The story that follows is fascinating for many reasons. Have drama as it takes place during the course of the Second World War. Involve notable scientists who joined forces for a cause innoble, how would the project (known by the name of Manhattan Project) that would end with the creation of the atomic bomb that was dropped twice on Japanese populations. The random that allows a mathematician to find out O to create a method that is still used today in almost all over the world where it is necessary to make estimates and modeling. That is, its use is as valid (or more) today than it was 70 years ago. In addition, it is used in almost all branches of science when it is necessary to reduce a problem that due to its magnitude would make it impossible to handle with other tools.

Imagine that you are playing some variant of the lonely, the card game. I sense that sometime in your life, even now in the digital age, you have tried to solve a solitaire. He may also have become frustrated thinking that he ‘already had it’ and yet he got away. Naturally, several questions arise: “If instead of having put the ‘seven’ in that place, you had put the ‘six’ in another column, would you have obtained a better result?” In other words, you are faced with situations where you have to choose and it is not clear that the one you chose was the most appropriate. Another (question): if instead of having been playing with this arrangement of the cards, there had been another person, would the result be the same?

Now move with me to 1946. I want to deal with the brilliant Polish mathematician Stanislaw Ulam. Ulam was one of the great mathematicians of the twentieth century, with results in both theoretical and applied mathematics and also in physics (very important). Left a lucky of autobiography where he tells multiple episodes of his life. One of the most interesting is related to a encephalitis he had shortly after joining the Los Alamos laboratory, in New Mexico, in the US, where the atomic bomb was built. Ulam had been invited by John von Neumann, a little the dad of computational mathematics, but it happened that Ulam -among other consequences- lost the power of speech. While recovering in the hospital, he entertained himself by doing solitaire as he needed to remain at rest and avoid physical and mental exertion. Unsurprisingly, after playing multiple times, a natural question was asked for a mathematician: What is the probability of solving a solitaire?

Ulam knew that the number of combinations was so huge that it would not take him several lifetimes (still healthy) to find the answer. Then he came up with a genius. He decided to start playing it, and write down the results (positive or negative, according to whether he could have solved it or not). He shuffled the cards well (to ensure they would be arranged randomly) and played randomly. lonely 100 times. Later, he was writing down the number of successful plays. When dividing the number of correct solutions per hundred, you would get a number that would give you an idea of ​​the probability of solving it. But Ulam knew well that one hundred was a very small number and it was not clear that it would give him even an approximation to reality.

But before continuing with the lonely, a pause. Ulam, who was already close friends with John von Neumann, asked him if he could use some free time who had the most sophisticated computers that existed at the time, to test his method. The reference computer bore the name ENIAC. It was a computer immense that filled a room. Von Neumann was enthusiastic about the idea and answered in the affirmative. “It is possible that we can do it using nothing more than a few hours.” It is important for me to include this data here: today, that calculation would take “microseconds”.

The method is tremendously effective and, as I said before, it is still used today (much more than yesterday) simply because computers are much faster, their memory capacity is astonishing, and they allow simulations not only of ten thousand cases but of millions. This allows to model the probability of success of a business, but also if a bridge with a certain design is going to fall or break, or even predetermine if it is worth shooting a movie with a certain budget (and I am choosing only a very small set of examples). Going back to the original story, the Monte Carlo method was used to solve neutron diffusion problems when physicists working on the Manhattan Project met stuck and they could not advance.[1]

By the way, since the project was secret, von Neumann and Ulam needed to give it a fictitious or code name. A colleague of both, Nicholas Metropolis, suggested that they use the name Monte Carlo, since at that time it was the name of the most recognized casino in the world (is it not even today, or did Las Vegas buy all the tickets?) .

To finish, I want to refer to how a problem can be solved using this method, taking advantage of an article that I wrote here (among other places) six years ago. That article can be found here: (https://www.pagina12.com.ar/diario/contratapa/13-275900-2015-06-28.html) and the accounts here: https://www.pagina12.com .ar / Diario / Contratapa / Subnotas / 275900-73282-2015-06-28.html. I briefly recall the situation (which, among others, allowed Manu Ginobili to win multiple bets with his teammates while playing in San Antonio, because no one believed him). How many people do you think needs to be gathered in a room for the chances that From birthday on the same day is greater than 50 percent? The answer – clearly counterintuitive – is that they reach nothing more than 23. Of course: people have to be chosen at random which is very important. In that article I wrote the demo doing some math. Here I want to show how it can be done using the Monte Carlo method to show that with 30 people in the room, the chances of two having their birthday on the same day is over 70 percent.

Do the following: number the days of the year from 1 to 365 (assuming it is not a leap year). For example, the number 1 is January 1 and 365 is December 31. Tell the program to choose 30 numbers out of those 365 randomly. This data is vital: they have to be chosen at random. Pick a number and replace it to the 365 you originally had. In this way, one of the 30 numbers may be repeated. When you have finished the process and you already have these 30 numbers, check -exactly- if there are at least one repeated pair, which would correspond to the same day of the year. Repeat the process 10 thousand times (of course, with the help of a computer). Notice how many of the 10,000 sample cases appear repeating numbers. Divide that number by 10,000. You will see that the number you get is (approximately) 0.7129… How do you interpret this? This means that with 30 people in a room, the chances of two having the same birthday is over 71 percent!

Isn’t this amazing? If you got here, and did not know the method, now you can say that understands what the Monte Carlo Method is all about. Naturally, no one would be very concerned about studying solitaires or coincidences on birthdays, but the remarkable thing is that once again -the game- is one of the ways to find the ‘triggers’ that allow generating ideas that have an unthinkable utility. And that, also it is make mathematics. And the good one!

[1] It was the physicists’ idea of ​​using the Monte Carlo Method that enabled them to design nuclear weapons when those same scientists failed to predict the fate of neutrons moving through uranium and other materials. Neutrons are the initiators of the nuclear chain reaction. When hitting the nucleus of an atom, a neutron can bounce off in multiple directions or it can be absorbed. In the latter case, the absorbed neutron may cause the nucleus to break. This process of ‘separation’ (fission) emits more neutrons, which can induce even more fissions. The crucial question is whether the amount of available neutrons is increasing or decreasing. To find the answer, the Los Alamos group decided to trace the paths of thousands of neutrons using computer simulations provided by the Monte Carlo method.

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