basic rules - mit mathematics

Basic rules

Two players: Blue and Red.

Perfect information.

Players move alternately.

First player unable to move loses .

The game must terminate.

Mathematical Games – p. 1

Outcomes (assuming perfect play)

Blue wins (whoever moves first): G > 0

Red wins (whoever moves first): G < 0

Mover loses: G = 0

Mover wins: G‖0

Mathematical Games – p. 2

Two elegant classes of games

number game : always disadvantageous tomove (so never G‖0)

impartial game : same moves alwaysavailable to each player

Mathematical Games – p. 3

Blue-Red Hackenbush

ground

prototypical number game:

Blue-Red Hackenbush : A player removes oneedge of his or her color. Any edges notconnected to the ground are also removed. Firstperson unable to move loses.

Mathematical Games – p. 4

An example

Mathematical Games – p. 5

A Hackenbush sum

Let G be a Blue-Red Hackenbush position (orany game). Recall:

Blue wins: G > 0

Red wins: G < 0

Mover loses: G = 0

G + H

HG

Mathematical Games – p. 6

A Hackenbush value

sum: 0 (mover loses), G= 0

sum: 2 (Blue is two moves ahead), G> 0

−1−2−223

−3−2313value (to Blue):

Mathematical Games – p. 7

1/2

value = ?clearly >0: Blue wins

mover loses!

x + x - 1 = 0, so x = 1/2

G

Blue is 1/2 move ahead in G.

Mathematical Games – p. 8

Another position

What about

?

Mathematical Games – p. 9

Another position

What about

?

Clearly G < 0.

Mathematical Games – p. 9

−13/8

8x + 13 = 0 (mover loses!)

x = -13/8Mathematical Games – p. 10

b and r

How to compute the value v(G) of any Blue-RedHackenbush position G?

Let b be the largest value of any position to whichBlue can move. Let r be the smallest value of anyposition to which Red can move. (We will alwayshave b < r.)

Mathematical Games – p. 11

The simplicity rule

The Simplicity Rule. (a) If there is an integer nsatisfying b < n < r, then v(G) is the closestsuch integer to 0.

(b) Otherwise v(G) is the (unique) rationalnumber x satisfying b < x < r whosedenominator is the smallest possible power of 2.

Mathematical Games – p. 12

The simplicity rule

The Simplicity Rule. (a) If there is an integer nsatisfying b < n < r, then v(G) is the closestsuch integer to 0.

(b) Otherwise v(G) is the (unique) rationalnumber x satisfying b < x < r whosedenominator is the smallest possible power of 2.

Moreover, v(G + H) = v(G) + v(H).

Mathematical Games – p. 12

Some examples

Examples.

b r x

23

461

23

−5 25

80

0 1 1

2

1

4

5

16

9

32

1

4

7

16

3

8

−27

8−2 3

32−21

2

Mathematical Games – p. 13

A Hackenbush computation

0 1/2 2

b = 1/2, r = 2, x = 1

1 = 0 (mover loses)1 -

xvalue

Mathematical Games – p. 14

Value of Blue-Red strings

1 + 1 . 0 1 1 0 1

= 2 + 1/4 + 1/8 + 1/32 = 2 13/32

.- 1 1 0 1 1 1

= - ( 1/2 + 1/4 + 1/16 + 1/32 + 1/64) = - 55/64

Mathematical Games – p. 15

Impartial Hackenbush

Now suppose there are also black edges, whicheither player can remove. A game with all blackedges is called an impartial (Hackenbush) game.At any stage of such a game, the two playersalways have the same available moves.

Mathematical Games – p. 16

∗1

*1

Mover wins! Not a number game.

Neither = 0, < 0, or > 0.

Two outcomes of any impartial game: moverwins or mover loses.

Mathematical Games – p. 17

∗n

Denote by ∗n (star n) the impartial game withone chain of length n.

*5Mathematical Games – p. 18

A nice fact

We can still assign a number with usefulproperties to an impartial game, based on thefollowing fact.

Fact. Given any (finite) impartial game G, there isa unique integer n ≥ 0 such that mover loses inthe sum of G and ∗n, i.e.,

G + ∗n = 0.

Mathematical Games – p. 19

The Sprague-Grundy number

Fact. Given any (finite) impartial game G, thereis a unique integer n ≥ 0 such that mover losesin the sum of G and ∗n, i.e.,

G + ∗n = 0.

Mathematical Games – p. 20

The Sprague-Grundy number

Fact. Given any (finite) impartial game G, thereis a unique integer n ≥ 0 such that mover losesin the sum of G and ∗n, i.e.,

G + ∗n = 0.

Denote this integer by N(G), the Sprague-Grundynumber of G.

NOTE: Mover loses (i.e., G = 0) if and only ifN(G) = 0.

Mathematical Games – p. 20

A simple example

N (*5) = 5

*5 *5

Mover loses (second player copiesfirst player).

*5 + *5 = 0

In general, N(∗n) = n.

Mathematical Games – p. 21

Nim addition

Nim addition . Define m⊕n by writing m and n inbinary, adding without carrying (mod 2 additionin each column), and reading result in binary.

Mathematical Games – p. 22

Nim addition

Nim addition . Define m⊕n by writing m and n inbinary, adding without carrying (mod 2 additionin each column), and reading result in binary.

Example. 13 ⊕ 11 ⊕ 7 ⊕ 4 = 5

8 4 2 113 = 1 1 0 111 = 1 0 1 17 = 1 1 14 = 1 0 05 = 0 1 0 1

Mathematical Games – p. 22

The Nim-Sum Theorem

Nim-sum Theorem. Let G and H be impartialgames. Then

N(G + H) = N(G) ⊕ N(H).

Mathematical Games – p. 23

Nim

Nim : sum of ∗n’s.

Mathematical Games – p. 24

Nim

Nim : sum of ∗n’s.

Last Year at Marienbad :

*1 + *3 + *5 + *7

Mathematical Games – p. 24

∗1 + ∗3 + ∗5 + ∗7

*1 + *3 + *5 + *7

4 2 11 = 13 = 1 15 = 1 0 17 = 1 1 10 = 0 0 0,

so mover loses!

Mathematical Games – p. 25

A Nim example

How to play Nim :

G = ∗23 + ∗18 + ∗13 + ∗7 + ∗5

23 = 1011118 = 10010

7 = 111 5 = 101

113 = 1 10 0111 = 7

Mathematical Games – p. 26

A Nim example

How to play Nim :

G = ∗23 + ∗18 + ∗13 + ∗7 + ∗5

23 = 1011118 = 10010

7 = 111 5 = 101

113 = 1 10 0111 = 7

Only winning move is to change ∗13 to ∗7.

Mathematical Games – p. 26

The minimal excludant

How to compute N(G) in general : If S is a setof nonnegative integers, let mex(S) (theminimal excludant of S) be the leastnonnegative integer not in S.

mex{0, 1, 2, 5, 6, 8} = 3

mex{4, 7, 8, 12} = 0.

Mathematical Games – p. 27

The Mex Rule

Mex Rule (analogue of Simplicity Rule). Let S bethe set of all Sprague-Grundy numbers ofpositions that can be reached in one move fromthe impartial game G. Then

N(G) = mex(S).

Mathematical Games – p. 28

A mex example

Can move to 4, 1 ⊕ 3 = 2, and 2 ⊕ 2 = 0. Thus

N(G) = mex{0, 2, 4} = 1.

Mathematical Games – p. 29

A mex example

Can move to 4, 1 ⊕ 3 = 2, and 2 ⊕ 2 = 0. Thus

N(G) = mex{0, 2, 4} = 1.

mover loses!

Mathematical Games – p. 29

The infinite

.

.

.

Clearly v(G) > n for all n, i.e., G is infinite . Sayv(G) = ω.

Mathematical Games – p. 30

The infinite

.

.

.

Clearly v(G) > n for all n, i.e., G is infinite . Sayv(G) = ω. (Still ends in finitely many moves.)

Mathematical Games – p. 30

Even more infinite

.

.

.

Clearly G − ω > 0, so G is more infinite than ω.Call v(G) = ω + 1.

Mathematical Games – p. 31

ω + 1 versus −ω

.

.

.

.

.

.

Blue takes the isolated edge to win.

Mathematical Games – p. 32

Ordinal games

Similarly, every ordinal number is the value of agame.

Mathematical Games – p. 33

ω − 1

.

.

.

We can do more! v(G) = ω − 1.

Mathematical Games – p. 34

A big field

Can extend ordinal numbers to an abeliangroup N .

Mathematical Games – p. 35

A big field

Can extend ordinal numbers to an abeliangroup N .

Conway defined the product of G · H of any twogames. This turns N into a field. We can extendthis to a real closed field, each element of whichis a number game . . . .

Mathematical Games – p. 35

The field I

The product G · H turns the set {∗0, ∗1, ∗2, . . . }into a field I! Since ∗n + ∗n = 0, char(I) = 2.

Mathematical Games – p. 36

The field I

The product G · H turns the set {∗0, ∗1, ∗2, . . . }into a field I! Since ∗n + ∗n = 0, char(I) = 2.

In fact, I is the quadratic closure of F2.

Mathematical Games – p. 36

Mixed games

Mathematical Games – p. 37

Mixed games

Much more complicated!

Mathematical Games – p. 37

Up

Not a Hackenbush game.

0

*1up

Outcome?

Mathematical Games – p. 38

Up

Not a Hackenbush game.

0

*1up

Outcome?

Blue wins, so ↑> 0.

Mathematical Games – p. 38

↑ is very tiny

up −1/ω

..

.

Red wins, so 0 <↑< 1/ω. In fact, for any numbergame G > 0, we have 0 <↑< G.

Mathematical Games – p. 39

Caveat

Caveat. ↑ is not a number game. Red can moveto ∗1, where it is not disadvantageous to move.

Mathematical Games – p. 40

Mathematical Games – p. 41

References

1. E. R. Berlekamp, The Dots and Boxes Game,A K Peters, Natick, Massachusetts, 2000.The classic children’s game Dots and Boxesis actually quite sophisticated and uses manydeep aspects of the theory of impartialgames.

2. E. R. Berlekamp, J. H. Conway, and R. K.Guy, Winning Ways, Academic Press, NewYork, 1982. The Bible of the theory ofmathematical games, with many generalprinciples and interesting examples.

Mathematical Games – p. 42

3. E. R. Berlekamp and D. Wolfe, MathematicalGo, A K Peters, Wellesley, MA, 1994.Applications of the theory to the game of Go.

4. J. H. Conway, On Numbers and Games,Academic Press, London/New York, 1976.The first development of a unified theory ofmathematical games.

Mathematical Games – p. 43

5. D. E. Knuth, Surreal Numbers,Addison-Wesley, Reading, MA, 1974. Anexposition of the numerical aspects ofConway’s theory of mathematical games andthe resulting theory of infinitesimal and infinitenumbers, in the form of a two-persondialogue.

Mathematical Games – p. 44

6. R. J. Nowakowski, ed., Games of No Chance,MSRI Publications 29, Cambridge UniversityPress, New York/Cambridge, 1996. Manyarticles on mathematical games, includingsome applications to “real” games such aschess and Go. There are two sequels.

Mathematical Games – p. 45

How can I get these slides?

Slides available at:

www-math.mit.edu/ ∼rstan/transparencies/games.pdf

Mathematical Games – p. 46