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.
dead, while
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 dierent generational steps oming from ea h of these initial states :
lambda
blinker
old man
Des ribe in a few words what you noti e in ea h situation.
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Swit h the omputer on and log in to your a
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 innite, 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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Toad
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 dierent 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 dierent 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 onguration 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 dened
max (jxj;jyj) n
:
Compute the speed of the Glider and of the LWSS.
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