Very recently I stumbled upon an interesting and quite surprising fact. I was performing an analysis with some colleagues, and we noticed that an apparently simple combination of questions (all of the types “yes/no” or “choose from this short list”) could lead to a stunning 10^19 number of possible outcomes. For those unfamiliar with scientific notation for numbers, 10^19 (ten to the nineteenth power) means the number written as 1 followed by nineteen zeroes: 10,000,000,000,000,000,000 (let’s call this number Joe, to make it shorter and less frightening!). Now, this is not a number the human mind can normally fathom, even a “simple” 1.000.000 is too much to be grasped by our intuitive mind. Sure, we can do formal math with one million, we can use that number, we can put it into spread- and balance-sheets, but we cannot enumerate one million things at a glance, nor we can imagine ourselves performing such a high number of simple actions. It transcends from our “everyday brain” into the world of abstraction. Yet, numbers such as Joe are perfectly realistic, and they are hidden in many simple problems. Let’s have a look at some of them: I’ll make some reflections later.
Imagine you are in New York, or in another square-streets city (what a mathematician would call a “taxicab geometry”). You can follow a street for a short while, and then you end up in a four-streets crossing so that you have three possible roads in front of you among which to choose. How many crossings do you have to pass to have one million possible combinations of paths? It will take just 13 to make up for 3^133=1.594.323 possible combinations. For Joe, you are done with 40. Forty random turns in New York lead you to such a high number of paths: well, the Big Apple is the city of possibilities!
Another classical example is the “traveling salesman problem”, or TSP in short. You are a salesman, and you have to pass through N cities (let’s assume, for simplicity, that you can always go from any city to any of the others). What is the shortest route to achieve your goal? Unless you are able to spot some hidden symmetry in the disposition of cities, or they are put in a trivial manner (like in a straight row, or around a circle/regular polygon), you are doomed to compute all the possible routes, in a so-called “brute force” approach to the problem. They add up to the factorial of N, written in short N!, and computed as N*(N-1)*(N-2)*…*3*2*1. Guess what you get with 21 cities? You get five times Joe. This kind of problems are called “NP-problems”, given the fact that they scale in a way which is Non-Polynomial: another way of saying “possibilities to take into account grow very fast”!
How many different pages setup can you use for your Word document? Let’s try to calculate it together, concentrating just on the order o magnitude. First of all, take into account different fonts: on an average Windows system, there are about 100. Font size: going from 5 to 200, it makes another 200. Let’s make 50 different indentations, 50 different spacings before and after words, four types of alignment, text in 1 to 10 columns, page vertical and landscape, ten million different font colors, 50 possibilities for horizontal margins and 50 for vertical margins. The result? 10^20 combinations, ten times more than Joe. Sure, you would never use some of the combinations very often, but nevertheless, they may have their use in the right context.
A fairly popular lottery in Italy goes by the name of Superenalotto. Six numbers are drawn among ninety, and if you are able to guess all six, you can get an enormous amount of money. However, what people don’t realize in a lottery is that the prize is not always comparable to the odds of winning. In this case, you have one winning set of numbers among 90*89*88*87*86*85 total case, for a whopping total of, roughly, 4*10^11. We are not at the level of Joe, but the number is no less astounding.
How many musical scales do exist? Can we run out of “music”? According to Nicolas Slonimsky, author of the “Thesaurus of Scales and Melodic Patterns”, there are 479,001,600 possible combinations of the 12 tones in the chromatic scales. Add different patterns, rhythmical variations, phrasing, sound, intention and different harmonies, and “there is no likelihood that new music will die of internal starvation in the next 1000 years”, in the words of Slonimsky. Truly, music is well beyond our Joe!
The question that I pose to myself after these few examples is: how do we choose? How can we perform rational, sensible, informed choices? How is it even possible? In the movie “The Legend of 1900” the protagonist, a virtuoso pianist who never left the ocean liner where he was born, is at one point about to disembark, but retreats from the stairs. Afterward, he tells his friend Max the reason (you can read the whole quote at the end of this post): in short, in the world outside the ship, there are just too many possibilities for him to choose from.
Another related question is: how do we know something? How can we test our moral, scientific, medical, practical beliefs among those possibilities? In my field of work, IT projects, it is important to have someone test your software, but how can someone certify the result of 10^19 possibilities in any reasonable amount of time and without some automated method? This problem is, luckily, solved in scientific contexts, and experiments are devised to discriminate different possibilities and theories, but the other areas leave me puzzled.
I have no answers to report, I’m still reeling at the sheer size of the numbers. How many ways are there to put together the 1,507 words of this post? How did I select them? How many different reactions will the 1,000… 100… sorry, 10 readers of my posts have? I don’t know, but I won’t be stuck with this dilemma for a long time.
It is a world of possibilities: embrace them, don’t be afraid, surprises are beyond every corner, and they are more likely to be enjoyable than not!
I’d love to hear your comments in the section below this post; please share your impressions and examples.
Until next time, revel in your opportunities!
(Here is the complete quote from 1900) “All that city… You just couldn’t see an end to it. The end! Please, could you show me where it ends? It was all very fine on that gangway and I was grand, too, in my overcoat. I cut quite a figure and I had no doubts about getting off. Guaranteed. That wasn’t a problem. It wasn’t what I saw that stopped me, Max. It was what I didn’t see. Can you understand that? What I didn’t see. In all that sprawling city, there was everything except an end. There was everything. But there wasn’t an end. What I couldn’t see was where all that came to an end. The end of the world. Take a piano. The keys begin, the keys end. You know there are 88 of them and no-one can tell you differently. They are not infinite, you are infinite. And on those 88 keys the music that you can make is infinite. I like that. That I can live by. But you get me up on that gangway and roll out a keyboard with millions of keys, and that’s the truth, there’s no end to them, that keyboard is infinite. But if that keyboard is infinite there’s no music you can play. You’re sitting on the wrong bench. That’s God’s piano. Christ, did you see the streets? There were thousands of them! How do you choose just one? One woman, one house, one piece of land to call your own, one landscape to look at, one way to die. All that world weighing down on you without you knowing where it ends. Aren’t you scared of just breaking apart just thinking about it, the enormity of living in it? I was born on this ship. The world passed me by, but two thousand people at a time. And there were wishes here, but never more than could fit on a ship, between prow and stern. You played out your happiness on a piano that was not infinite. I learned to live that way. Land is a ship too big for me. It’s a woman too beautiful. It’s a voyage too long. Perfume too strong. It’s music I don’t know how to make. I can’t get off this ship. At best, I can step off my life. After all, it’s as though I never existed. You’re the exception, Max. You’re the only one who knows that I’m here. You’re a minority. You’d better get used to it. Forgive me, my friend. But I’m not getting off.”