### TL;DR
The idea of Monte Carlo (MC) methods is to study a sys...

Nicholas Metropolis is a Greek-America physicist. He is know for hi...

The **Alamogordo test refers to the first detonation of a nuclear b...

### ENIAC
***(Electronic Numerical Integrator And Computer)***
...

The ***lucky numbers sieve*** was invented in the 1950s by a group ...

The term Monte Carlo was coined by N. Metropolis and refers to the ...

A pseudorandom number is a number that appears to be random but is ...

### FERMIAC
FERMIAC was an analog computer invented by physicist...

#### MANIAC
***(Mathematical and Numerical Integrator and Calcula...

THE BEGINNING

of the

MONTE CARLO METHOD

by N. Metropolis

T

he year was 1945. Two earth-

shaking events took place: the

successful test at Alamogordo

and the building of the first elec-

tronic computer. Their combined impact

was to modify qualitatively the nature of

global interactions between Russia and

the West. No less perturbative were the

changes wrought in all of academic re-

search and in applied science. On a less

grand scale these events brought about a

renascence of a mathematical technique

known to the old guard as statistical sam-

pling; in its new surroundings and owing

to its nature, there was no denying its new

name of the Monte Carlo method.

This essay attempts to describe the de-

tails that led to this renascence and the

roles played by the various actors. It is

appropriate that it appears in an issue ded-

icated to Stan Ulam.

Los Alamos Science

Special Issue 1987

Some Background

Most of us have grown so blase about

computer developments and capabilities

-even some that are spectacular—that

it is difficult to believe or imagine there

was a time when we suffered the noisy,

painstakingly slow, electromechanical de-

vices that chomped away on punched

cards. Their saving grace was that they

continued working around the clock, ex-

cept for maintenance and occasional re-

pair (such as removing a dust particle

from a relay gap). But these machines

helped enormously with the routine, rela-

tively simple calculations that led to Hi-

roshima.

The ENIAC. During this wartime pe-

riod, a team of scientists, engineers, and

technicians was working furiously on the

first electronic computer—the ENIAC—

at the University of Pennsylvania in Phil-

adelphia. Their mentors were Physicist

First Class John Mauchly and Brilliant

Engineer Presper Eckert. Mauchly, fa-

miliar with Geiger counters in physics

laboratories, had realized that if electronic

circuits could count, then they could do

arithmetic and hence solve, inter

alia, dif-

ference equations—at almost incredible

speeds! When he’d seen a seemingly

limitless array of women cranking out

firing tables with desk calculators, he’d

been inspired to propose to the Ballistics

Research Laboratory at Aberdeen that an

electronic computer be built to deal with

these calculations.

John von Neumann, Professor of Math-

ematics at the Institute for Advanced

Study, was a consultant to Aberdeen and

to Los Alamos. For a whole host of

125

Monte Carlo

reasons, he had become seriously inter-

ested in the thermonuclear problem being

spawned at that time in Los Alamos by

a friendly fellow-Hungarian scientist, Ed-

ward Teller, and his group. Johnny (as he

was affectionately called) let it be known

that construction of the ENIAC was near-

ing completion, and he wondered whether

Stan Frankel and I would be interested

in preparing a preliminary computational

model of a thermonuclear reaction for the

ENIAC. He felt he could convince the

authorities at Aberdeen that our problem

could provide a more exhaustive test of

the computer than mere firing-table com-

putations. (The designers of the ENIAC

had wisely provided for the capability of

much more ambitious versions of firing

tables than were being arduously com-

puted by hand, not to mention other quite

different applications.) Our response to

von Neumann’s suggestion was enthusi-

astic, and his heuristic arguments were

accepted by the authorities at Aberdeen.

In March, 1945, Johnny, Frankel, and I

visited the Moore School of Electrical En-

gineering at the University of Pennsylva-

nia for an advance glimpse of the ENIAC.

We were impressed. Its physical size

was overwhelming—some 18,000 double

triode vacuum tubes in a system with

500,000 solder joints. No one ever had

such a wonderful toy!

The staff was dedicated and enthusi-

astic; the friendly cooperation is still re-

membered. The prevailing spirit was akin

to that in Los Alamos. What a pity that a

war seems necessary to launch such revo-

lutionary scientific endeavors. The com-

ponents used in the ENIAC were joint-

army-navy (JAN) rejects. This fact not

only emphasizes the genius of Eckert and

Mauchly and their staff, but also suggests

that the ENIAC was technically realizable

even before we entered the war in Decem-

ber, 1941.

After becoming saturated with indoc-

trination about the general and detailed

structure of the ENIAC, Frankel and I re-

turned to Los Alamos to work on a model

126

that was realistically calculable. (There

was a small interlude at Alamogordo!)

The war ended before we completed our

set of problems, but it was agreed that we

continue working.

Anthony Turkevich

joined the team and contributed substan-

tially to all aspects of the work. More-

over, the uncertainty of the first phase of

the postwar Los Alamos period prompted

Edward Teller to urge us not only to com-

plete the thermonuclear computations but

to document and provide a critical review

of the results.

The Spark. The review of the ENIAC

results was held in the spring of 1946

at Los Alamos. In addition to Edward

Teller, the principals included Enrico Fer-

mi, John von Neumann, and the Direc-

tor, Norris Bradbury. Stanley Frankel,

Anthony Turkevich, and I described the

ENIAC, the calculations, and the con-

clusions. Although the model was rel-

atively simple, the simplifications were

taken into account and the extrapolated

results were cause for guarded optimism

about the feasibility of a thermonuclear

weapon.

Among the attendees was Stan Ulam,

who had rejoined the Laboratory after

a brief time on the mathematics faculty

at the University of Southern California.

Ulam’s personality would stand out in

any community, even where “characters”

abounded. His was an informal nature; he

would drop in casually, without the usual

amenities. He preferred to chat, more or

less at leisure, rather than to dissertate.

Topics would range over mathematics,

physics, world events, local news, games

of chance, quotes from the classics—all

treated somewhat episodically but always

with a meaningful point. His was a mind

ready to provide a critical link.

During his wartime stint at the Labora-

tory, Stan had become aware of the elec-

tromechanical computers used for implo-

sion studies, so he was duly impressed,

along with many other scientists, by the

speed and versatility of the ENIAC. In ad-

Stanislaw Ulam

dition, however, Stan’s extensive mathe-

matical background made him aware that

statistical sampling techniques had fallen

into desuetude because of the length and

tediousness of the calculations. But with

this miraculous development of the

ENIAC—along with the applications Stan

must have been pondering—it occurred to

him that statistical techniques should be

resuscitated, and he discussed this idea

with von Neumann. Thus was triggered

the spark that led to the Monte Carlo

method.

The Method

The spirit of this method was consis-

tent with Stan’s interest in random pro-

cesses—from the simple to the sublime.

He relaxed playing solitaire; he was stim-

ulated by playing poker; he would cite

the times he drove into a filled parking

lot at the same moment someone was ac-

commodatingly leaving. More seriously,

he created the concept of “lucky num-

bers,” whose distribution was much like

that of prime numbers; he was intrigued

by the theory of branching processes and

Monte Carlo

contributed much to its development, in-

cluding its application during the war to

neutron multiplication in fission devices.

For a long time his collection of research

interests included pattern development in

two-dimensional games played according

to very simple rules. Such work has lately

emerged as a cottage industry known as

cellular automata.

John von Neumann saw the relevance

of Ulam’s suggestion and, on March 11,

1947, sent a handwritten letter to Robert

Richtmyer, the Theoretical Division lead-

er (see “Stan Ulam, John von Neumann,

and the Monte Carlo Method”). His let-

ter included a detailed outline of a pos-

sible statistical approach to solving the

problem of neutron diffusion in fission-

able material.

Johnny’s interest in the method was

contagious and inspiring. His seemingly

relaxed attitude belied an intense interest

and a well-disguised impatient drive. His

talents were so obvious and his coopera-

tive spirit so stimulating that he garnered

the interest of many of us. It was at that

time that I suggested an obvious name

for the statistical method—a suggestion

not unrelated to the fact that Stan had an

uncle who would borrow money from rel-

atives because he “just had to go to Monte

Carlo.” The name seems to have endured.

The spirit of Monte Carlo is best con-

veyed by the example discussed in von

Neumann’s letter to Richtmyer. Consider

a spherical core of fissionable material

surrounded by a shell of tamper material.

Assume some initial distribution of neu-

trons in space and in velocity but ignore

radiative and hydrodynamic effects. The

idea is to now follow the development

of a large number of individual neutron

chains as a consequence of scattering, ab-

sorption, fission, and escape.

At each stage a sequence of decisions

has to be made based on statistical prob-

abilities appropriate to the physical and

geometric factors. The first two decisions

occur at time

t = O, when a neutron is se-

lected to have a certain velocity and a cer-

tain spatial position. The next decisions

are the position of the first collision and

the nature of that collision. If it is deter-

mined that a fission occurs, the number of

emerging neutrons must be decided upon,

and each of these neutrons is eventually

followed in the same fashion as the first.

If the collision is decreed to be a scatter-

ing, appropriate statistics are invoked to

determine the new momentum of the neu-

John von Neumann

tron. When the neutron crosses a material

boundary, the parameters and characteris-

tics of the new medium are taken into ac-

count. Thus, a genealogical history of an

individual neutron is developed. The pro-

cess is repeated for other neutrons until a

statistically valid picture is generated.

Random Numbers. How are the vari-

ous decisions made? To start with, the

computer must have a source of uni-

formly distributed psuedo-random num-

bers. A much used algorithm for gener-

ating such numbers is the so-called von

Neumann “middle-square digits.” Here,

an arbitrary n-digit integer is squared,

creating a 2n-digit product. A new in-

teger is formed by extracting the middle

n-digits from the product. This process

is iterated over and over, forming a chain

127

Monte Carlo

I

I

1

example, see the section entitled “The

Monte Carlo Method” in “A Primer on

Probability, Measure, and the Laws of

Large Numbers.”)

Since its inception,

many international conferences have been

held on the various applications of the

method.

Recently, these range from

the conference,

“Monte Carlo Methods

and Applications in Neutronics, Photon-

ics, and Statistical Physics,” at Cadarache

Castle, France, in the spring of 1985 to

the latest at Los Alamos, “Frontiers of

Quantum Monte Carlo,” in September,

1985.

Putting the Method into Practice

Let me return to the historical account.

In late 1947 the ENIAC was to be moved

to its permanent home at the Ballistics

Research Laboratory in Maryland. What

a gargantuan task! Few observers were

of the opinion that it would ever do an-

other multiplication or even an addition.

It is a tribute to the patience and skill

of Josh Gray and Richard Merwin, two

fearless uninitiated, that the move was a

success. One salutary effect of the inter-

ruption for Monte Carlo was that another

distinguished physicist took this occasion

to resume his interest in statistical studies.

Enrico Fermi helped create modern

physics.

Here, we focus on his inter-

est in neutron diffusion during those ex-

citing times in Rome in the early thir-

ties. According to Emilio Segre, Fermi’s

student and collaborator, “Fermi had in-

vented, but of course not named, the

present Monte Carlo method when he was

studying the moderation of neutrons in

Rome. He did not publish anything on

the subject, but he used the method to

solve many problems with whatever cal-

culating facilities he had, chiefly a small

mechanical adding machine.”*

In a recent conversation with Segre, I

Company from From X-Rays to Quarks by Emilio

Segre.

128

learned that Fermi took great delight in

astonishing his Roman colleagues with

his remarkably accurate, “too-good-to-be-

lieve” predictions of experimental results.

After indulging himself, he revealed that

his “guesses” were really derived from

the statistical sampling techniques that he

used to calculate with whenever insomnia

struck in the wee morning hours! And

so it was that nearly fifteen years earlier,

Fermi had independently developed the

Monte Carlo method.

Enrico Fermi

It was then natural for Fermi, during

the hiatus in the ENIAC operation, to

dream up a simple but ingenious ana-

log device to implement studies in neu-

tron transport. He persuaded his friend

and collaborator Percy King, while on a

hike one Sunday morning in the moun-

tains surrounding Los Alamos, to build

such an instrument—later affectionately

called the FERMIAC (see the accompa-

nying photo).

The FERMIAC developed neutron ge-

nealogies in two dimensions, that is, in a

plane, by generating the site of the “next

collision. ”

Each generation was based

on a choice of parameters that charac-

terized the particular material being tra-

versed. When a material boundary was

crossed, another choice was made appro-

priate to the new material. The device

could accommodate two neutron energies,

referred to as “slow” and “fast.” Once

again, the Master had just the right feel

for what was meaningful and relevant to

do in the pursuit of science.

The First Ambitious Test. Much to

the amazement of many “experts,” the

ENIAC survived the vicissitudes of its

200-mile journey. In the meantime Rich-

ard Clippinger, a staff member at Ab-

erdeen, had suggested that the ENIAC

had sufficient flexibility to permit its con-

trols to be reorganized into a more conve-

nient (albeit static) stored-program mode

of operation. This mode would have a

capacity of 1800 instructions from a vo-

cabulary of about 60 arithmetical and log-

ical operations. The previous method of

programming might be likened to a gi-

ant plugboard, that is to say, to a can

of worms. Although implementing the

new approach is an interesting story, suf-

fice it to say that Johnny’s wife, Klari,

and I designed the new controls in about

two months and completed the implemen-

tation in a fortnight. We then had the

opportunity of using the ENIAC for the

first ambitious test of the Monte Carlo

method—a variety of problems in neu-

tron transport done in collaboration with

Johnny.

Nine problems were computed corre-

sponding to various configurations of ma-

terials, initial distributions of neutrons,

and running times.

These problems, as

yet, did not include hydrodynamic or ra-

diative effects, but complex geometries

and realistic neutron-velocity spectra

were handled easily. The neutron histo-

ries were subjected to a variety of statisti-

cal analyses and comparisons with other

approaches.

Conclusions about the effi-

cacy of the method were quite favorable.

It seemed as though Monte Carlo was

here to stay.

Not long afterward, other Laboratory

Monte Carlo

staff members made their pilgrimages to

ENIAC to run Monte Carlo problems.

These included J. Calkin, C. Evans, and

F. Evans, who studied a thermonuclear

problem using a cylindrical model as well

as the simpler spherical one. B. Suydam

and R. Stark tested the concept of artifi-

cial viscosity on time-dependent shocks;

they also, for the first time, tested and

found satisfactory an approach to hydro-

dynamics using a realistic equation of

state in spherical geometry. Also, the dis-

tinguished (and mysterious) mathemati-

cian C. J. Everett was taking an inter-

est in Monte Carlo that would culminate

in a series of outstanding publications in

collaboration with E. Cashwell. Mean-

while, Richtmyer was very actively run-

ning Monte Carlo problems on the so-

called SSEC during its brief existence at

IBM in New York.

In many ways, as one looks back, it

was among the best of times.

Rapid Growth. Applications discussed

in the literature were many and varied

and spread quickly. By midyear 1949 a

symposium on the Monte Carlo method,

sponsored by the Rand Corporation, the

National Bureau of Standards, and the

Oak Ridge Laboratory, was held in Los

Angeles. Later, a second symposium was

organized by members of the Statistical

Laboratory at the University of Florida in

Gainesville.

In early 1952a new computer, the MA-

NIAC, became operational at Los Ala-

mos. Soon after Anthony Turkevich led

a study of the nuclear cascades that result

when an accelerated particle collides with

a nucleus. The incoming particle strikes

a nucleon, experiencing either an elastic

or an inelastic scattering, with the latter

event producing a pion. In this study par-

ticles and their subsequent collisions were

followed until all particles either escaped

from the nucleus or their energy dropped

below some threshold value. The “exper-

iment” was repeated until sufficient statis-

tics were accumulated. A whole series of

target nuclei and incoming particle ener-

gies was examined.

Another computational problem run on

the MANIAC was a study of equations

THE FERMIAC

The Monte Carlo trolley, or FERMIAC, was

invented by Enrico Fermi and constructed

by Percy King. The drums on the trolley

were set according to the material being tra-

versed and a random choice between fast

and slow neutrons. Another random digit

was used to determine the direction of mo-

tion, and a third was selected to give the dis-

tance to the next collision. The trolley was

then operated by moving it across a two-

dimensional scale drawing of the nuclear

device or reactor assembly being studied.

The trolley drew a path as it rolled, stopping

for changes in drum settings whenever a

material boundary was crossed. This infant

computer was used for about two years to

determine, among other things, the change

in neutron population with time in numerous

types of nuclear systems.

of state based on the two-dimensional

motion of hard spheres. The work was

a collaborative effort with the Tellers,

Edward and Mici, and the Rosenbluths,

Marshall and Arianna (see “Monte Carlo

at Work”). During this study a strategy

was developed that led to greater com-

puting efficiency for equilibrium systems

obeying the Boltzmann distribution func-

tion. According to this strategy, if a sta-

tistical “move” of a particle in the sys-

tem resulted in a decrease in the energy

of the system, the new configuration was

accepted. On the other hand, if there was

an increase in energy, the new configu-

ration was accepted only if it survived a

game of chance biased by a Boltzmann

factor. Otherwise, the old configuration

became a new statistic.

It is interesting to look back over two-

score years and note the emergence, rather

early on, of experimental mathematics,

a natural consequence of the electronic

computer.

The role of the Monte Carlo

method in reinforcing such mathematics

seems self-evident. When display units

were introduced, the temptation to exper-

129

Monte Carlo

iment became almost irresistible, at least

for the fortunate few who enjoyed the lux-

ury of a hands-on policy. When shared-

time operations became realistic, exper-

imental mathematics came of age. At

long last, mathematics achieved a certain

parity-the twofold aspect of experiment

and theory-that all other sciences enjoy.

It is, in fact, the coupling of the sub-

tleties of the human brain with rapid

and reliable calculations, both arithmeti-

cal and logical, by the modern computer

that has stimulated the development of

experimental mathematics. This develop-

ment will enable us to achieve Olympian

heights.

The Future

So far I have summarized the rebirth

of statistical sampling under the rubric

of Monte Carlo. What of the future—

perhaps even a not too distant future?

The miracle of the chip, like most mir-

acles, is almost unbelievable. Yet the fan-

tastic performances achieved to date have

not quieted all users. At the same time we

are reaching upper limits on the comput-

ing power of a single processor.

One bright facet of the miracle is the

lack of macroscopic moving parts, which

makes the chip a very reliable bit of

hardware. Such reliability suggests par-

allel processing.

The thought here is

not a simple extension to two, or even

four or eight, processing systems. Such

extensions are adiabatic transitions that,

to be sure, should be part of the im-

mediate, short-term game plan. Rather,

the thought is massively parallel opera-

tions with thousands of interacting pro-

cessors-even millions!

Already commercially available is one

computer, the Connection Machine, with

65,536 simple processors working in par-

allel. The processors are linked in such

a way that no processor in the array is

more than twelve wires away from an-

other and the processors are pairwise con-

nected by a number of equally efficient

routes, making communication

both flex-

ible and efficient. The computer has been

used on such problems as turbulent fluid

flow, imaging processing (with features

analogous to the human visual system),

document retrieval, and “common-sense”

reasoning in artificial intelligence.

One natural application of massive par-

allelism would be to the more ambitious

Monte Carlo problems already upon us.

To achieve good statistics in Monte Carlo

calculations, a large number of “histories”

need to be followed. Although each his-

tory has its own unique path, the under-

lying calculations for all paths are highly

parallel in nature.

Still, the magnitude of the endeavor

to compute on massively parallel devices

must not be underestimated. Some of the

tools and techniques needed are:

●

●

●

●

A high-level language and new archi-

tecture able to deal with the demands

of such a sophisticated language (to the

relief of the user);

Highly efficient operating systems and

compilers;

Use of modern combinatorial theory,

perhaps even new principles of logic,

in the development of elegant, compre-

hensive architectures;

A fresh look at numerical analysis and

the preparation of new algorithms (we

have been mesmerized by serial com-

putation and purblind to the sophistica-

tion and artistry of parallelism).

Where will all this lead? If one were

to wax enthusiastic, perhaps—just per-

haps—a simplified model of the brain

might be studied. These studies, in turn,

might provide feedback to computer ar-

chitects designing the new parallel struc-

tures.

Such matters fascinated Stan Ulam. He

often mused about the nature of memory

and how it was implemented in the brain.

Most important, though, his own brain

possessed the fertile imagination needed

to make substantive contributions to the

very important pursuit of understanding

intelligence.

■

Further Reading

S. Ulam, R. D. Richtmyer, and J. von Neumann.

1947. Statistical methods in neutron diffusion. Los

Alamos Scientific Laboratory report LAMS–551.

This reference contains the von Neumann letter dis-

cussed in the present article.

N. Metropolis and S. Ulam. 1949. The Monte

Carlo method. Journal of the American Statistical

Association 44:335-341.

S. Ulam. 1950. Random processes and transforma-

tions. Proceedings of the International Congress of

Mathematicians 2:264-275.

Los Alamos Scientific Laboratory. 1966. Fermi in-

vention rediscovered at LASL. The Atom, October,

pp. 7-11.

C. C. Hurd. 1985. A note on early Monte Carlo

computations and scientific meetings. Annals of the

History of Computing 7:141–155.

W. Daniel Hillis. 1987. The connection machine.

Scientific American,

June, pp. 108–1 15.

N. Metropolis received his B.S. (1937) and his

Ph.D. ( 1941) in physics at the University of Chi-

cago. He arrived in Los Alamos, April 1943, as

a member of the original staff of fifty scientists.

After the war he returned to the faculty of the

University of Chicago as Assistant Professor. He

came back to Los Alamos in 1948 to form the

group that designed and built MANIAC I and II. (He

chose the name MANIAC in the hope of stopping

the rash of such acronyms for machine names, but

may have, instead, only further stimulated such use.)

From 1957 to 1965 he was Professor of Physics

at the University of Chicago and was the founding

Director of its Institute for Computer Research. In

1965 he returned to Los Alamos where he was made

a Laboratory Senior Fellow in 1980. Although he

retired recently, he remains active as a Laboratory

Senior Fellow Emeritus.

130

A pseudorandom number is a number that appears to be random but is not (unlike dice rolls or lottery tickets). Pseudorandom sequences typically exhibit statistical randomness while being generated by a deterministic causal process. Pseudorandom number generators are mathematical algorithms that produce sequences of numbers that appear to be random, their outputs are periodic, but the period is so big that it's very hard to notice. Modern algorithms for generating pseudorandom numbers are so good that the numbers look like they were really random.
Learn more here:
- [Khan Academy - Pseudorandom number generators](https://www.khanacademy.org/computing/computer-science/cryptography/crypt/v/random-vs-pseudorandom-number-generators)
- [Wikipedia - Pseudorandom number generator](https://en.wikipedia.org/wiki/Pseudorandom_number_generator)
The **Alamogordo test refers to the first detonation of a nuclear bomb**. The code name of the first test was **Trinity** and was carried out by the US Army in the Jornada del Muerto desert on July 16th 1945, as part of the Manhattan Project. You can learn more here: [Trinity (nuclear test).](https://en.wikipedia.org/wiki/Trinity_nuclear_test)
The **first electronic computer was called Electronic Numerical Integrator and Computer - ENIAC**. It was built from 1943 to 1945 by J. Presper Eckert and John V. Mauchly at the University of Pennsylvania and funded by the US army which needed a way to compute ballistics during World War II. The machine wasn't completed until 1945, but then it was used extensively for calculations during the design of the hydrogen bomb.
Learn more here: [ENIAC](https://en.wikipedia.org/wiki/ENIAC)
Nicholas Metropolis is a Greek-America physicist. He is know for his contributions to the Monte Carlo method and is one of the first “computer programmers”. He headed a team that carried out the first actual Monte Carlo calculations on the ENIAC computer. The Monte Carlo method originated from a collaboration between Metropolis and Stanislaw Ulam.
Learn more here: [Atomic Heritage Foundation - Nicholas Metropolis](https://www.atomicheritage.org/profile/nicholas-metropolis)
![metropolis](https://i.imgur.com/BpytOoc.jpg)
The term Monte Carlo was coined by N. Metropolis and refers to the Monte Carlo Casino in Monaco. It was a nod to a colleague, Stanislaw Ulam, who had an uncle that was a gambling man and would borrow money from the family by saying that “I just have to go to Monte Carlo”. He chose this name because of the apparent element of ‘chance’ in gambling and in Monte Carlo methods.
The ***lucky numbers sieve*** was invented in the 1950s by a group in the Mathematics Division at the Los Alamos National Laboratory.
How it works:
1. Start with a sequence of positive integers listed in the natural order.
2. Remove every other number. The first left-over number is 3.
3. Mark 3 and remove every third number among the remaining ones. The first left-over number past 3 is 7.
4. Mark 7 and remove every seventh number among the remaining ones.
5. Continue this way.
![lucky numbers gif](https://i.imgur.com/tZ2Klex.gif)
Learn more here:
- [Cut The Knot - Lucky Numbers](https://www.cut-the-knot.org/Curriculum/Algorithms/LuckyNumbers.shtml)
- [Lucky Number](http://mathworld.wolfram.com/LuckyNumber.html)
- [Wikipedia - Lucky Number](https://en.wikipedia.org/wiki/Lucky_number)
### ENIAC
***(Electronic Numerical Integrator And Computer)***
The ENIAC was the first programmable general-purpose electronic computer build during WWII and finished in 1945. The ENIAC was funded by the US army and the total cost was about 487,000 dollars equivalent to 6,887,000 in 2017 dollarS. It was built in secret by the University of Pennsylvania’s Moore School of Electrical Engineering from 1943 to 1945. It was first put to use at the end of 1945.
The specs of the ENIAC:
![specs](https://i.imgur.com/t7SVH12.png)
![eniac image](https://i.imgur.com/OYCPC3m.jpg)
*Fugure: Programming the ENIAC*
Learn more here:
- [ENIAC Overview](https://www.thocp.net/hardware/eniac.htm)
- [Wikipedia - ENIAC](https://en.wikipedia.org/wiki/ENIAC)
### FERMIAC
FERMIAC was an analog computer invented by physicist Enrico Fermi to aid in his studies of neutron transport. It resembles a trolley car and in order to use it, several adjustable drums are set using pseudorandom numbers.
These random numbers represent:
1. the material being traversed, a random choice is made between fast and slow neutrons
2. the direction of neutron travel
3. the distance traveled to the next collision
Once these settings are chosen, the device is physically driven across a 2-D scale drawing of the nuclear reactor and the paths of neutrons through the various materials are plotted. Whenever a material boundary is crossed, the appropriate drum is adjusted to represent a new pseudorandom digit.
Learn more here: - [Wikipedia - FERMIAC](https://en.wikipedia.org/wiki/FERMIAC)
![fermiac](https://i.imgur.com/3D9kT1J.jpg)
*FERMIAC, or Monte Carlo trolley is an analog device invented by Enrico Fermi*
#### MANIAC
***(Mathematical and Numerical Integrator and Calculator)***
The MANIAC was one of the earliest computers. It was developed at Los Alamos under the direction of Nicholas Metropolis. It was smaller than the ENIAC and was able to store programs (the ENIAC couldn't).
The MANIAC was used by physicists like E. Fermi and E. Teller allowing them to perform simulations much faster in order to gain a better understanding of the behavior of particles. It was used for other innumerable experiments and discoveries. MANIAC became the first computer to play and beat a human, at a chess-like game.
FUN FACT: Metropolis chose the name MANIAC in the hope of stopping the rash of silly acronyms for machine names. It is one of the ancestors of many modern computers.
![maniac chess](https://i.imgur.com/NVHfIvk.jpg)
*Picture: Paul Stein and Nick Metropolis playing modified chess with the MANIAC I*
Learn more here:
- [Metropolis, Monte Carlo and the MANIAC](http://permalink.lanl.gov/object/tr?what=info:lanl-repo/lareport/LA-UR-86-2600-05)
- [MANIAC I](https://chessprogramming.wikispaces.com/MANIAC+I)
### TL;DR
The idea of Monte Carlo (MC) methods is to study a system by simulating it with the repeated generation of random events in a computer model. Monte Carlo methods are often used in simulations of physical and mathematical systems. The development of MC methods is linked with the development of the general purpose machines and the need to solve problems related to the development of atomic weapons.
The general premise of Monte Carlo methods is remarkably simple:
1. Randomly sample input to the problem
2. Compute an output for each sample
3. Approximate the solution by aggregating the outputs
MC methods in a nutshell:
- the core idea of the Monte Carlo is to learn about a system by simulating it with random sampling
- the development of these methods is linked to the necessity of solving problems related to atomic weapons
- it's the simplest way or even the only feasible way to solve complex problems
- these methods constitute a powerful and flexible approach
- they are applied to physical sciences (widely used at the LHC), engineering, statistics, finance, machine learning