## The Game of Life Lesson 2 - Euromath

The game starts with a set of living cells, the initial state. Each cell ... To familiarize with these rules, we will first work on some initial states by hand . Then we will ...
The Game of Life The

Game of Life

themati ian John

Lesson 2 was devised in 1970 by British ma-

Conway. It is a zero-player game

taking pla e in a grid made of square ells that an be lled or empty. An empty ell is said to be a lled ell is said to be

alive

or

living.

The game starts with a set of living ells, the initial state. Ea h ell will intera t with its 8 neighbors so that the status of ea h ell in the grid may hange when moving to the next step (or next generation). Births and deaths o

ur following four pre ise rules : 1. a living ell with exa tly two or three neighbors stays alive ; 2. a living ell with less than two neighbors dies of loneliness ; 3. a living ell with more than three neighbors dies of rowding ; 4. a dead ell with exa tly three living neighbors omes to life.

I.

First examples

To familiarize with these rules, we will rst work on some initial states by hand. Then we will see how to use a software to study more omplex situations. 1. Observe the dierent generational steps oming from ea h of these initial states :

lambda

old man

Des ribe in a few words what you noti e in ea h situation.

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ount. Laun h a web browser and go to

http://www.bitstorm.org/gameoflife/.

You should see a large empty grid

with a few simple options at the bottom. A living ell is yellow and a dead one is grey. A few initial states are available in the rst pull-down menu. We will study them along the way. Right now the only option you will need is Clear, that empties the grid and brings down the generations ounter to 0. The Next button moves from one generation to the next. The Start button advan es the game automati ally ( ounted at the bottom right). You an stop the generations at any moment with this same button. The Game speed is regulated by the Slow-Fast-Hyper pull-down menu. Finally you an zoom in or out of the grid using the Big-Medium-Small pull-down menu. 2. Clear the grid and he k, using only the Next button, if the steps of the three

previous examples were orre t.

The grid should be innite, unfortunately in this software it is not. Therefore, some strange things may happen next to the borders. Make sure there is always room enough for your experiment to go on un onstrained. II. 1. A

Still-life and os illators

still-life

is a stable pattern, that does not hange from one generation to the

next. You have en ountered one in the rst se tion : whi h one ? Find as many still-lifes as you an : don't hesitate to experiment as mu h as you want on the omputer !

2. An

os illator

is a pattern that repeats itself after a number of generations. This

number is alled the

period

of the os illator.

You have also en ountered one in the rst se tion. What was its period ? Che k that the following sets of living ells are (or be ome) os illators and give the period of ea h one. Try to nd two more.

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III. A

Cross

Pulsar

Polyominoes

polyomino

is a polyform with the square as its base form. It is a onne ted shape

formed as the union of one or more identi al squares, su h that every square an be

onne ted to every other square through a sequen e of shared edges (

i.e.,

shapes

onne ted only through shared orners of squares are not permitted). 1. A pentomino is a polyomino omposed of ve ongruent squares. There are

twelve dierent pentominos named after 12 letters of the Latin alphabet.

O

P

Q

R

S

T

U

V

W

X

Y

Z

a) Observe the behavior of the pentominoes in the Game of Life. Classify them into four families. b) The behavior of one pentomino is very dierent from the others. Whi h one is it and what is spe ial about it ? 2. A tetromino is made of four squares.

a) Find all the tetrominoes and draw them on the grid, not too lose to one another. b) Observe their behavior in the Game of Life. What do you noti e ?

) Is there one tetromino that behaves like the spe ial pentomino we've found.

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IV. A

Spa eships

spa eship

is a pattern that, after a number of generations, translates itself along

the board. The simplest example is the

glider.

1. Choose the Glider option in the rst pull-down menu and observe the behavior

of this onguration step by step. After how many steps do we get a translation of the inital motif ? What translation has o

ured at this point ? 2. Another spa eship is available in the pull-down menu. What is its name ?

After how many steps do we get a translation of the inital motif ? What translation has o

ured at this point ? 3. The

speed

of a spa eship is often expressed in terms of , the metaphori al speed

of light (one ell per generation) whi h is the fastest any spa eship an move. If a spa eship is translated by

(x;y ) after

as : v

=

n generations, then its speed v is dened

max (jxj;jyj) n

:

Compute the speed of the Glider and of the LWSS.

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