1.05. Large Numbers in Probability

(back to 1.04)

Introduction

We're now quite familiar with numbers that measure things in the real world: number of gallons of water on Earth, number of atoms in the observable universe, and so on. But we haven't really gone into detail on probability; what realms of insanity can that bring us to? Let's go find out, this time starting all the way from the bottom. Are you ready for a journey through insane odds and amounts of possibilities? If you are, then let's get started ...

1 to 100

Like I said, we'll start at the bottom of probability. As I said in the previous article, probability deals not only with how likely things are but also with numbers of possibilities. People deal with probability every day: the lottery, weather forecasts, insurance, batting averages, and so on. Numbers of possibilities are commonly dealt with as well, largely in games. Perhaps the most fundamental example of numbers of possibilities is:

2 - the number of ways a coin flip can go. Everyone's familiar with flipping coins: a coin has two sides, heads or tails, and when you flip it and it lands on the ground, either one side or the other shows. Flipping coins is common when you want a random decision (for example, if you have two projects to work on and can't decide which to do first) because in general the coin is equally likely to land heads or tails. This is boring, we need something new. How about adding a second coin to the picture? This brings us...

4 - the number of ways two coin flips can go. If you flip two coins at once instead of just one, then you get 2*2 = 4 ways the coins can land. Another way to think of this is that if you flip 2 coins, there's a 1 in 4 chance they'd both land heads. Not quite as much of a dud as 2 is, but still something you learned in grade school. Let's dump coins and get something a bit more interesting:

6 - the number of ways a dice can land. Dice, like coins, are objects everyone's familiar with. They are cubes with 6 faces, each with a number from 1 to 6 written on it (either the glyph or as a number of dots). They are commonly used in board games, and because they have 6 faces, they can land 6 different ways when you roll them. Still nothing out of the ordinary though.

16 - the number of ways 4 coins can land. Sixteen still isn't a very big number, as 16 dots looks like this:

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but at least it can feel big in certain circumstances such as this one. Imagine flipping four coins at once and having them all land heads (that has a 1 in 16 chance of happening). That sounds pretty lucky, doesn't it? If you flip four coins 16 times, on average one of those flips would have all land heads. Imagine a series of sixteen movies, or giving birth to sixteen kids. Somehow 16 seems kind of big again ... but it's still too small. We can certainly do better with probability. Let's go to:

24 - the number of ways to arrange four objects, books or whatever else. You probably know that to calculate the number of ways to arrange n objects: take the factorial of n (denoted n!, the product of all integers 1 through n). Perhaps you already thought of factorials when thinking of ways to generate large numbers; those are common in elementary attempts to make large numbers, but not as relevant to googology as you may think. Still, factorials are a very common occurrence when dealing with probability. In 4's case, the number of ways to arrange four objects is 4! = 4*3*2*1 = 4*3*2 = 4*6 = 24. A little better, but still not even a little intimidating.

36 - the number of ways two dice can land. Some board games like Monopoly use two dice instead of one; those two dice together can land 36 different ways. Like before, to get this value you just multiply 6 by 6 and get 36. 36 dots looks like this:

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Still modest as far as numbers go, even the ones we use every day. Let's go to something a little more interesting:

44 - the number of ways to arrange five objects, such that none are in their original place. Another way to imagine this is to imagine five people trading hats. If no one wants to end up with the same hat they started with, then there are 44 ways that trade could go. The number of ways to arrange n objects such that none are in their original place (also called the number of derangements of n objects) is known as n subfactorial, sometimes denoted !n. n subfactorial is a little harder to directly calculate than n factorial, but there are several nice formulas that make it easier. A surprisingly elegant one which I like is [n!/e], where [x] is the nearest integer to x, and e is of course the legendary Euler's number.^[1]

64 - the number of ways six coins can land. In other words, if you flip six coins at once (or one coin six times) you have a 1 in 64 chance of getting all heads. Once again it's weird how getting that kind of 6-heads streak seems quite a lucky feat, although its chance is merely 1 in 64. To visualize this, here's 64 dots:

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I think that's about it for 1 to 2 digit numbers. Nothing too exciting, though all these everyday probabilities serve as a foundation for something much more epic.

100 to 1,000,000

120 - the number of ways to arrange 5 objects, or 5! = 5*4*3*2*1. Compare this to 44, the number of ways to arrange 5 objects such that none are in their original place. To visualize this, look at each of those dots:

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Each dot represents one of the arrangements of 5 objects, and the red dots are the arrangements that are derangements (no objects are in their original place). Moving on:

1:179 - the odds of a random person being blind as of August 2014,^[2A] that is, 1 in 179 people are blind. To visualize that, here's 179 dots:

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Only one dot, colored red, would be a blind person.

1:583 - the odds of a person being a millionaire as of 2014.^[2A] In other words, 1 in 583 people in the world are millionaires:

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720 - the number of ways to arrange 6 objects. This equals 6! or 6*5*4*3*2*1. Sbiis Saibian, a figure in googology whose work we will discuss more later in this site, discovered this on his own as a kid. He was curious about how many ways you could arrange the 6 books on top of his shelf, and he reasoned that "in the first position you can put any of the 6 books, in the second there are 5 books to choose from, in the third there are 4 books to choose from, and so on up to the last position". In his own words, he had "rediscovered factorials".^[3]

Hmm... still not hitting the real big guns. Let's pick up the pace and get to numbers like:

1024 - the number of ways 10 coins can land, and the odds against getting heads 10 times in a row. By now this seems to be quite a lucky streak, which makes sense because a thousand is pretty much always considered a big number. 1024 dots looks like this:

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1:4096 - odds of getting a 300 (the largest possible score) in bowling, assuming a 50% strike rate. A score of 300 is when you knock down all ten pins in one go every round, and in the tenth and final round, you score all your bonus strikes perfectly, making for twelve strikes total. Calculating this number is simple: it's just two to the twelfth power. Previously, I had listed 1:11,500 as the supposed odds for getting a 300 in bowling, but the webpage I cited gave no explanation for that value.^[4] Since I still wanted a number around that range here, I decided to replace it with a value calculated with proper math.

Anyway, here's 4096 dots:

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Imagine if you're good enough at bowling to get a strike 50% of the time, and play as many bowling games as there are dots shown above. Chances are that hidden somewhere in this storm of dots, one of them will represent a game where you score a perfect 300.

1:649,740 - the odds of getting a royal flush in poker. In poker, when you draw 5 cards from a 52-card deck, you have a 1 in 649,740 chance of getting a royal flush, that is, the five highest cards (ten, jack, queen, king, ace) all in the same suit. This number comes from the fact that there are only four different royal flushes but 2,598,960 hands of 5 cards you can have, and so we divide 4 into 2,598,960 to get a 1 in 649,740 chance of getting a royal flush.

As a kid I remember reading about poker probabilities like this in a book, and they quickly caught my interest. Some time in 3rd grade or so I remember asking my dad if he's ever gotten a royal flush (at the time he played poker a lot); I think he said that he has gotten one once.

These odds are starting to get quite large, but still everyday—time to take things up a notch by going into the millions...

1,000,000 to 10¹²

1:2,598,960 - odds of getting a royal flush in spades. In poker, when you draw 5 cards from a 52-card deck you have a 1 in 2,598,960 chance of getting a spades royal flush (10♠ J♠ Q♠ K♠ A♠).

3,746,160 - the number of arrangements of a 2x2x2 Rubik's cube, often called a Pocket Cube. Most people know what a Rubik's cube is (we'll examine them in more detail a little bit later). This is the number of ways you can arrange a 2x2x2 version of that cube without disassembling it. A 2x2x2 Rubik's cube looks like this:

It's made of eight modest little cubes, and yet it can be arranged 3.746 million ways.

If you want to know how this number is calculated, think of it like so. One of the cubes is considered to have a fixed orientation and position, and then the other seven can be arranged 7! = 5040 ways; additionally, each can have 3 different rotations, giving you 7!*3⁶ = 3,746,160.^[5A] Moving on:

1:16,777,216 - the odds of rolling all eights with eight eight-sided dice. An eight-sided dice is like a normal dice but shaped like an octahedron (a polyhedron with eight triangular faces, looks like two square pyramids stuck together at the bases). You'd need a lot of luck—ALLLLLLLL the luck, perhaps—to achieve such an outcome, but we still haven't really blasted into the stars yet.

1:175,711,336 - the odds of winning a recent Mega Millions lottery. These odds are insanely rare, and Tim Urban on his blog post "From 1,000,000 to Graham's Number" gave a nice way to put those odds in perspective:^[2B] 175,711,336 is about the number of seconds in 6 years, so the lottery is like knowing a hedgehog will sneeze once and only once in the next 6 years, and putting your hard-earned money on one second (say the 36th second of 2:52am on March 19th, 2017); you only win if that sneeze does happen exactly on that second. In other words, don't buy a Mega Millions ticket.

1:76,017,000,000 - the odds of having an IQ over 200, based on 100 being the average IQ and the standard deviation being 15.^[6] That means that by this measure, nobody in the world has this IQ or higher! Now, the whole concept of IQ is very disputed (and often touted as meaningless), but this still serves as a useful benchmark for extreme probabilities. Moving on...

10¹² to 10³⁰

1:30,874,473,598,904,896 - the odds of winning the Mega Millions lottery, twice in a row. Those odds are 1 in 30.8 quadrillion, which as we saw in an earlier article is pretty insane; feel free to refer back to this article to get a feel of how big those numbers are. It would take 6500 years for this many gallons of water to go down Niagara Falls.

43,252,003,274,489,856,000 - this well-known figure is the number of ways you can arrange the classic 3x3x3 Rubik's Cube. As you probably know, the Rubik's cube is a famous puzzle which is a cube made from 26 little cubes, with 9 stickers on each face of the cube, and the goal is to solve it so that all faces of the cube have the same color. It's famously challenging to solve a Rubik's cube if you don't know how to do it. When solved, the Rubik's cube looks like this:

This little cube above turns out to have 43 quintillion different combinations, which is pretty insane—and yet the puzzle is quite simple, being made up of only 26 little cubes! Of those 43 quintillion combinations, only one of them is the solution! If you had one Rubik's cube (assuming dimensions 6x6x6 centimeters) for each of the 43 quintillion combinations, then those cubes could cover the entire surface of Earth in 305 layers. This means that the whole layer of Rubik's cubes covering Earth would be as thick as a 6-story building is tall! Now think about it: a 6-story building is probably bigger than your house is. Now that's quite insane alright.

It's a little trickier to derive the number of combinations for the Rubik's cube than it is for its little brother the Pocket Cube, but Robert Munafo discusses how the number is derived here.

519,024,039,293,878,272,000 - the number of ways you can arrange a Rubik's cube if you're allowed to disassemble and reeassemble it. This number is exactly 12 times larger than the previous number, and it's equal to about 519 quintillion. You can imagine this many cubes covering Earth in a layer about as thick as the size of a skyscraper with 72 floors.

6,670,903,752,021,072,936,960 - the number of possible Sudoku grids with each number filled. Sudoku is another puzzle which most everyone is familiar with. You are given a 9x9 grid of numbers in some of the spaces, and the goal is to fill up the grid such that each row, column, and 3x3 grid has each number 1 through 9 exactly once. An example of a solved Sudoku grid is:

This number is 6.671 sextillion, or 6.671 billion trillion in terms of more familiar illions.

5,524,751,496,156,892,842,531,225,600 - the number of possible Sudoku grids, but without the restriction that each 3x3 grid must have each number from 1 to 9. This is therefore the number of 9x9 Latin squares. A Latin square is a x-by-x grid of numbers 1 through x where each row and column has each number 1 through x exactly once. This insane number is equal to about 5.5 octillion.

With these insane combinations and odds, now we're really getting somewhere! Let's continue our journey by ascending to reach a googol...

10³⁰ to 10¹⁰⁰

1.2676*10³⁰ - the number of ways 100 coins can land, and the odds of getting heads 100 times in a row. This is equal to 2¹⁰⁰, or 1.2676 nonillion. By now we can be quite certain such a lucky streak has never happened in human history; even if a billion people flipped 100 coins every second since the beginning of the universe, that would only take us to 432 septillion 100-flip combos, and there's only a 1 in 3000 chance that among all those combinations there would be a 100-heads streak.

7.4012*10⁴⁵ - the number of arrangements for a 4x4x4 Rubik's Cube, also known as Rubik's Revenge, which looks like this:

Solving this Rubik's cube is only a little more complicated than the regular Rubik's cube, but this cube has A LOT more combinations: 1.72 septillion times more! It's almost impossible to even comprehend that. This number is equal to about 7.4012 quattuordecillion, or 7.4 quadrillion quadrillion quadrillions, which is pretty insane. Written out in full this number is 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000.

To put those combinations in perspective, first consider a cube of 3.7 million 2x2x2 Rubik's cubes (3.7 million is the number of ways to arrange a 2x2x2 Rubik's cube). That cube would be 3.8 meters tall, more than twice as tall as the average person and a good bit taller than anyone on record! But that's much smaller than the cube of 43.252 quintillion 3x3x3 Rubik's cubes, which would be about 200 kilometers or 124 miles wide. This itself pales in comparison the cube of 7.4012*10⁴⁵ (7.4 quattuordecillion) 4x4x4 Rubik's cubes. Such a cube of Rubik's Revenges would be 148 billion kilometers wide, which, if centered at the sun, would exceed the orbit of Pluto at its furthest point, 20.3 times!! That's how many different ways you can arrange something just one step up from a regular Rubik's cube.

2.2836*10⁴⁶ - Randall Munroe's estimate of the number of "meaningfully different" English tweets, from his "What If?" blog where he answers strange scientific questions people submit to him.^[7A] The question we're looking at here, in the asker's own words, is:

How many unique English tweets are possible? How long would it take for the population of the world to read them all out loud?

Munroe goes on to explain how you can get an estimate on the number of English tweets there can be, and he explains it something like this:

The number of tweets you can make with the 26 letters and spaces would be 27¹⁴⁰ (140 was the character limit in Twitter at the time) ~ 2.46*10²⁰⁰. However, of course most of those tweets would be meaningless jumbles of text such as "asdkfaohirgo kxngorrighoiqwenkrlwenmsdof iawuehfoeawnkfadsokfonweokr", or "gkosiedktwpqknzzvejo kodjefpwoegjpmwpm wewianhreignoakdpgpalwe". The question doesn't ask how many possible tweets are there, it asks, how many tweets can we have that say something in English?

Even among the not entirely meaningless tweets, there would be many distinct tweets that would not reasonably be considered meaningfully different. For example, Hi I'm Mxyztplk would not really be meaningfully different from Hi I'm Mxzkqklt, but something like Hi I'm xPoKeFaNx would definitely be meaningfully different from either, even though xPoKeFaNx is by no means an English word. So things are now really complicated! We need to turn to a different approach.

Imagine a language with only two valid sentences, and all of Twitter being in that language. Then, when you scroll through your Twitter feed all you'd see is either the one sentence or the other sentence. In this case, there are only two possible tweets, and each tweet you see is essentially a bit of data, as it's either the one sentence or the other sentence.

Now imagine this idea being applied to the English language, and that's where things get tricky. To figure out how many meaningfully different tweets there are, Munroe turned to the experiments of a computer scientist named Claude Shannon. In Shannon's experiments, he showed people samples of English text that were cut off after a random point (e.g. Oh my god the volcano is eru), and then people guessed which letter came next. Based on the amounts of correct guesses, Shannon arrived at the estimate that each letter holds an average of 1.1 bits of data.

If a piece of text holds n bits of data, then there are 2ⁿ meaningfully different messages that it can convey. This means that to estimate the number of possible meaningfully different English tweets, we calculate 2^1.1*140 ~ 2.2836*10⁴⁶ meaningfully different English tweets.

Randall Munroe then goes to give an analogy on how long it would take for someone to read all of those (read source [7a] if you're interested), but I will instead discuss on how long it would take the whole world to send all those tweets.

It takes about 30 seconds (for me at least) to type a 140-character tweet, and processing the tweet to be viewed on the Internet is nigh instantaneous. So we can estimate that sending any single 140-character tweet (most of the 2.28*10⁴⁶ tweets would have 140 characters) would take 30 seconds. Now imagine all 7.1 billion people on Earth, sending tweets for 16 hours a day. Dreariness from sending those tweets aside, sending all those tweets would take 4.584 nonillion years—suffice to say that such a time is unimaginably long. If you want to get a feel of how long that is, consider the time span from the Big Bang to the present, and have as many such times pass as seconds have passed since the Big Bang! That time span, which is itself incredibly long, would need to pass 739 times before humanity finishes sending all those tweets. As Munroe himself put it: 140 characters may not seem like a lot, but we will never run out of things to say. Moving on...

5.2320*10⁴⁹ - Robert Munafo's estimate of the number of possible chess positions.^[5B] This is actually his upper-bound on how many positions there can be. This estimate allows anywhere from 2 to 32 pieces on play with no more than 16 of one color, which includes the standard count of chess pieces but is higher than some other estimates because it allows pawn promotion. This is equal to about 52 quindecillion, and it's comparable to a googolism by Sbiis Saibian known as a gogol which is equal to 10⁵⁰. Up next we have...

8.0658*10⁶⁷ - this is 52! = 52*51*50 ... *4*3*2*1, the number of ways to shuffle a deck of 52 playing cards. It's equal to about 81 unvigintillion, or 80,000 vigintillion using only canonical -illions. Imagine shuffling a deck of 52 cards and somehow having the deck return exactly back to how it was originally. We can be quite certain this has never unwittingly happened in human history, because this figure is so ridiculously huge.

1.0066*10⁶⁸ - the number of arrangements of the Megaminx (12-color version), a puzzle similar to a Rubik's Cube but shaped like a dodecahedron. The Megaminx looks like this:

It's quite a complicated puzzle alright, with 50 movable pieces in comparison to 20 for the Rubik's cube. The puzzle has two variants, one with 12 colors and one with 6 colors. The 6-color Megaminx has exactly 16,384 = 2¹⁴ times fewer arrangements than the 12-color Megaminx.

2.8287*10⁷⁴ - the number of ways to arrange a 5x5x5 Rubik's cube. It's equal to about 283 billion vigintillion or 283 trevigintillion. This figure is starting to really blast into the stars, into the realm of utterly unfathomable giants, as even this many atoms in a cube would be 880 billion kilometers wide—way bigger than what we can really wrap our mind around. Such a sphere would be only a little smaller than the center part of the Cat's Eye Nebula (to get a handle of how big it is, check out this article by Sbiis Saibian). Numbers like this are just CRAZY HUGE ... but quite modest in comparison to what we'll see next! Need I remind you once again, it only gets worse ...

10¹⁰⁰ to 10^10,000

1:2.7369*10¹¹⁰ - the odds of getting all questions on the SAT right by randomly guessing, as of July 2012.^[7B] This is another figure from Randall Munroe's What If? blog. The question asked in that post is:

What if everyone who took the SAT guessed on every multiple-choice question? How many perfect scores would there be?

Munroe says that nobody will get a perfect score that way, because the odds are incredibly slim. The SAT at the time had 158 questions, and each has 5 choices; therefore, the odds are 1 in 5¹⁵⁸ ~ 2.7369*10¹¹⁰. Using Conway and Guy's -illions (we'll look at them in a later article) that's 1 in 273.69 quinquatrigintillion^{note 1} (also known as quintrigintillion in simplifications to the system). Even if everyone who has ever lived (about 100 billion people) took the SAT by guessing, nobody would get it right; the same can be said if every atom in the observable universe took the test!!! This is just a number too big for us to really wrap our minds around. Munroe says that these odds are worse than the odds that every living former American president and every member of the main cast of Firefly (at the time, that was 13 people) all being independently struck by lighting, on the same day.

It's worth noting that for the first time in this article we have surpassed a googol, which is quite a milestone. This number is about 27 billion googol, which is simply incomprehensible. It's comparable to the number of atoms that would fill up the observable universe (10¹¹⁰).

1.5715*10¹¹⁶ - the number of ways to arrange a 6x6x6 Rubik's cube. This is a far crazier number than previous, as a sphere of this many atoms would dwarf the observable universe by a factor of a million! A cube of this many atoms would have a diameter 100x greater than the observable universe. I find it pretty crazy that a 6x6x6 Rubik's cube already has enough combinations to surpass a googol. This number is about 15.7 quadrillion googol, and we've moved way beyond numbers we can really appreciate—but we've examined much bigger numbers earlier, so let's continue with some even more insane giants ...

1.4781*10¹¹⁸ - a common estimate of all possible chess games, mentioned by Sbiis Saibian in his article on large numbers in probability.^[3] For obvious reasons, it's really hard to calculate the number of possible chess games, but we can get a simple estimate as Sbiis Saibian discusses. Assume that on average there are 30 moves you can make at any turn, and on average each player will get 40 turns. Thus we calculate 30⁸⁰ ~ 1.48*10¹¹⁸ as an estimate of how many possible chess games there are. If you have not been impressed by this number, consider that this insane number comes from a board of merely 32 pieces, and this number is more than the number of atoms that would fill up the observable universe, which is something whose size is impossible for us to really wrap our minds around.

Although this may seem like a pretty sensible guess, it's probably a substantial underestimate of all possible chess games though. Consider that a game can easily last more than just 80 turns, and that there are many more possible games longer than 80 turns than there are games shorter than 80 turns. In not too long we'll look at my upper-bound on all possible games which is likely closer to the real value.

9.3326*10¹⁵⁷ - the number of ways to arrange 100 objects in order. 100 objects (books or whatever else) can be ordered 100! = 100*99*98 ... *4*3*2*1 ways. This number is somewhat larger than a googol.

1.9501*10¹⁶⁰ - the number of ways to arrange a 7x7x7 Rubik's cube. This number is just INSANE ... no known particle is small enough that this many of them could fit in the observable universe. If there was a particle small enough to squeeze this many into the observable universe, here's how small it would be: Imagine shrinking the entire planet Earth down to the size of an atom, then filling the universe with all those shrunken-down Earths. Each particle would be the size of a single atom from the shrunken-down Earth. Is that scary? If so, good, we're just about to get started with truly insane numbers!

A YouTuber who goes by the name "Sharkee" brought up this number in his popular video "What is the Largest Number",^[8] a video which talks about large numbers in order of size that's notable because it introduces viewers to numbers way way beyond the so-called "world's largest number" Graham's number (we'll examine it in detail in a later article). He compares this number, a number with real-world meaning as the number of ways to arrange the 7x7x7 Rubik's cube, to the googol, to show that there are numbers that are in some way useful that are bigger than a googol.

9.1198*10²⁶² - the number of arrangements of the Gigaminx, a dodecahedral puzzle which is a more complicated version of the Megaminx. It is to the Megaminx as the 5x5x5 Rubik's cube is to the 3x3x3 Rubik's cube, and it has 230 movable pieces. This number is about a googol times bigger than the previous number!!! We have now left the volume of the observable universe in Planck volumes in the dust, by a factor of about 3*10⁷⁷! We are now zooming through the numbers at simply incomprehensibly fast speeds, and ... uh ... get ready for some more insanity?

1.0715*10³⁰¹ - the number of ways 1000 coins can land (2¹⁰⁰⁰), and also the odds of getting heads a thousand times in a row. Imagine flipping a coin a thousand times, keeping track of whether it lands heads or tails each time. Doing that would take about an hour. Imagine that in the whole hour you spend flipping a coin, every single time it lands heads. That is indescribably rare, and those odds are quite close to a centillion (10³⁰³). Actually this number is about 100 times smaller, though on this scale it's considered pretty close.

1.41*10⁴⁰⁸ - Robert Munafo's "single-perturbation count".^[5C] This is another probability figure that requires a bit of explanation, so I'll explain this figure, what it means, and how it was calculated.

If you choose a random moment in the history of the universe (Big Bang to the present), then choose a random particle and move it to a random location somewhere else in the universe, then you have 1.41*10⁴⁰⁸ different choices you can make. This number is finite by assuming intervals below Planck units the same (we discussed Planck units in an earlier article). For example, for our purposes any random location (let's call it location X) and a location a millionth of a Planck length away from location X (let's call it location Y) are not considered to be different places.

How can you get this figure? Robert Munafo calculated by multiplying the quartic space-time hypervolume of the universe (Big Bang to present) in Planck units (1.75*10²⁴⁵) by the number of particles in the observable universe (10⁹⁷ according to Munafo) by its age in Planck times (8.03*10⁶⁰), giving us this enormous number.

This is the single-perturbation count, which you can think of as the number of ways you can make a single adjustment to the universe. Robert Munafo uses this in his estimate of the number of possible alternate universes, which we'll get to later.

1.7989*10⁵⁷¹ - the number of arrangements of the Teraminx, which is to the Megaminx as the 7x7x7 Rubik's cube is to the normal Rubik's cube. That puzzle is crazy complicated, with 530 movable pieces, and yet it does indeed exist, as Teraminxes have been made by enthusiasts of Rubik's cubes and related puzzles.

10⁸⁰⁰ - Randall Munroe's estimate of all possible tweets if you're allowed to use whatever Unicode characters you want.^[7A] The estimate is rough because the way Twitter counts characters is complicated (Twitter has a page on how it counts characters). Needless to say, virtually all of those tweets would be jumbles of characters from virtually any language, the kind you'd see when opening a file that is not a text document or similar with Notepad (like ™.çrù‡´ê|ýI•b!.ulÂ.Íe©¿.ÌÃ3¿+»in) but even worse, as it can use characters or symbols from literally any script that is available in Unicode. This figure isn't quite as interesting as the number of meaningfully different English tweets, but it still is definitely a clear meaningful one.

7.7623*10⁹⁹² - the number of arrangements of the Petaminx. The Petaminx is to the Megaminx as the 9x9x9 Rubik's cube (yes that cube does exist, and it's even mass-produced) is the the 3x3x3 Rubik's cube. This number is quite close to 10¹⁰⁰⁰ (which is often known as a googolchime, name coined by Sbiis Saibian). Actually it's about 13 million times smaller, though once again on this scale that's considered quite close.

6.6909*10¹⁰⁵⁴ - the number of arrangements of a 17x17x17 Rubik's Cube, currently the world record largest Rubik's Cube. This number written out in full is:

66,­909,­260,­871,­052,­009,­626,­140,­831,­457,­599,­196,­711,­140,­812,­269,­154,­070,­729,­060,­136,­529,­449,­625,­780,­211,­961,­895,­693,­820,­570,­513,­604,­163,­602,­868,­942,­801,­633,­627,­363,­413,­148,­772,­664,­738,­570,­971,­988,­412,­147,­490,­850,­469,­267,­091,­069,­898,­537,­146,­037,­768,­890,­069,­934,­919,­884,­249,­763,­818,­629,­080,­668,­367,­898,­685,­033,­459,­370,­133,­844,­075,­322,­446,­474,­048,­403,­397,­592,­421,­266,­564,­641,­031,­053,­781,­182,­835,­951,­043,­902,­666,­703,­934,­718,­275,­733,­629,­773,­072,­428,­119,­603,­386,­280,­810,­232,­743,­294,­106,­725,­017,­906,­015,­726,­602,­505,­404,­809,­355,­600,­713,­515,­400,­760,­343,­408,­510,­054,­774,­806,­467,­063,­695,­824,­637,­124,­911,­945,­446,­317,­465,­833,­055,­520,­836,­975,­861,­238,­244,­940,­397,­333,­234,­336,­971,­270,­687,­092,­383,­804,­133,­631,­886,­114,­309,­853,­819,­332,­336,­282,­986,­834,­777,­948,­178,­464,­656,­888,­802,­372,­250,­927,­074,­981,­140,­246,­608,­824,­577,­036,­094,­710,­201,­099,­095,­240,­641,­256,­513,­217,­598,­802,­423,­874,­027,­822,­421,­584,­587,­650,­039,­125,­516,­202,­912,­205,­481,­540,­427,­864,­199,­947,­576,­722,­221,­866,­866,­102,­507,­350,­876,­922,­115,­628,­881,­880,­203,­115,­212,­216,­766,­503,­665,­426,­445,­956,­786,­264,­399,­133,­302,­962,­649,­600,­884,­736,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000

At this point I'd like to stop and take a moment to discuss how INCREDIBLY HUGE this number is. It has exactly 1055 digits, and it's certainly more than measly figures such as the number of atoms or Planck volumes in the observable universe, even the number of Planck volumes in the observable universe raised to the sixth power!! Compare this to a googolchime, or 10¹⁰⁰⁰. A googolchime is a number that is pretty close to this number, in the sense that it has only slightly fewer (1001 as opposed to 1055) digits. However it's tricky to remember that a googolchime is 6.7 septendecillion times smaller! Now a septendecillion is itself a number that is difficult to really wrap your mind around, but putting it in terms of atoms might help a bit. A septendecillion atoms in a sphere would fall somewhere between Jupiter and the sun size-wise. Let's not forget that a septendecillion is only the difference between the 17x17x17 Rubik's cube number and a googolchime, two relatively close numbers!!

2.8692*10²⁹⁵⁰ - the number of arrangements of the "Yottaminx" puzzle.^[10] It is a dodecahedral puzzle which is to the Megaminx as the 15x15x15 Rubik's cube (I'm not sure if such a cube exists) is to the normal Rubik's cube. The Yottaminx comes from the SI prefix yotta- meaning one septillion, and the name is, of course, extrapolated from Megaminx, Gigaminx, etc. Believe it or not, this crazy puzzle has actually been made. This is what it looks like:

That is insane alright (it's about as big as a basketball), as is this number. It's a pretty big jump from the previous number, being more than its square, multiplied by a centillion and then having THAT number multipled by a centillion!! This number is fairly close to a number often known as a millillion (10³⁰⁰³), from Latin mille meaning 1000, indicating that this is the 1000th -illion. Actually it's 3*10⁵² times smaller than a millillion, but you get the point. Now for a HUGE jump among the numbers to ...

10^10,000 to 10^{10^12}

1.1057*10^29,689 - my upper-bound of all possible chess games

Earlier we encountered a simple rough estimate of all possible chess games, based on 30 possible moves per turn and a game lasting 80 turns. However, as I said this is probably a severe underestimate. So to get a better idea of how many possible chess games there might be, I will try my hand and come up with an upper-bound on that value.

First off, although 30 is a reasonable estmate of the average number of moves a player can make in any turn, for the sake of upper-bounding it's possible to get positions with well over 30 moves. For example, take this position:

This position has 218 possible moves for white! It's been the record holder for most moves any single attainable position allows since the 1960s,^[16] and since no position with more moves has ever been found, we'll use 218 as an upper-bound for how many moves any player can make in a single turn. This is reasonable largely because positions like this are more or less impossible to obtain, as they require that white manages to successfully promote all his pawns to queens. Such a feat would require black to play poorly on purpose while white eschews every easy opportunity to put black into checkmate. As such, this more than suffices for coming up with an upper-bound.

Now calculating the maximum duration of a chess game is trickier. But by the rules of chess, a game cannot last forever. This is because of the fifty-move rule, which states that if fifty moves (in chess, a move is white moving a piece followed by black moving a piece) go by without a pawn move or a capture, then the game ends with a draw (neither player wins). Since there are only a finite number of pawn moves or captures that can happen in a game of chess, it quickly follows that with the fifty-move rule, all games must end at some point even if checkmate or stalemate doesn't happen.

So what is the longest a game can last? First we need to calculate the total maximum number of pawn moves or captures that can happen in a game. Each player can do a maximum of 48 pawn moves in a game—48 because a player has eight pawns, and each can move a maximum of six times before either getting captured or making it to the back row and promoting. Then, each player can do a maximum of 15 captures, since each player has 15 pieces (all 16 except the king) that can be captured. This gives us a maximum of 63 pawn moves/captures per player.

Now how many turns can a game last? It's tricky but still possible to find an upper-bound for that. The game can begin with white and black moving their knights around the board (not taking any pieces), until on black's turn in the 49th move, black does a pawn move or capture. White and black then continue aimlessly moving pieces around the board until in the 99th move black does another pawn move or capture. At 99 moves we've already gotten to a game much longer than most strategically played chess games ... but this is just the beginning. Continuing, black does another pawn move or capture on the 149th, 199th, 249th ... all the way up to the 3149th move (63 such moves total), when he has done all possible pawn moves or captures. After that, white will do the same as black did on the 3198th, 3248th, 3298th, 3348th ... all the way up to the 6298th move (63 total moves). After that, both players can spend an additional 50 moves moving their kings around the board, leading to a game lasting 6348 moves. Doubling that gives a chess game lasting a maximum of 12,696 turns.

Now 12,696 is not the maximum duration of a chess game but actually an upper-bound of the maximum duration of a chess game. Why is that? A chess game cannot actually have black doing all possible pawn moves or captures every 50 moves and white doing the same, because if black does all possible pawn moves or captures, by the time white starts to do his pawn moves or captures he will have no pawns to move! Therefore this is an upper-bound of how many moves a game of chess can last.

To upper-bound the total number of possible chess games this way, we take the maximum number of moves (218) to the power of the upper-bound maximum number of turns a chess (12,696) to get about 1.1057*10^29,689 as an upper-bound of all possible chess games.

Although this is an upper-bound, it should still give us a good idea of how many chess games there can be. Even though 30⁸⁰ may seem like a reasonable estimate at first glance, it ignores the fact that chess games can EASILY last over 80 moves, giving a much much larger reasonable estimate than just 30⁸⁰.

Now here we'll stop again, to discuss how extremely BIG this number is. Let's compare this number, about 10^29,689, to the previous number, about 10²⁹⁵⁰. Recently we've been comparing numbers in terms of ratios, which has been a handy way to get an idea of the gaps between numbers, so we'll try doing this to these two numbers.

This number is about 10^26,739 times larger than the previous number! Even the ratio is much much larger than the previous Yottaminx number, as it's 10^23,789 times larger—even THAT ratio is itself 10^20,839 times larger than the Yottaminx, and 10^20,839 is still nowhere near the Yottaminx number! We'd need to divide my upper-bound of all possible chess games by the Yottaminx number nine times to get a number comparable to the Yottaminx number!!

A better way to compare this number to the Yottaminx number is imagine a sphere with a Yottaminx number of particles. You'd need to dwarf it by a factor of itself nine times, then have 10¹⁸⁹ of those spheres, to get to my upper-bound of all possible chess games. Now this dwarfing is radically different from all the simple ratios we've previously encountered! All this goes to show that this number is a definite tipping point for us, as we are now moving at an alarming rate now ... up next we have ...

1:1.0391*10^307,383 - odds of a monkey randomly typing Hamlet on its first try with a 48-key typewriter

You've probably heard of the infinite monkey theorem before. It's a famous theorem which states that if a monkey at a typewriter were to press random keys for a time of length N, the probability that the monkey will type any given text approaches 100% as N approaches infinity.

Now imagine that a monkey was indeed placed in front of a typewriter and started typing random letters (doing this with a real monkey probably wouldn't work), and let's assume each key has an equal probability of being hit. Typewriters vary in the number of keys they have, but for our purposes we'll use a typewriter with 48 keys: the lowercase alphabet, space, numbers, and the symbols .,?!;:'"()-. With those keys it would be difficult for the monkey even type short words. For example, if our monkey typed a million letters on a 48-key keyboard, how likely is it that among those letters you'd find the word "banana"? The probability of a string of text of length N occurring among M characters the monkey types is calculated with the formula 1-(1-1/47^N)^M/N, and by plugging in 6 (length of the word "banana") for N and 1,000,000 for M gives us 0.000015462 as the probability, or a 1 in 64,676 chance for the 1,000,000 letters the monkey types to contain the word "banana". These probabilities escalate quickly. For example, the sentence "this is a sentence." has a 1 in 6.6 decillion chance of occurring among a million characters the monkey types.

Now imagine the monkey on the typewriter, as he started typing, produced not a meaningless jumble of letters and symbols, but all of the Shakespearean play Hamlet, perfectly! To calculate just how unlikely this would be, we just need to calculate the number of possibilities for typing as many characters as Hamlet has, since among those possibilities only one is a perfect reproduction of Hamlet. Calculating the number of possibilities is simple. There are 48 different characters the monkey can type, and according to Robert Munafo there are 182,831 characters (letters and symbols, not fictional characters!) in the text of Hamlet. Therefore we just calculate 48^182,831 ~ 1.0391*10^307,383 as the number of possibilities. The odds of our little monkey perfectly typing Hamlet on his first try are therefore 1 in 1.0391*10^307,383.

Now this number right here is ABSURDLY HUGE!!! Let's compare this number, about 10^307,383, to the previous number, about 10^29,689. It's not just ten times larger, or even ten orders of magnitude larger! Ten times larger than 10^33,201 would only be 10^29,690, which seems to be barely an improvement on that, and ten orders of magnitude larger would be 10^29,699; neither of these seem like much of an improvement to the previous number. Even going 100,000 orders of magnitude away from the previous number will take us only to 10^129,689, which is STILL SMALLER than the monstrous Hamlet monkey number we're talking about here!! So how much larger is this number than the weeny little upper-bound of all chess games? About 10^274,182 times larger! That number seems to be barely smaller than the Hamlet monkey number itself!! It represents a jump of about 274,000 orders of magnitude, and 274,000 orders of magnitude is something way way way beyond anything we can come near understanding ... but we can try.

Imagine that you somehow had a comprehension, a clear mental image, of how much a googol is. Now imagine dwarfing that perception of a googol by a factor of itself, and continuing 2740 times total, to get a kind of feel of how incomprehensible this monstrous number is. Even dwarfing two by a factor of itself 2740 times already creates an unimaginable number equal to about 6.63*10⁸²⁴, so imagine how insane doing the same to a googol would be!!! Hopefully you get the idea that this number is utterly incomprehensibly enormous ... suffice to say that we are yet to encounter anything near the Hamlet monkey number within our known universe.

This number is utterly unfathomable, and from here on out these numbers will accelerate EVEN MORE INCREDIBLY quickly ... up next we have:

1.9650*10^1,834,097 - number of books in the Library of Babel

This is the number of books in a gigantic fictional library known as the Library of Babel, a number sometimes known as Borges' number. What exactly is the Library of Babel? It came from a famous short story of the same name by Argentinian author Jorge Luis Borges, which was first published in Spanish in 1941, and in English in 1962. You can read it here. The story is about a universe which is a giant library which contains every possible book with 410 pages, 40 lines per page, 80 characters per line, and 25 possible different characters (22 letters, space, comma, period). That is what the Library of Babel is.

Among those books would be information on literally everything: your own biography, anyone else's biography, any report on anything that has happened, phrased however you choose, translated into any language, and so on. Initially for people in the library this seems glorious and amazing ... until you consider the ramifications of all this. There would be even more, if not many more, books with false information than books with true information, from books that mess up a fact or two of the correct information to books whose information is complete and utter bogus! And that's not even the worst part: the vast majority of all those books would consist entirely of meaningless nonsense, so it could well take very long to find a book with any meaningful information! Borges, in his story, goes on to discuss how people in the library would react to such a scenario ... some people decided to destroy the books that are meaningless, while others searched for a book that would be an index to all books or similar, suspecting that there was a god-like man who indeed found such a book. Finding such a man would be very much like finding the holy grail! Now the idea of a Library of Babel seems not glorious, but horrifying!!

Because of this, the Library of Babel to this day has a famous cult significance, and is valued greatly as a thought experiment. But what may be even more interesting about that crazy library is that it's HUGE!!! To get an idea of how huge it is, we can first calculate the number of books that the Library of Babel contains. As with the monkey on the typewriter, the figure is easy to calculate: the number of characters in each book in the Library of Babel is 410*40*80 = 1,312,000, and there are 25 possible different characters. Thus we calculate 25^1,312,000 ~ 2*10^1,834,097, and that's how many books there are in the Library of Babel.

To get an idea of how insanely huge this number must be, let's compare this number, which is about 10^1,834,097, to the Hamlet monkey number which is about 10^307,383. Once again, the exponent is deceiving: the number is not 6 times larger as the exponent may suggest, and it isn't just 6 orders of magnitude bigger either. 6 orders of magnitude bigger than 10^307,383 would be 10^307,389 which seems to be barely a difference from the Hamlet monkey number! Increasing the number by 6 orders of magnitude would be the same as increasing by a factor of a million ... but going a million orders of magnitude away from the Hamlet monkey number still doesn't quite take you to the enormity of Borges' number!! To get to Borges' number from the Hamlet monkey number you need to take an enormous leap by a factor of 1.76 million orders of magnitude. This is utterly impossible to even comprehend, but I have another way to get an idea of the massive gulf between these two numbers, which might help you to get a feel of how unbelievable huge Borges' number really is.

Imagine a sphere made of 10^307,383 books all clumped together. That sphere would be incomprehensibly massive, about as big as dwarfing the universe by a factor of its volume in Planck volumes, 539 times!!! Call that sphere S1. But now imagine dwarfing the unimaginably gigantic S1 by a factor of 10^307,383, having as many S1's as there are books in S1. That sphere would be S2. Now imagine repeating the process, dwarfing S2 by a factor of 10^307,383, to get S3, then S4, then S5, then S6. S6 would have roughly Borges' number of particles. I say "roughly" because the number of particles in S6 10^1,844,298 in comparison to 10^1,834,097, which is approximately Borges' number. Those two numbers appear quite close, and true, on this scale those two numbers are considered quite close. However, the first of them is roughly a googoltoll (10^10,000) times larger than the second. To get a feel for how insanely huge a googoltoll is, imagine a sphere of a googol particles, then dwarfing it by a factor of a googol, 99 times. As you can see this is utterly insane ... but once again you ain't seen nothing yet!

3.1874*10^2,205,788 - number of books in the Library of Babel, allowing 48 characters instead of 25 characters

I find it a little weird that the Library of Babel has such a constrained set of characters to use, particularly having 22 letters you can choose from instead of 26. By allowing a bigger set of allowed characters (here I'm using the same set of characters as the 48-key typewriter I discussed earlier) we can make larger versions of the Library of Babel. To calculate this number we calculate 48^1,312,000 ~ 3.8174*10^2,205,788.

How much larger would this expanded library be than the original Library of Babel, which has about 10^1,834,097 books? It would be roughly 371,000 orders of magnitude bigger. That's more orders of magnitude than it takes to get from 1 to the Hamlet monkey number. By now we have lost all hope of fathoming any such numbers ... let's jump to a more interesting figure:

3.1060*10^{3,576,838,408} - Robert Munafo's number of possible human beings^[5D]

Consider your DNA, the code within your body that defines who exactly you are. Your DNA is a code made of four different bases, A, T, C, and G; for example, a strip of it could be ATACGAGCTAGA. How many different possible codes of DNA can you have? It is often assumed that since "everyone is unique" absolutely everyone (except for twins, triplets, quadruplets, etc.) has different DNA. But in reality, the number of possible DNA combinations someone can have is very huge, but NOT infinite.

Acording to Robert Munafo, there are about 5,941,000,000 bases in a typical human's DNA. Because there are 4 different bases of DNA, we calculate 4^{5,914,000,000} ~ 3.1*10^{3,576,838,408} possible different human beings. Now think about how huge this number is. The population of the world is 7.1 billion, and the number of people who have ever lived is estimated to be 100 billion—that's only eleven orders of magnitude away from one! The number of possible humans, however, is 3.58 billion orders of magnitude away from one, and hardly fewer orders of magnitude away from the number of humans who have ever lived! It's scary because the number of humans who have ever lived would only be an unimaginably small fraction of all possible combinations. Even if humanity lasts for a googol years, that would only add roughly 100 orders of magnitude to the number of human beings, which still doesn't take us ANYWHERE NEAR this number!

Note that not all those possible humans each be distinct. Why is that? About 98% of human DNA is noncoding DNA, DNA that is not used as code for who you are but for other purposes. Much of the noncoding DNA is repetitive duplicate DNA, used to decrease your odds of getting a genetic mutation, and some of it is specially used as a trigger to begin or end genetic coding. Noncoding DNA can also be used to make special regulatory molecules. So not all of your DNA really defines who you are, and many of these possible combinations would distinguish from one another only from a slight difference in repetitive DNA. But still, as Sbiis Saibian put it, the bottom line is: unless you have an identical twin, triplets, or what have you, you really don't have to worry about bumping into someone with the same DNA as you.^[3]

10^{10^12} to 10^{10^100}

10^{10^16} - Linde and Vanchurin's number of distinguishable parallel universes^[11]

Physicists Andrei Linde and Vilaty Vanchurin published a paper in 2009 (revised 2010) which discusses how many parallel universes there are. In the paper, they explain that the total amount of information a human can absorb in their lifetime is on the order of 10¹⁶ (ten quadrillion) bits of data. By applying this estimate, they came up with 10^{10^16} as an estimate of the number of parallel universes a human observer can distinguish.

This figure is among the most conservative estimates for the number of distinguishable parallel universes; there are many more liberal estimates people have made as I will discuss later in this article. This is not to be confused with the total number of parallel universes, a value that seems impossible to calculate at all; however, Linde and Vanchurin, in their paper, managed to arrive at an estimate of 10^{10^10,000,000} total parallel universes, using models of inflation. I'm not going to give that number a separate entry in this article because it doesn't really relate to probability.

How would you compare this number to the previous number? You'd need to multiply the previous number by itself 2.8 million times (i.e. raise it to the 2.8 millionth power) to get this number! Even the number of digits in this number is much larger than the number of digits in the previous number (the number of possible human beings). The previous number has 3.6 billion digits, but this one has 10 quadrillion digits! To put that in perspective, the digits of the previous number would be able to fit in about 358 dictionaries (assuming each page can fit 10,000 digits and each dictionary has 1000 pages), but to store the digits of this number we'll need a billion dictionaries! That alone would require ten million bookshelves to store (assuming a bookshelf can store 100 dictionaries). But we're just getting started with some REALLY big numbers!

10^{10^29} - distance in meters between you and an identical copy of you^[12]

This is Max Tegmark's as an estimate of the distance between you and an identical copy of you, assuming the universe is infinite and ergodic (ergodic means that everything that could happen at any point has happened at some previous point in time), with unending cosmic inflation. The estimate is given in meters, but can be put in terms of any length unit (Planck lengths or yotta-light-years) and it wouldn't really make a difference on the number. This is close to 10 to the power of the number of protons, neutrons, and electrons in the human body (about 6.32*10²⁸), and that's not a coincidence.

This figure may seem like more of a length (a physical quantity) than a probability or number of possibilities, but it reflects the vast number of possible things that can be created in a randomly generated area of space, and how far you would have to go to find an identical copy of yourself. Indeed, it is impossible for you to observe the copy of you, since that copy of you would be well beyond the portion of the universe you'd be able to observe.

You would need to raise the previous number to the power of ten trillion to get this number. You'd need about 10 sextillion (10²²) dictionaries to store all the digits of this number! By now putting the digits of a number in terms of a number of dictionaries makes it no easier to picture, but this analogy may help: if those 10 sextillion dictionaries were packaged into a cube it would be 2320 km wide, which could cover roughly half of the continental United States.

1:10^{10^36} - odds of a person living at least 1000 years^[5D]

Robert Munafo, on his number list, explains that life insurance companies have tables of the odds of a person living past certain ages, and that above a certain point they give an extrapolation formula. By applying this formula, Munafo came up with this estimate of how likely someone is to live at least a thosuand years.

Now this number is truly insane. You'd need to raise the previous number to the ten-millionth power to get to this number. If all the digits of this number were stored in a bunch of dictionaries and those dictionaries were put into a cube then the cube would be 232,000 km wide, which would be about 3.32 times wider than Jupiter, and 33 times wider than Earth!! And I think we've heard enough from this number. Up next we have another really big leap...

10^{10^100} to 10^{10^1,000,000}

10^{10^115} - distance in meters between the observable universe and an identical copy of the observable universe^[12]

In the same paper where Max Tegmark discusses the distance between you and an identical copy of you, he describes a similar figure but for the observable universe, once again assuming the universe is infinite and ergodic with unending cosmic inflation. The observable universe is not to be confused with the whole universe; the observable universe is simply the portion of the universe we are able to observe, which may or may not be the entire universe. If the observable universe is not the entire universe and the entire universe is infinite (and ergodic), an identical copy of the observable universe would be extremely far away, about 10^{10^115} meters. That figure is fairly close to 10 to the power of the number of atoms that would fill up the observable universe (about 10¹¹⁰); it's about as close as things like this usually get.

For the first time now, we have surpassed a googolplex! This is another milestone, and a good time to take a closer look at how insanely huge these numbers are—specifically, we'll try to compare this number with the previous number.

You'd need to raise the previous number to the power of 10⁷⁹ to get this monstrous number. Here's a way to get an idea of what that means. Consider a universe with a volume of 10^{10^36} Planck volumes. Now, imagine that every Planck time that universe's volume multiplied by 10^{10^36}. Then, it would take about 1.7 octillion years to get a universe with a volume of 10¹⁰¹¹⁵ Planck volumes. Up next we have another even more insane number:

10^{10^245} - Old value of promaxima (Sbiis Saibian's alternate-universe count)

Remember Linde and Vanchurin's number of possible universes? Well, as it turns out it is a rather conservative estimate, considering that they would have to be distinguishable by a human observer. There are definitely other universes that would be reasonably distinguishable. For example, two parallel universes could be considered the same by Linde's estimate because they are the same during the time a human has to observe them, but after the observer dies the universes could take entirely different paths. So what would happen if we were to make a more liberal estimate?

On the "Really Big Numbers" forum page, one of the earliest examples of googology on the Internet, an anonymous contributor asked this very question.^[13] Then, in 2004 Sbiis Saibian responded by giving an upper-bound of how many different parallel universes there can be. That was his very first large numbers post on the Internet.

Sbiis Saibian calculated this number by first calculating the volume of the observable universe in Planck volumes (10¹⁸³). With that we can imagine the universe filled with the hypothetical Planck-sized particles known as strings, making 10¹⁸³ a reasonable upper-bound of the number of particles of any type in the observable universe. Then, we can take the factorial of 10¹⁸³ (about 10^{10^185}) to get the count of arrangements of all those particles. After that, Sbiis Saibian decided to let the universe run for 500 quadrillion years, an estimate of how long it will take until all the stars burn out. He raised 10^{10^185} to the power of the number of Planck times in 500 quadrillion years, to get 10^{10^245} as an upper-bound for the number of possible 500-quadrillion-year universes.

This number is interesting because it is indeed a "practical number" in that it has a kind of real-world meaning, and surpasses a googolplex. Saibian named this number "promaxima", short for "probability maximum", indicating that this is in a sense the largest number that represents something in probability, as a meaningful number of possibilities. However, Sbiis Saibian later gave an improved (and bigger) estimate:

10^{10^343} - New value of promaxima (Sbiis Saibian's alternate-universe count)^[3]

This is the new value of Sbiis Saibian's promaxima, a recalculated value that changes some conditions. For one thing, Sbiis Saibian allowed the universe to run 10³⁴ years, an estimate of how long it would last until it fades away from proton decay (if the proton decay theory is correct). But he also made some more interesting changes in conditions.

Instead of assuming that the universe has the same size throughout its history, he treats the universe as though it were a glome (a 4-dimensional analog of a sphere), treating time as the fourth dimension. To simplify things, he then bases the calculation upon an average volume of the universe throughout its history, ending up with 10^{10^343} as his alternate-universe count. This is the new value of the promaxima, his estimate of the total number of possible histories of the universe from birth to death.

Is this the largest "physically meaningful" number? That depends on what you mean by "physically meaningful". It seems that it would make sense that this is the largest number that can represent something in the real world, considering that it represents the total number of ways absolutely anything can happen. After all, Sbiis Saibian himself considers the promaxima the boundary between "corporeal numbers" (numbers with a real-world counterpart) and "ethereal numbers" (numbers without a real-world counterpart). So this is the end ... or is it? It's possible to create larger speculative numbers using much more abstract scenarios, but it's debatable whether they're "physically meaningful". Nonetheless, we're not quite at the end of the line just yet...

10^{10^410} - Factorial of the single perturbation count (Robert Munafo's alternate-universe count)^[5E]

This is Robert Munafo's alternate-universe count. It is the factorial of the single-perturbation count, which as you may recall is the number of choices you have if you want to move any particle to any location at any point in time. It is an alternate-universe count notable for being larger than the promaxima; however, it is debatable whether this number would be considered anything more than an upper-bound, so let's go into a bit more detail on what this number really is.

Imagine a universe where each possible single-perturbation happened once and only once. This means that each Planck time from the Big Bang to the present, every particle moved everywhere at once. Now imagine every way that history of the universe could be shuffled. This number of ways would necessarily be less than the factorial of the single-perturbation count, since some perturbations are simultaneous and therefore swapping those wouldn't change the history. Even that number of ways would be greater than the actual alternate-universe count. That is because it is impossible for every single-perturbation to actually happen: for example, two perturbations might be moving a particle to location A and moving the same particle at the same time to location B, and those perturbations can never actually both happen. Therefore, this number really is an upper-bound on the number of possible alternate universes.

However, what's also unusual about this figure is that it only accounts for the first 13.7 billion years of the universe, not anywhere near its entire run. This all goes to show, once again, that this figure really is an upper-bound, but still not quite the largest number that could even theoretically represent something in probability. Up next are some numbers way out there that we can generate using very sketchy and theoretical scenarios...

10^{10^1,000,000} to ???

10^{10^1,834,103} - Number of ways to arrange the books in the Library of Babel

By taking the factorial of the number of books in the Library of Babel, we get this monstrous number. It's debatable whether this quantity would be considered a real-world example of a number or not, but nonetheless this is another number that probability can let us form. This number has a 1.8-million-digit number of digits—even the number of digits is an incomprehensible number now! To get an idea of the size of the number of digits, it's approximately the number of Planck volumes in the universe, multiplied by itself 9888 times.

This number is often cited as 10^{10^33,014,740} from the book "The Unimaginable Mathematics of the Library of Babel", but that value is clearly too big. It's much larger than the factorial of the number of books in the Library of Babel. In fact you need to raise the factorial to the power of about 31.18 million to get the erroneous value! Even more confusingly, book where this value came from says that the number has only 33 million digits, while in reality it has a 33-million-digit number of digits! All this is a good example of human innumeracy, the difficulties people have with comprehending very large numbers.

10^{10^10^12} - My estimated alternate universe count

One glaring quirk in many of the alternate universe counts is that they only consider the observable universe. This is notable because there is quite a lot of evidence that the universe goes way beyond just the observable universe. It is not known how big it is, but a common estimate is 10^{10^12} meters (or whatever other unit, it doesn't make much of a difference). This suggests that the number of possible parallel universes that way is on the order of 10^{10^10^12}. This is a very rough estimate, but it still allows us to get an idea of how many different parallel universes there might really be.

This number is simply unimaginable ... it may well be approximately the number of parallel universes that are at all distinguishable. Is this the real limit of probability?! Believe it or not, this is still not the largest "physically meaningful" number if we consider certain strange theories that can take us EVEN FURTHER!

10^{10^10^82} - "Chess" with the known universe

Sbiis Saibian, in his article on Skewes' numbers, brought up this number as a large number in probability that can compare to the Skewes' numbers.^[15] He credits the value to "[a] physicist". The Skewes' numbers are two famous large numbers which were large upper-bounds in a problem in mathematics. We'll go over them in the next article, but for now know that the first is about 10^{10^10^34} and the second is about 10^{10^10^963}, and this number is between the two in size. But how can probability lead you to such an enormous number?

Imagine playing "chess" with the entire observable universe, where each move is simply swapping the positions of two of its atoms. You keep doing this until the exact same position has occurred three times! Then, the number of possible games is a number very much like Skewes' number. As Sbiis Saibian said, that's got to make your head spin!

Here's how this number is derived: First off, the number of ways you can arrange the all the atoms in the observable universe is 10⁸⁰!, approximately 10^{10^82}. This is roughly the number of moves it would take to have the same position to occur 3 times. Next, how many moves can you make? You can choose any of the 10⁸⁰ particles, and swap it any of the other 10⁸⁰-1 particles. However, you'd need to divide the product of the two numbers by 2, since swapping particle A with particle B is the same as swapping particle B with particle A. This gives you 10⁸⁰*(10⁸⁰-1)/2 ≈ 10¹⁶⁰ moves you could make. So now we have a game with 10¹⁶⁰ choices every turn, which goes on for roughly 10^{10^82} turns. We then raise the number of choices to the power of the number of turns, to get about 10^{10^10^82} possible games of chess with the known universe.

This number is just INSANE!!! At this point numbers generated with probability are the only way you can get a feel for the size of these numbers. However, probability can still take us somewhat further.

10^{10^10^120} - Poincaré recurrence time for a black hole with the mass of the observable universe^[5E]

In physics, there is a theory known as the Poincaré recurrence theorem which states that certain systems will, after a long but finite time, return to their initial state. Sbiis Saibian gave a way to get an idea of what this means:^[3] Imagine one of those lottery machines with all the numbered balls shuffling around, and imagine that the balls were to shuffle around for an indefinite amount of time. If they were to shuffle randomly forever, it would only make sense that at some point the balls would be back to the position they started at, right? After all, there's only a finite number of possible positions for all these balls. That's basically what the theorem says.

Now imagine the same idea, but with a black hole the mass of the observable universe. Consider the state the black hole starts out as, and then imagine the black hole having its contents constantly changing. Physicist Don Page calculated that after about 10^{10^10^120} Planck times, millenia, or whatever else, the black hole would return to how it started. This may seem more like a duration than a probability, but it actually reflects the vast number of states our universe (or a black hole that size) can take, and how unlikely it is to, at any random point in time, get back to where it started.

We have now transcended a googolduplex!! That's quite insane alright, but you may be wondering: why not use a black hole with the mass of the entire universe beyond just what is observable? That's exactly what we'll get to next.

10^{10^10^10^13} - Poincaré recurrence time for a black hole with the mass of the entire inflationary universe^[5E]

Consider applying the Poincaré recurrence theorem to a bigger black hole ... a black hole the size of the entire inflationary universe by Andrei Linde's theory(with the estimate of size on the order of 10^{10^12}). Then, the Poincaré recurrence time for this black hole, calculated by Don Page, would be about 10^{10^10^10^13} Planck times, millennia, or whatever else. There are about 10 trillion digits in the number of digits in the number of digits in this number. In other words, it's really fucking huge. Once again, this may seem more like a time than a probability, but it actually reflects the vast amount of states a black hole of this size can take.

This is the largest number that I have ever seen touted as having any sort of physical meaning!! It is often considered the largest number to have appeared in science, and that makes sense. Physicists don't just throw around large numbers for the sake of it, and if they do the numbers would have to be in some way meaningful. But is this a meaningful number? This all depends what you mean. This is often touted as the largest "physcially meaningful" number, as a boundary between "physical" and "abstract" numbers, but it is arguably too hypothetical to be a clear-cut boundary. Poincaré recurrence times really are a rather "abstract" theory, especially when they're applied to things we don't know at all to exist! Really, for anything after a promaxima it's debatable whether these numbers are "real" or not.

And that brings us to a point: there is more of a gray area between "real" and "abstract" numbers than a clear boundary. Though there most definitely are numbers that we can't make have any sort of physical meaning, we can't really define a clear-cut boundary. Maybe the promaxima could be a boundary, but that's a shaky idea since probability can generate larger numbers than that! In any case, we will soon be leaving the realm of numbers that we can connect to the real world well behind us, accelerating towards the truly gargantuan!

Conclusion

So, what are we to learn from all this? First off, we've learned that probability can easily generate numbers a lot more mind-blowing than you probably thought! But also, once we go past a promaxima, the connections between numbers and the real world become shaky, hard to really give a real-world counterpart. I hope you found this an enriching journey through numbers probability can make. Up next we'll examine two numbers I had mentioned towards the end of this article that arose from a math problem: the Skewes' numbers.

Sources / Footnotes

1.06. Skewes' Numbers