machine

requires two states. To ac complish this, we leverage phasing:

an object with phasing can ‘phase in’ or ‘phase out’, and

while it’s phased out, it’s treated as though it doesn’t exist.

We can grant phasing to our Rotlung Reanimators usin g the

enchantment Cloak of Invisibility (“Enchanted creature has

phasing and can’t be blocked except by Walls”) and create a

second set of Ro tlung Reanimat ors to encode th e program

4

Throughout this paper, card text that has been modiﬁed using cards such

as A rtiﬁcial Evolution is written in italics.

3

TABLE I

GAME STATE WHEN THE (2, 18) UTM BEGINS

Card

Controller Changed text / choices / attachment

29 Rotlung Reanimator Bob See Table II

7 Xathrid Necromancer Bob See Table II

29 Cloak of Invisibility

Alice attached to each Rotlung Reanimator

7 Cloak of Invisibility Alice attached to each Xathrid Necromancer

Wheel of Sun and Moon Alice attached to Alice

Illusory Gains Alice attached to latest tape token

Steely Resolve

Alice Assembly-Worker

2 Dread of Night Alice Black

Fungus Sliver Alice Incarnation

Rotlung Reanimator

Alice Lhurgoyf, black, Cephalid

Rotlung Reanimator Bob Lhurgoyf, green, Lhurgoyf

Shared Triumph Alice Lhurgoyf

Rotlung Reanimator Alice Rat, black, Cephalid

Rotlung Reanimator

Bob Rat, white, Rat

Shared Triumph Alice Rat

Card Controller

Wild Evocation Bob

Recycle Bob

Privileged Position

Bob

Vigor Alice

Vigor Bob

Mesmeric Orb Alice

Ancient Tomb

Alice

Prismatic Omen Alice

Choke Alice

Blazing Archon

Alice

Blazing Archon Bob

q

2

. At the moment we read the current cell, exactly one set of

Rotlung Reanimato rs will be phased in.

Objects with phasing phase in or out at the beginning of

their c ontroller’s turn, effectively toggling between two states.

Accordingly we will arr a nge for the turn cycle to last 4 turns

for each player when no state chan ge occurs, but just 3 turns

when we need to chan ge state.

IV. THE FULL CONSTRUCTION

Now we will provide the full construction of the Magic: The

Gathering Turing machine and walk thr ough a computational

step. The outline of one step of the comp utation is as follows

(Bob’s tur ns ar e omitted as nothing happens during them):

T1 Alice c asts Infest. Turing processing occurs: a white or

green token dies, a new white or gre en token is c reated.

T2 Alice casts Cleansing Beam, putting two +1/ + 1 c ount-

ers on the side of the tape we are moving away fro m.

T3 If the Turing machine is remaining in the same state,

Alice casts Coalit ion Victor y. If it is changing state,

Alice casts Soul Snuffers, putting a

−

1/

−

1 counter on

each creature.

T4 If the Turing machine is remaining in the same state, this

is the point where Alice casts Soul Snuffer s. Otherwise,

the next computational step begins.

A. Beginning a Computational Step and Casting Spells

At the beginning of a computational step, it is Alice’s turn

and she has the card Infest in hand. Her library consists of the

other cards she will cast during the co mputation (Cleansing

Beam, Coalition Victory, and Soul Snuffers, in tha t order).

Bob’s hand and library are both empty. The Turing machine

is in its starting state and th e tape has already been initialised.

At the start of each of Alice’s turns, she has one card in

hand. She’s for ced to cast it due to Bob controlling Wild

Evocation, which reads “At the beginning of each player’s

upkeep, that player reveals a card at random from their hand.

If it’s a land card, the player pu ts it onto th e battleﬁeld.

Otherwise, the player casts it without paying its mana cost

if able.” When the card resolves, it would normally be put

into her graveyard, but Alice is enchanted by Wheel of Sun

and Moon, which causes it to be placed at the bottom of her

library instead, allowing her to redraw it in the future and

keeping the cards she needs to cast in order. After her upkeep

step, Alice proceeds to her draw step and draws the c ard that

she will ca st next turn.

Alice has no choices throughout this process: she do es

control one land, but it remains permanently tapped because

of Choke (“Islands don’t untap during their con trollers untap

steps”), so she is unable to cast any of the spells she draws

except via Wild Evocation’s ability. Neither player is able to

attack because they both control a Blazing Archon, “Creatures

can’t attack you.”

Bob has no cards in hand and controls Recycle, which reads

(in par t) “Skip your draw step”. This prevents Bob from losing

due to drawing fro m an empty library.

B. Reading the Current Cell

On the ﬁrst turn of the cycle, Alice is forced to cast Infest,

“All creatures get

−

2/

−

2 until end of turn.” This kills one

creature: the tape token at the position of the current read

head, controlled by Bob. This will cause prec isely o ne creature

of Bob’s to trigger – either a Rotlung Reanimator or a

Xathrid Necromancer. Which precise one triggers is b ased

on that token’s cre ature type and the machine’s current state,

correspo nding to the a ppropriate rule in the deﬁnition of the

(2, 18) UTM . This Reanimator or Necromancer will create a

new 2/2 token to replace the one that died. T he new token’s

creature type represents the symbol to be written to the current

cell, and the new token’s colour indicates the direction for the

machine to move: white for left or green for right.

Alice controls Illusory Gains, an Aura which reads “You

control enchanted creature. Whenever a creature enters the

battleﬁeld under an opponent’s control, attach Illusory Gains to

that creature.” Each time one of Bo b’s Rotlung Reanimato r s

or Xathrid Necromancers creates a new token, Illusory

Gains triggers, granting Alice co ntrol of the newest token o n

the tape, and reverting contr ol of the previous token to Bob.

4

TABLE II

FULL TEXT OF THE ROTLUNG REANIMATORS AND XATHRID NECROMANCERS ENCODING THE (2, 18) UTM

Rogozhin’s program Card text

q

1

1 c

2

Lq

1

Whenever an Aetherborn dies, create a 2/2 white Sliver

q

1

−→

1

←−

1

1

Rq

1

Whenever a Basilisk dies, create a 2/2 green Elf

q

1

←−

1 c

2

Lq

1

Whenever a Cephalid dies, create a 2/2 white Sliver

q

1

−→

1

1

1 Rq

1

Whenever a Demon dies, create a 2/2 green Aetherborn

q

1

←−

1

1

−→

1

1

Lq

1

Whenever an Elf dies, create a 2/2 white Demon

q

1

b

←−

b Rq

1

Whenever a Faerie dies, create a 2/2 green Harpy

q

1

−→

b

←−

b

1

Rq

1

Whenever a Giant dies, create a 2/2 green Juggernaut

q

1

←−

b b Lq

1

Whenever a Harpy dies, create a 2/2 white Faerie

q

1

−→

b

1

b Rq

1

Whenever an Illusion dies, create a 2/2 green Faerie

q

1

←−

b

1

−→

b

1

Lq

1

Whenever a Juggernaut dies, create a 2/2 white Illusion

q

1

b

2

b

3

Lq

2

Whenever a Kavu dies, create a tapped 2/2 white Leviathan

q

1

b

3

−→

b

1

Lq

2

Whenever a Leviathan dies, create a tapped 2/2 white Illusion

q

1

c

−→

1 Lq

2

Whenever a Myr dies, create a tapped 2/2 white Basilisk

q

1

−→

c

←−

c Rq

1

Whenever a Noggle dies, create a 2/2 green Orc

q

1

←−

c

−→

c

1

Lq

1

Whenever an Orc dies, create a 2/2 white Pegasus

q

1

−→

c

1

←−

c

1

Rq

2

Whenever a Pegasus dies, create a tapped 2/2 green Rhino

q

1

←−

c

1

HALT

Whenever a Rhino dies, create a 2/2 blue Assassin

q

1

c

2

←−

1 Rq

1

Whenever a Sliver dies, create a 2/2 green Cephalid

q

2

1

←−

1 Rq

2

Whenever an Aetherborn dies, create a 2/2 green Cephalid

q

2

−→

1

←−

1 Rq

2

Whenever a Basilisk dies, create a 2/2 green Cephalid

q

2

←−

1

−→

1 Lq

2

Whenever a Cephalid dies, create a 2/2 white Basilisk

q

2

−→

1

1

←−

1

1

Rq

2

Whenever a Demon dies, create a 2/2 green Elf

q

2

←−

1

1

1 Lq

2

Whenever an Elf dies, create a 2/2 white Aetherborn

q

2

b b

2

Rq

1

Whenever a Faerie dies, create a tapped 2/2 green Kavu

q

2

−→

b

←−

b Rq

2

Whenever a Giant dies, create a 2/2 green H arpy

q

2

←−

b

−→

b Lq

2

Whenever a Harpy dies, create a 2/2 white Giant

q

2

−→

b

1

←−

b

1

Rq

2

Whenever an Illusion dies, create a 2/2 green Juggernaut

q

2

←−

b

1

−→

b Lq

2

Whenever a Juggernaut dies, create a 2/2 white Giant

q

2

b

2

b Rq

1

Whenever a Kavu dies, create a tapped 2/2 green Faerie

q

2

b

3

←−

b

1

Rq

2

Whenever a Leviathan dies, create a 2/2 green Juggernaut

q

2

c

←−

c Rq

2

Whenever a Myr dies, create a 2/2 green Orc

q

2

−→

c

←−

c Rq

2

Whenever a Noggle dies, create a 2/2 green Orc

q

2

←−

c

−→

c Lq

2

Whenever an Orc dies, create a 2/2 white Noggle

q

2

−→

c

1

c

2

Rq

2

Whenever a Pegasus dies, create a 2/2 green Sliver

q

2

←−

c

1

c

2

Lq

1

Whenever a Rhino dies, create a tapped 2/2 white Sliver

q

2

c

2

c Lq

2

Whenever a Sliver dies, create a 2/2 white Myr

So at any point Bob controls all of the tape except for the

most rece ntly written sy mbol, which is controlled by Alice .

C. Moving Left or Right

If the new token is white, the Turing machine needs to

move left. To do this we n eed to take two actions: put a

+1/ + 1 counter on a ll white creatures (move the tape away

from white), and put a

−

1/

−

1 counter on all green creatures

(move the tape towards green). We rephrase this instead as:

put two +1/ + 1 counters on a ll white creatures, and put a

−

1/

−

1 counter on all creatur es.

On Alice’s second turn, she ca sts Cleansing Bea m, which

reads “Cleansing Beam deals 2 d a mage to target creature and

each other crea ture that shares a color with it.” Bob controls

Privileged Position so none of Bob’s creatures can be targeted

by any spell Alice casts. Alice controls some creature s other

than the tape token, but they have all been gr a nted creature

type Assembly-Worker by a hacked Olivia Voldaren, and

Alice controls a Steely Resolve namin g Assembly-Worker

(“Creatures of the chosen type have shroud. (They can’t b e

the targets of spells or abilities.)”) This makes it so that the

only legal target for Cleansing Beam is the one tape token

that Alice controls thanks to her Illusory Gains.

Recall that this token is white if we’re moving left, or green

if we’re moving right. Cleansing Beam is about to deal 2

damage to ea ch white crea ture if we’re moving left, or to

each green creature if we’re moving rig ht. Alice an d Bob

each control a copy of Vigor – “If damage would be dealt

to another creature you control, prevent tha t damage. Put a

+1/ + 1 counter on that creature for each 1 damage prevented

this way.” So Cleansing Beam ends up putting two +1/ + 1

counters on eith e r each white cr e ature or each gr e en creature.

On the last tu rn of the cycle, Alice casts Soul Snuffers, a

3/3 black creature which reads “When Soul Snuffers enters the

battleﬁeld, put a

−

1/

−

1 counter on eac h creature.” Th e re are

two copies of Dread of Night hacked to each say “Black

creatures get

−

1/

−

1”, which mean that the Soul Snuffers’

triggered ability will kill itself, as well as shrinking every other

5

creature. The creatures comprising the tape have n ow received

either a single

−

1/

−

1 counter, or two +1/ + 1 counter s and a

−

1/

−

1 counter.

To ensure that the creatures providing the infrastructure

(such as Rotlung Rea nima tor) aren’t killed by the succession

of

−

1/

−

1 counters each computational step, we arrange that

they also have game colours gre en, white, red and black,

using Prismatic Lace, “Target permanent becomes the color

or colors of your choice. (This effect lasts indeﬁnitely.)”

Accordingly, each cycle Cleansing Beam will put two +1/+1

counters on them, growing them faster than the

−

1/

−

1 counters

shrink them. This applies to each creature excep t Vigor

itself; to keep each player’s Vigor from dwindling, there is

a Fungus Sliver hacked to read “All Incarnation creatures

have ‘Whenever this crea ture is dealt damage, put a +1/ + 1

counter on it.’ ”

D. Changing State

The instruction to change state is handled by replacing seven

of Bob’s Rotlung Reanimators with Xathrid Necromancer.

These two c ards have very similar text, except that Xathrid

Necromancer only notices Bob’s creatures dying (this is not

a problem, as the active cell of the tape is always contro lled

by Bob), and that Xathrid Necromancer creates its token

tapped.

For example, when the q

1

program (State A) sees symbol 1,

it writes symbol 18, m oves left, and remains in state A. This

is represented by a p hasing Rotlung Reanimator under Bob’s

control saying “Whenever Rotlung Reanimator or another

Aetherborn dies, c reate a 2/2 white Sliver creature token.”

By contrast, when the q

1

program sees symbol 11, it

writes symbol 12, m oves left, and changes to state B. This is

represented by a phasing Xathrid Necromancer under Bob’s

control saying “Whe never Xathrid Necromancer or another

Kavu creature you control dies, create a tapped 2/2 white

Leviathan creature token.”

In both cases this token is cr eated under Bob’s control on

turn T1, but Alice’s Illusory Gains triggers and grants her

control of it. In the case where it’s tappe d, that means at

the beginning of turn T2, it will untap. This causes Alice’s

Mesmeric Orb’s trigger to be put on the stack at the same time

as Bob’s Wild Evocation’s trigger ( sin ce no pla yer receives

priority during the untap step). Alice is the active pla yer, so

Alice’s tr igger is put on the stack ﬁrst and then Bob’s [16]; so

the Wild Evocation trigger resolves, forcing Alice to ca st and

resolve Cleansing Beam, before the Mesmeric Orb trigger

resolves.

When the Mesmeric Orb trigger does reso lve, it tries to

put the Coalition Victory from the top of Alice’s library into

her graveyard. But Wheel of Sun and Moon modiﬁes this

event to put Coalition Victory onto the bottom of her library,

just underneath the Cleansing Beam that’s just resolved.

Once all these triggers are resolved, Alice proceeds to her

draw step. When the state is not c hanging, she will draw

Coalition Victory at this point, but when the state is changing,

that card is skipped and she moves on to draw Soul Snuffers

in turn T2’s draw step, so she will cast it on turn T3.

The net result of this is that the computation step is 3

turns long for each player when the state is changing, but

4 turns long f or each player when the state is not changing. In

the normal 4-turn operation, Bob’s p hasing Reanima tors and

Necromancers will phase in twice and ph ase out twice , and be

in the same state on one cycle’s turn T1 as they were in the

previous cycle’s turn T1. But when changing state, they will

have changed phase by the next cycle’s turn T1, switching the

Turing machine’s state.

E. Out of Tape

The Turing tape can be initialised to a ny desired length

before starting processing. But it is preferable to allow the

machine to run on a simulated inﬁnite tape: in other words,

to assume that any uninitialised tape space contains symbol

3 (the blank symbol in the (2, 18 ) UTM), represented by

creature type Cephalid. This is accomplished by having the

ends of the currently-initialised tape marked by two special

tokens, one green Lhurgoyf and one white Rat. Su ppose we’ve

exhausted all the initialised tape to the left. This means that

the casting of Infest on turn T1 kills the Lhurgoyf rather than

one of the normal tape types. This does not directly trigg er

any of the normal Reanimators/Necromancers. Instead, Bob

has another Rotlung Re animator hacked to read “Wh e never

Rotlung Reanimator or another Lhurgoyf dies, create a 2/2

green Lhurgoyf creature token”, and A lice has a Rotlung

Reanimator hacked to read “Whenever Rotlung Reanimator

or another Lhurgoyf dies, create a 2/2 black Cephalid creature

token.” Bob’s trigger will resolve ﬁrst, then Alice’s.

First, Bob’s Reanimator trigger creates a new Lhurgoyf just

to the left of the current head. (Alice’s Illusory Gains triggers

and gives her control of this new Lhurgoyf, but that will

soon change.) We have one c opy of Shared Tr iumph set to

Lhurgoyf (“Creatures of th e chosen type get +1/+1”) so this

token arrives as a 3/3.

Second, Alice’s Reanimator trigger now creates a 2/2 black

Cephalid under Alice’s control. The same two copies of Dread

of Night as before are giving all black cre atures

−

2/

−

2, so the

black Cephalid w ill arrive as a 0/0 and immediately die.

The death of this Cephalid triggers one of th e regular

phasing Reanimators o f Bob’s just as if a tape cell containing

symbol 3 had been read: a new 2/2 token is created and

Illusory Gains moves to that new token. The green Lhurgoyf

token ser ving as an end-of-tape marker has be en recreated one

step over to th e left.

The situation for the white Rat representing the right-hand

end o f the tape is exactly equivalent. Bob has a Rotlung

Reanimator hacked to read “Whenever Rotlung Reanimator

or another Rat dies, crea te a 2/2 white Rat creature token”;

Alice has a Rotlung Reanimator hacked to read “Whenever

Rotlung Reanim ator or another Rat dies, create a 2/2 black

Cephalid creature token”; and we have ano ther Shared Tri-

umph set to Rat.

6

(This algorithm would be a little more complex if reading

symbol 3 could cause a state change in the (2, 18) UTM , but

thankfully it ca nnot.)

F. Halting

We choose to encode halting as making Alice win th e game.

When the Turing machine doesn’t change state, Alice casts

the card Coalition Victory on her third turn. It reads “You win

the game if you control a lan d of eac h basic land type and a

creature of e ach color.” This normally accomplishes nothing

because she controls no blue cr e atures (Prismatic Lace has

been used to give her creatures of all the other colours). She

does, however, control one land, and also controls Prismatic

Omen, which re ads “L ands you control are eve ry basic land

type in addition to their other types.” Recall that Choke is in

play, preventing he r from activating the mana ability of this

land.

When the halt symbol is read (symbol 17 in state A), the

appropriate phasing Reanimator of Bob’s read s “Whenever

Rotlung Reanimator or another Rhino dies, create a 2/2 blue

Assassin creature token.” Alice’s Illusory Gains takes control

of this Assassin token in th e usual way in turn T1. She now

meets the condition for Coalition Victory when she casts it

on turn T 3, and wins the game.

If the encoded machine does not in fact h a lt then th e game

has entered an unbreakable deterministic in ﬁnite loop , which

is speciﬁed as a draw by r ule 104.4b [16].

V. DISCUSSION

A. Consequen ces for Computational The ories of Ga m e s

This construction establishes that Magic: The Gathering is

the most computationally complex real-world game known in

the literature. I n addition to showing that optimal strategic

play in Magic is non-computable, it also shows that merely

evaluatin g the deterministic consequences of past moves in

Magic is non-computable. The full complexity of optimal

strategic play remains an open qu estion, as do ma ny other

computational aspects of Magic. For example, a player ap-

pears to have inﬁnitely many moves available to them from

some board states of Magic. Whether or not there exists a

real-world game o f Magic in which a player has inﬁnitely

many meaningfully different moves available to them has the

potentially to highly impact the way we understand and model

games as a form of computation.

Indeed , this result raises several interesting philosophical

questions about games as a form of computa tion. Some

authors, such as Demaine and Hearn [9], have sought a formal

framework for modelling games that is strictly sub-Turing.

Unlike the open-world, non-strategic game s in which Turing

machines have been constructed be fore, Magic: The Gathering

is unam biguously a two-playe r strategic gam e like such models

attempt to represen t. Therefore this result shows th a t any sub-

Turing model is n ecessarily inadequate to capture all gam es.

Quite the opposite: it seems likely that a super-Turing model

of game s would be necessary to exp la in Magic. The na¨ıve

extension of Demaine and Hea rn’s Constraint Logic to allow

for unbounded memory appears to be meaningless, although

it’s possible that a clever appr oach would bring success.

Open Problem 3: Does there exist a generalisation of

Constraint Logic that explains the computational complexity

of Magic: The G athering?

Although our construction is reducible to the Halting Prob-

lem, the fact that even evaluating a board is non-computable

is strongly suggestive that th e complexity of strategic play is

greater than that. We be lieve there is strong evidence that the

true computation al complexity is far highe r. In particular, we

conjecture:

Conjecture 4: Playing Magic: The Gathering optimally is

at least as hard as 0

(ω)

.

Whether or not it is possible for there to be a real-world

game whose computational complexity is strictly harder than

0

(ω)

is an interesting p hilosophical question. If not, then this

conjecture would imply that Magic: The Gathering is as har d

as it is possible for a real-world game to be.

B. Real-world Playab ility a nd Legality

While ther e are practical difﬁculties invo lved with correctly

setting up the necessary board state, such as running out of

space on y our table, a sufﬁciently tena cious player could set

up and execute this construction in a re al-world tournament

TABLE III

60-CARD DECKLIS T TO PLAY THE TURING MACHI NE I N A LEGACY TOURNAMENT

Card

Purpose Card Purpose Card Purpose

4 Ancient Tom b Bootstrap 1 Rotlung Reanimator Logic processing 1 Xathrid Necromancer Change state

4 Lotus P etal

Bootstrap 1 Cloak of Invisibility Logic processing 1 Mesmeric Orb Change state

4 Grim Monolith Inﬁnite mana device 1 Infest Logic processing 1 Coalition Victory Halting device

4 Power Artifact Inﬁnite mana device 1 Cleansing Beam Logic processing 1 Prismatic Omen Halting device

4 Gemstone Array

Inﬁnite mana device 1 Soul Snuffers Logic processing 1 Choke Halting device

4 Staff of Domination Draw rest of deck 1 Illusory Gains Logic processing 1 Recycle Remove choices

1 Memnarch Make token copies 1 Privileged Position Logic processing 1 Blazing Archon Remove choices

1 Stolen Identity Make token copies 1 Steely Resolve Logic processing 1 Djinn Illuminatus Simplify setup

1 Artiﬁcial Evolution

Edit cards 1 Vigor Logic processing 1 Reito Lantern Simplify setup

1 Olivia Voldaren Edit cards 1 Fungus Sliver Logic processing 1 Claws of Gix Simplify setup

1 Glamerdye Edit cards 1 Dread of Night Logic processing 1 Riptide Replicator Set up tape

1 Prismatic Lace

Edit cards 1 Wild Evocation Forced play device 1 Capsize Set up tape

1 Donate

Edit card control 1 Wheel of Sun and Moon Forced play device 1 Karn Liberated Cleanup after setup

1 Reality Ripple Edit card phase 1 Shared Triumph Inﬁnite tape device 1 Fathom Feeder Cleanup after setup

7

game of Magic: The Gathering. An example 60-card deck that

is ca pable of executing this construction on the ﬁrst turn of

the game and which is legal in the competitive Legacy format

can be seen in Table III.

With the correct d raw, the deck uses Ancient Tomb and

three Lotus Petals to play Grim Monolith a nd Power Arti-

fact and generate unlimited colour less mana, at which point

Staff of Domination draws the rest of the deck and Ge m-

stone A rray gene rates unlimited coloured mana. T he deck

casts most of the permanents immediately, and uses Stolen

Identity to make to ken copies of them (using Memnarch

ﬁrst on the en chantments like Cloak of Invisibility). The

tape is initialised with Riptide Replicator and Capsize. Dj inn

Illuminatus or Reito Lantern allow repeated casting of the

text-modiﬁcation card s, as well as Reality Ripple whic h sets

the phase of the Rotlung Reanimators and Donate which

gives most permanents to Bob. Once everything is set up,

Steely Resolve is cast, and then Karn Liberated and Capsize

are used to exile all setup permanen ts and all cards from

Bob’s han d, eventually exiling Capsize and Karn Liberate d

themselves. Now no player has any remainin g choices except

to let the Turing machine execute.

In addition to the Comprehensive Rules [16], play at sanc-

tioned Magic: The Gathering tourna ments is also governed by

the Tournament Rules [17]. Some of these rules, most notably

the ones involving slow play, may effect an individual’s ability

to successfully execute the combo due to concerns a bout the

sheer amount of time it would take to manually move the

tokens ar ound to simulate a computation on a Tu ring machine.

This would not be a concern for two agents with sufﬁciently

high computational power, a s the Tournament Rules also

provide a mechanism called “shortcuts” for players to skip

carrying out laborious loops if both players agree on the game

state at the beginning and the end of the shortcut.

VI. CONCLUSION

We have presented a methodology for embeddin g Ro-

gozhin’s (2, 18) universal Turing machine in a two-player

game of Magic: The Gathering. Consequently, we have shown

that identifying the outcome of a game of Magic in which all

moves are forced for the r e st of the gam e is undecidable. In

addition to solving a decade-old outstanding open problem, in

the process of arriving at our resu lt we showed that Magic:

The Gathering d oes not ﬁt assumptions commonly made by

computer scientists while modelling games. We conjec ture that

optimal play in Magic is far harder than this result alon e

implies, and leave the true complexity of Magic and the

reconciliation of Magic with existing theories of games fo r

future research.

ACKNOWLEDGEMENTS

We are grateful to C-Y. Howe for help simplifying our Tur-

ing machine constructio n considerab ly and to Adam Yedidia

for conversations about the design and construction of Turing

machines.

REFERENCES

[1] David Auger and Oliver Teytaud. The frontier of decidability in partially

observable recursive games. International Journal of Foundations of

Computer Science, 2012.

[2] Stella Biderman and Bjørn Kjos-Hanssen. Non-comparable natu-

ral numbers. Theoretical Computer Science Stack Exchange, 2018.

https://cstheory.stackexchange.com/q/41384(version:2018-08-16).

[3] Krishnendu Chatterjee and Rasmus Ibsen-Jensen. T he complexity of

deciding legality of a single step of Magic: The Gathering. In 22nd

European Conference on Artiﬁcial Intelligence, 2016.

[4] Alex Churchill. Magic: The Gathering is Turing complete v5, 2012.

https://www.toothycat.net/

∼

hologram/Turing/.

[5] Alex Churchill et al. Magic is Turing complete (the Turing machine

combo), 2014. http://tinyurl.com/pv3n2lg.

[6] Peter I. Cowling Colin D. Ward and Edward J. Powley. Ensemble deter-

minization in Monte Carlo tree search for the imperfect information card

game Magic: The Gathering. In IEEE Transactions on Computational

Intelligence and AI in Games, volume 4, 2012.

[7] Michael J. Coulombe and Jayson Lynch. Cooperating in video games?

Impossible! Undecidability of team multiplayer games. In 9th Interna-

tional Conference on Fun with Algorithms, 2018.

[8] Erik D. Demaine and Robert A. Hearn. Playing games with algorithms:

algorithmic combinatorial game theory. In 26th Symp. on Mathematical

Foundations in Computer Science, pages 18–32, 2001.

[9] Erik D. Demaine and Robert A. Hearn. Constraint logic: A uniform

framework for modeling computation as games. In 2008 23rd Annual

IEEE Conference on Computational Complexity, pages 149–162, 2008.

[10] Erik D. Demaine and Robert A. H earn. Games, Puzzles, and Computa-

tion. CRC Press, 2009.

[11] Alexander Esche. Mathematical Programming and Magic: The Gather-

ing. PhD thesis, Northern Illinois University, 2018.

[12] Eugenio Fortanely. Personal communication, 2018.

[13] H. G. Rice. Classes of recursively enumerable sets and their decision

problems. Trans. Amer. Math. Soc., 74:358366, 1953.

[14] Yurii Rogozhin. Sm all universal Turing machines. Theoretical Computer

Science, 168(2):215–240, 1996.

[15] Colin D. Ward and Peter I. Cowling. Monte Carlo search applied to

card selection in Magic: The Gathering. In CIG’09 Proceedings of the

5th international conference on Computational Intelligence and Games,

pages 9–16, 2009.

[16] Wizards of the Coast. Magic: The G ath-

ering comprehensive rules, Aug 2018.

https://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules.

[17] Wizards of the Coast. Magic: The

Gathering tournament rules, Aug 2018.

https://wpn.wizards.com/sites/wpn/ﬁles/attachements/mtg

mtr 21jan19 en.pdf.

8

![](https://i.imgur.com/UmfRdn6.jpg)
![](https://i.imgur.com/TkQmL50.jpg)
![](https://i.imgur.com/cz2y2uY.jpg)
![](https://i.imgur.com/NmRaCyd.jpg)
![](https://i.imgur.com/KXbMR8g.jpg)
![](https://i.imgur.com/QmIDXIC.jpg)
![](https://i.imgur.com/4tOqPt4.jpg)
#### What is a Creature Token?
Creature tokens are effectively markers that represent a creature card with specific properties. Creature tokens can be created as the result of a spell for instance. Creature Tokens are not actual Magic cards. They cannot exist in your hand or deck. While playing the game, players can use whatever is more player friendly for them to represent a creature token with specific properties (e.g. die, piece of torn paper, etc).
Example of a Turing Machine built by a player in Minecraft
![](https://i.imgur.com/Ri3E1aW.png)
![](https://i.imgur.com/x7qX86o.jpg)
![](https://i.imgur.com/dITxN0c.jpg)
![](https://i.imgur.com/JzoBkQm.jpg)
![](https://i.imgur.com/KOgOpih.jpg)
![](https://i.imgur.com/mfrzt5F.jpg)
![](https://i.imgur.com/JzQtbet.jpg)
![](https://i.imgur.com/Ode71ig.jpg)
Caulombe and Lynch were able to reason about those video games by structuring them into an abstract game on graphs that serves as a simplification of the Team Computation Game they mention (which is undecidable).
Here is an example of how they built those graphs leveraging elements of a game (Super Smash Bros in this case):
![](https://i.imgur.com/dCuwf4F.png)
If you are curious to learn more about it, here is the [paper](https://drops.dagstuhl.de/opus/volltexte/2018/8805/pdf/LIPIcs-FUN-2018-14.pdf) by Caulombe and Lynch.
Rengo Kriegspiel is a "blindfolded", team variant of Go. There are two teams with two players each, a referee and five Go sets. The players move alternatingly. Each player keeps track of their own moves on their own board; they are not informed about teammates' or opponents' moves. The referee keeps track of the complete game and informs a player if their move was illegal, in which case they can try again. The referee removes captured stones from all affected boards.
![](https://i.imgur.com/UlI4r8W.png)
*A game of Rengo Kriegspiel*
Unlike games like Chess and Go, researchers still haven't been able to develop an AI capable playing magic at the level of the best human Magic players.
## Playing Magic
The full rules of Magic are extensive - you can find the official document [here (https://media.wizards.com/2021/downloads/MagicCompRules%20202109224.pdf), which is over 250 pages long.
#### Overview of game
A typical game of Magic usually involves two players. Each player has their own deck of cards, either one previously constructed or made from a limited pool of cards for the event. A player starts the game with a "life total" of twenty and loses the game when their life total is reduced to zero. A player's life total is typically reduced via an attack by an unblocked creature or by direct damage caused by spells. A player can also lose if they must draw from an empty deck. Some cards specify other ways to win or lose the game. Additionally, the "Golden Rule of Magic" states that "whenever a card's text directly contradicts the rules, the card takes precedence".
If you wish to learn how to play Magic, [this video](https://www.youtube.com/watch?v=RZyXU1L3JXk) is a good resource.
![](https://i.imgur.com/j7hi0HL.png)
![](https://i.imgur.com/bKSaDmW.jpg)
![](https://i.imgur.com/nlR78ll.jpg)
![](https://i.imgur.com/S3zGlk7.jpg)
Rice's Theorem states that all nontrivial semantic properties of programs are undecidable. A semantic property is one about the program's behavior for a given set of inputs (e.g. does the program terminate for all inputs), unlike a syntactic property (e.g. does the program contain a go-to statement)
Rogozhin's (2,18) Turing Machine is a Turing machine with 2 states and 18 different symbols (that can be written to the "tape")
### Rogozhin Turing Machines
When Alan Turing came up with the idea of a universal machine he had in mind the simplest computing model powerful enough to calculate all possible functions that can be calculated.
Turing Machines are generally defined with a set of symbols (which can be written to the "tape") and a set of states that the Turing Machine can be in. The natural question is: What is the smallest possible Turing Machine? Claude Shannon showed that two symbols were sufficient so long as enough states were used (or vice versa), and that it was always possible to exchange states for symbols.
Yurii Rogozhin worked on finding the smallest universal Turing Machines. Generally, the syntax for Universal Turing Machines is $UTM(m; n)$ - which denotes the class of universal Turing machines with $m$ states and $n$ symbols.
As the authors state, there are a number of real world games that show non-trivial computational complexity:
- **Tetris** - If the sequence of pieces is given, it is NP-complete to determine whether it is possible to keep the stack below a finite height
- **Crossword puzzle construction** - NP-complete
- **Dots-and-boxes** - NP-hard
- **Mahjong Solitaire** - NP-complete
- **Minesweeper** - coNP-Hard
- **FreeCell** - NP-Complete
- **Jigsaw puzzles** - Jigsaw puzzles can be formalized in a variety of ways, different edge matching rules, different boundary conditions, etc. Many versions are NP-Complete
- **Lemmings** - NP-Complete
Cards mentioned by the authors:
![](https://i.imgur.com/KktkX2L.jpg)
![](https://i.imgur.com/Muk7kBO.jpg)
![](https://i.imgur.com/ns2jzg2.jpg)
![](https://i.imgur.com/R0hYhxL.jpg)
Zero-sum games are 2-player games where the sum of the payoffs of the players is 0 - these are games of pure competition between the players. Zero-sum stochastic games are dynamic zero-sum games played in discrete time with probabilistic transitions between states. In games with imperfect information (like Poker) there might be certain parts of hidden state that are not visible to one or more players
### Magic The Gathering
Magic: The Gathering is a trading card game created by the American mathematician and inventor Richard Garfield (great-grandson of U.S. President James A. Garfield) and released in 1993. Garfield was influenced by his passion for the roleplaying game Dungeons & Dragons.
Magic is considered to be the first trading card game. It successfully combined elements of board games and card collecting. During a game of Magic, two players compete by leveraging their homebuilt deck of cards to defeat their opponent by spending "mana" to play creatures, spells and other items. Players can customize their deck before each battle, choosing cards from their own library. More than 20,000 different cards have been released for Magic: The Gathering since its release and there are an estimated 20 million players all over the world.
![](https://i.imgur.com/j2HEE0p.jpg)
*Two players playing Magic: the Gathering*