A number NOBODY has thought of - Numberphile
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- čas přidán 16. 05. 2022
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A ten digit number could just be somebody's phone number
Ye I guess there’s a 10 digit number of people alive today, the majority with unique phone numbers. But I do still think 1.76x10^67 is way too big.
@@finnwilde thats the point, though. He wanted it to be big so that he could say with certainty that this number is unique
Yeah, to test whether brady's estimate was good, I googled a couple random 10 digit numbers to see if there were results, and loads of phone numbers came up. No results for any 11 digit numbers I tried, though!
Agreed. And a 16 digit number could easily be someone's credit card number.
I guess BUT do you think of your phone number as a 10 digit number or 10 one digit numbers
I like that Brady doesn't just accept 99% probability at face value.
And he's right
There was another numberphile video about category of numbers. Specifically covered was the transcendental number based on the combination of the Fibonacci sequence put into a decimal format, as in 1123581321345589... Simply putting that breaks the premise that nobody has ever though of, or even used, a 67 digit number. And in a more practical application, cryptographic permutations far exceed this limit as well.
There are some videos that don't need to be made....this is one of them....can you think of another?
I mean it’s clearly wrong
@@edme8865 It is not being argued in the video that no one has ever thought of or used a 67 digit number. Rather, that if you pick a *random* 67 digit number, you are likely the only person who will ever think of it.
I think Tony mis-stated the question and I think that's why Brady was so confused. So he stated that if you pick a number larger than 10^67, there is a 99% chance that it has never been thought of before. But in the mathematics, he then shows that there is a 99% chance that NOT A SINGLE number above 10^67 has EVER been thought of before when thinking of numbers according to that 1/n^1.3 distribution. That's a big difference. The 99% isn't a probability for that exact number, it's the probability that every number ever thought of following that distribution is less than that number.
At 10:58 he states this. But then he says "so if you go farther, there's a 99% chance you'll find a number that's never been thought of before." But really what the math means is "if you go farther, there's a 99% chance that you'll never again come across a number that has been thought of by following the distribution"
Thats what I thought too. Great explanation
Also that's easily disproven as people love to think of large numbers (googolgoy), as he himself did on this channel (googol, graham's number, tree (3),etc)
@@johannesh7610 yea but there's basically about 100 numbers with so many digits or more that people have ever thought of. Which is literally a 10^-67 or whatever probability
That seems likely. Well said!
yeah he integrated from 1 to N when that doesnt correspond to the question asked. The answer should just be when N*P(x) (10^20) = x^1.3 --> x ≈ 10^15
so if you choose a number on the scale of 10^15 there should be a 1% chance someone has thought that exact number before
Seems like there is a really big difference between "no one has ever thought of this number" and "all the numbers that have ever been thought of are less than this number". I think the latter is an amazingly loose bound on the former.
We also know for certain people have thought about Graham’s Number, so all numbers that have ever been thought of are less then 10^70 is definitely wrong.
@@fejfo6559 A lot of people missed the probability part.
@@smurfyday But the probability is that "all numbers in the set (of every number someone has ever thought of) are smaller than this number", not that "this particular number is not in the set", and no reasoning was presented to why they could be related. On the surface, they seem completely different concepts that would have unrelated probabilities.
And while "every number in the set is smaller than this number" would by definition mean that number is not in the set, that doesn't really work if we already know for a fact that there are larger numbers in the set.
But... when we say "this number" we thought of it. So it should be "equal or less than" perhaps.
@@fejfo6559 Except no one has ever actually thought of a number with 10^70, and there is no need to go as far in the digits for a number to never have been thought of by anyone, by the simple fact that people don't think most of those big numbers as a chain of digits. It'd be like claiming because you thought about PI, you were able to think the number as an actual infinite chain of digits, which is not the case. Or the wording for "thinking of a number" on this video should be changed, as clearly, thinking about something implies some degree of mental representation, and no one has ever been mentally representing a 100 digit number as a whole.
Sorry to ruin your day, but I have just thought of all numbers from 0 to infinity.
5:35
But then the question you asked at the start is different to the one you just answered.
Start: "How big a number do you have to generate for it to be likely to be *different to* any other number ever thought of?"
Answered: "How big a number do you have to generate for it to be likely to be *bigger than* any other number ever thought of, ignoring anomalous occurrences?"
Yes you are right, the way he answered it, replies to your second question.
Exactly. I'm pretty sure you get the actual answer by solving:
0.3 / n^1.3 * 1.5 * 10^18 = 0.01
Which gives:
n ≈ 1.3 * 10^15
i really hope they see your comment. the whole video is basically a mistake 🤦🏼♂️
@@jebbush3130 Theres no way 1.3*10^15 can reach 0.01 probability, due to all the 15 digit phone numbers. It definitely needs to be higher than that
@@z-beeblebrox There are less than 10^10 people in the world. Even if every single person owned one thousand unique phone numbers, that would be less than 10^13 total phone numbers. A random 15-digit number has a less than 1% chance of being one of those numbers.
10^67 is a huge over estimate. Consider a 30 digit number. To have a greater than 1% chance of repeating a 30 digit number, then more than 10^28 of those 10^30 numbers would have to been "thought of". With a total population of 10^11 people, that would mean that every person would have to have thought of 10^17 30 digit numbers in their lifetime. So if everyone lived for 80 years, then you would have to come up with 40 million 30 digit numbers every second of your life (60 million if you want some sleep).
I agree with your reasoning. And producing all those 30-digit numbers is quite a challenge especially as I would need to cross-check each one against the ones I've previously generated to avoid duplicates! 😮 (And that still wouldn't guarantee no duplicates with the rest of humanity's lists).
I thought of a number with 24 digits, but it turned out that some guy named Avogadro thought of it first back in the 19th century. (Yes, technically it would have been Loschmidt to think it first)
@@ingulari3977 32 Digits hex (128bit) is a larger address space than 10^30 though.
So your point is arguing for a higher cap
I don't see a problem with that
Remember when IPv4 protocol was made, and they though "who would ever use 256^4 different IP addresses"?
For those that wondering, the spike on the graph at 7:09 is 2004, the year the data was gathered. The reason 2003 isn't as high, is many pages are updated to the current year
the shape in the thought bubble at 4:16 is a Calabi Yau manifold. some physics theories postulate at least 10^500 different ones of those (a number with 501 digits).
i recognize it as the world's most complex tortilla.
It's called a Lufa
At first I read this as "The shape...is a Calabi, you manifold!" and I just thought it was a clever dig at a physicist.
@@Popbot If you made a taco with that, there's a 100% chance the stuff will fall out.
There's probably only 10.1^500 shapes of any kind, so it sounds like that one makes up the majority of them
"That number is yours forever" isn't too far-fetched for those of us who use computational hashes
Yeah, flip 160 coins and interpret the result as binary and you're _done_
I think I've solved this in a fairly reasonable method, starting from the same assumptions and formula that Tony used. I might be slightly bodging the explanation but this should explain the ideas for anyone that wants to quickly replicate my math:
With N being his 1.5e18 estimate, and p(x) calculating the percentage of numbers above x
And 'amount of numbers' meaning some defined portion of N because I don't want to write 'amount of numbers thought of by humanity' every time
Calculating the amount of numbers BELOW some N* is simply the total N minus the fraction of N above N*, so N-(p(10)*N)
So below 10 would be N-(p(10)*N)
With the ability to calculate the number above and below some range of N*, you can then get the amount of numbers in that range by subtracting the number below the minimum and the number above the maximum from the original N
Or Calculate the number above the minimum, and subtract the amount above the maximum. This is how I actually implemented it in the test spreadsheet.
And the 'normal size' of any range its maximum-minimum.
So I think it's fair to propose that you can roughly estimate the likeliness of a random number in some range being unique by simply dividing the 'amount of numbers' in a range into the 'size' of that range. Of course to be REALLY precise about this you'd want to do some further stats to account for the birthday-paradox type errors that my simple estimation leaves in.
Using a spreadsheet to test these formulas on power of 10 ranges(0-10,10-100,100-1000,etc) gives fairly intuitive results:
For a 15 digit number the amount of numbers that land there is 47.2 trillion, and the size of that range is 900 trillion, so you'd be at ~5% odds that your number has never occurred in that N dataset before.
For a 16 digit number the amount of numbers that land there is 23 trillion, and the size of that range is 9 quadrillion, so you'd be at ~.2% odds that your number has never occurred in that N dataset before.
17 digit - .013%
20 digit - .00000165%
So Brady's 10 digits is definitely an underestimation, especially considering other commenter's examples such as ip addresses and phone numbers. But you definitely don't have to go too much further to reach near-certainty even from these assumptions.
I'm sceptical of his reasoning. There is no point in looking for some number that, with some probability, is larger than every number ever thought of (which is easily invalidated, anyway, by a single person thinking of a larger number). It is, indeed, far simpler: Take a (uniformly) random number less than or equal to 1.5*10^20. Even if all numbers ever thought of are distinct and less than 1.5*10^20 and his ridiculous estimate of 1.5*10^18 is anywhere near accurate, there are only those 1.5*10^18 numbers below 1.5*10^20 that have ever been chosen, making the probability that yours is one of them only about 1.5*10^18/(1.5*10^20)=1%. In fact, you can likely go far lower, yet: you really just need to find some n, such that there are at most n*(1-sqrt(0.99)) distinct random numbers less than or equal to n with probability at least sqrt(0.99) which is really awful to actually work out so I won't even attempt to do so, here.
Yeah when he estimates the total numbers to be 10^18 or so, ending on 10^66 or so should set off some alarm bells lol
He has massively overestimated it, but for the stated purpose, that is fine.
I agree with Brady.
If picking a random 67 digit number gives a 99% chance of having a new number, doesn't that imply that 1% of all 67 digits numbers have already been thought of? Which would be many MANY orders of magnitude more numbers than humanity has ever thought of...
But most of that 1% would be close after 1E67. So for example, numbers past 1E80 would be at the 0.001%
I do wonder if picking a 67 digit number would be random, considering how some numbers appear more regularly. As discussed in the 6 to 7 min range in the video of how some numbers are more commonly picked, and considering how likely people are to 'build' a number from smaller numbers, all would indicate that the 67 digit numbers might not be so unique.
@@edme8865 So if you'd like *yours* to be unique, your best bet is rolling a d10 67 times to get 67 random digits
no, it means that there is a 1% chance that a human has EVER thought of a single 67 digit number.
@@bzw77 An interesting thing that people can use a 67 digit number without even thinking about it, in the context of crytpo permutations.
This kind of reminds me of a question I keep thinking about: what's the largest known prime consecutively (i.e. where we know all primes up to that point)?
Sure, there's all the Mersenne primes with millions of digits, but that's skipping a lot of primes inbetween.
It's 5 and 7. It's the largest know prime consecutively........for a 3 yr old considering the child only know to count till 10
The trouble with this is that, whatever prime you say, it'd only take seconds for someone to find the next few after that prime.
@@rosiefay7283 no
Edit: yes
@@MrMctastics why do you say "no"? Remember, we are talking consecutive primes here, which cuts down the effort required you are probably imagining by orders of magnitude(s of orders of magnitude).
@@rosiefay7283 Very true! Knowing at least the order of magnitude for sure would still be cool though.
There's a huge gaping problem with Tony's assumptions: he's trying to find how large a number must be such that there's only a 1% chance humanity has thought of a larger number. That is, the cumulative distribution function up to that point is 0.99. The original problem, however, wasn't that no human has thought of a bigger number. It's that no human has ever thought of that EXACT SPECIFIC number. He needs to evaluate the probability mass function. He needs to find a number such that the probability of that number is no more than 1%, and the probability of any individual number after it is no more than 1%. Not combined probabilities, mind you, individual probabilities.
Ohhh, I thought he meant the lowest number where there was a ninety nine percent chance that any whole number contained within it would have never been though of again, to which my reply would be a billion.
Now knowing what he was really trying to say than I'd say about I0^30's seems about a decent call figure in finding such a number from a random educated guess of mine.
im with brady on this one. I think hes tricking himself with his own math and confirmation bias. Brady's argument about 66 digit numbers makes sense if you instead say how many numbers of length 66 have been thought of? which surely is waay less than a majority so you should have good odds of picking a random number that if within that set. not to mention decimals.
He was also completely wrong about how long people lived in the past. Their life expectancy wasn't much different from ours, humans were exactly the same. We live slightly longer due to modern medicine, but not notably so.
He fell in to the common trap of thinking average is the same as mean. The average life expectancy before modern medicine was low because child mortality was high, not because people didn't live to 70.
@@VikingTeddy
1) That doesn't change how many people were born. If anything at all, it overestimates
2) It doesn't matter in this case, because the life expectancy in the past did not affect the numbers a single little bit.
@@Heartsii_ Sure, but I wasn't commenting on the math, just the common misconception.
@@VikingTeddy He wasn't completely wrong. Some data from wikipedia:
Paleolithic (old stone age): total life expectancy (at 15y): 54y with 60% chance of reaching 15
Neolithic (late stone age): total life expectancy (at 15y): 28-33 years
Bronze and iron age: total life expectancy (at 15y): 28-36 years
Classical Greece: total life expectancy (at 15y): 37-41 years
Classical Rome: When infant mortality is factored out [i.e. counting only the 67-75% who survived the first year], life expectancy is around 34-41 more years [i.e. expected to live to 35-42]. When child mortality is factored out [i.e. counting only the 55-65% who survived to age 5], life expectancy is around 40-45 [i.e. age 45-50]. The ~50% that reached age 10 could also expect to reach ~45-50; at 15 to ~48-54; at 40 to ~60, at 50 to ~64-68; at 60 to ~70-72; at 70 to ~76-77.
Europe (5th-10th century): total life expectancy (at 20y): male: 45y, female: 37y
Late Medieval England: total life expectancy (at 25y): 48.3y (high ranking male)
England 15th-16th century: total life expectancy (at 15y female): 48 years
@@kvarts314 Thanks for the numbers, they're fascinating.
And you're right. I was thinking about biological life expectency only, and completely forgot about the actual one. Oops...
2:38 The shadow moves across the Earth is rotating in the wrong direction! But correct at 3:50 and subsequently.
For encryption purposes, we use super large numbers - primes! - all the time! But I suppose no one will think of them if they are larger than 10^73.
"Prime!" as in 3! = 6?
Keep in mind though that the prime density of primes above 10^73 is very low. E.g. proportion of primes below 10^73 is about 1/log(10^73) = 1/(2.73 x 10^27), way below 1 percent. So even assuming all primes have been though about, we are still way below the 1 percent threshold of this video if we take random numbers above 10^73.
@@__Brandon__ True enough ... but then there's Rayo's number ... which I think is a cheat ... plus the path to get there -- 11 !!!!!!!!!!!!...!!! -- was actually an extremely tiny number, so there's all sorts of games we play for enjoyment.
@@__Brandon__ close, but 10^73 is only ~243 bits of entropy
I'm with Brady, this seems like a HUGE overestimate. Let's see if we can lower the bound a little, and still be conservative.
Let's say there have been 117 billion people. And on average, each person uses/thinks of a number once per second. We'll say the average lifespan across all of human history is 100, to simplify the math and be even more conservative. So over all of human history, a total of about 3.7 * 10^20 numbers.
Now let's say every single person who has ever lived has conspired to think of YOUR number. And not just that, they're all psychic and know exactly how many digits are in your number, and can coordinate with each other across time and space to ensure nobody ever uses the same number twice. How big does your number need to be to have a 99% chance of nobody successfully using your number?
Well, if we assume a uniform distribution of "used" numbers, and your own number is random, then you just need to multiply their number of guesses by 100. Now the set of possible numbers is exactly 100 times larger than the set of their guesses, so the chance of any particular number in the set of all numbers being in the set of guessed numbers is 1/100.
So 23 digits should be more than enough. If you think people think of/use numbers more often than once a second, add one additional digit for each factor of 10 increase. So if you think 10 numbers per second, 24 digits. 10,000 numbers per second, 27 digits.
It is just a probability
Not all no with 23 digit or higher follow that
Like if i go to a party of like 10 billion people
But 3 chief guests are missing
U combine all their 3 no and like 10 billion people think of that at the same time
A very large proportion of the 117 billion people who have ever lived did not inhabit a world where number systems even existed. That means any estimate based on the whole 117 billion people is way too big.
A few years ago I was thinking about this exact problem. In the process I ended up defining the set of forgotten numbers - the set of numbers that no one will ever think of. It has a few interesting properties, like how almost all numbers are forgotten numbers, however you can't ever name any one of them.
Everybody gets one MASSIVE thing wrong about life expectancy of the past: They don't take into account that the average was SKEWED.
The way we calculate life expectancy is just a mean average. But when you have a high rate of infantile deaths, the average age gets WAY skewed towards lower numbers. If you made it into adult hood, you'd probably be making it into your 60s. But the "life expectancy" would be 40 because SO MANY children died at young ages compared to now.
But in this case, dead infants are included as "humans who ever lived", so the numbers still work out.
Thank you so much for pointing this out. I've always wondered why I hear about so many famous people from 2, 3, 5 hundred years ago who lived to 65 or 70, when that doesn't jive with the common conception of life expectancy. Well yeah, when half of all people born don't live past age 5, that's gonna skew things downward. Never even thought of that, so thanks.
And since the amount of pregnancies was higher in the past, and the high infant mortality rates, means there's a whole lot more born to die, which skews the numbers even more.
@@lunacouer and being rich would generally let you live as long as modern humans do today, I believe I read that Ramses the Great died at 90-something age, likely from infection from tooth decay (ancient egyptian dentistry wasn't great)
In this case it doesn't matter if the 117 billion number counts that infant mortality.
A ten digit number has been a credit card number or a phone number already it needs to be bigger.
The Doomsday argument always makes me laugh. Imagine a particularly intelligent cro-magnon figuring it out and concluding that humans will go extinct in the next few hundred years. It's basically "well if we're roughly in the middle then we must be roughly in the middle."
The Doomsday Argument is bad statistics. In order for the distribution to be at all fair, you need to be able to select a random human from any point in human history (past AND future) and figure out what birth rank they are, and THEN you can make a statistical analysis of roughly how long humanity will exist, with increasing accuracy the more birth ranks for random humans you collect. And then what they end up doing is saying "right now is a sufficiently random point in human history" which... it objectively is not.
You could do a similar argument with populations.
If I am a typical human living today, I live in the United States a population of 330 million. From this you can conclude there is only a 10% chance the world contains more than 3.3 billion people.
Wait a second! He's calculating something different though. He's looking at the probability that nobody has thought of a LARGER number, NOT that the number is unique. Notice that his probability equation he uses is to calculate "what's the probability that a randomly chosen number from this distribution is LOWER than the number n* ?" (See timestamp 7:39 )
And then his number clearly fails, because of all of the obvious examples... Tree(3), Graham's number, etc.
For unique, you need to look at the finite differences of the CDF, and calculate (1-FDCDF)^ (1.5*10^18) >= 0.99. I get about 1.16*10^28 which seems a lot more reasonable. Still a huge overestimate from going on his assumptions of how many numbers have been considered though. I'm sure you could confidently bring it down a few more orders of magnitude.
Strange, I think you made a mistake. I got only 662 million, ie 7*10^8
Tree and Graham’s number have never been written down in integer form, which is what the question was about
Encryption keys could be thought of as huge numbers "we use". A 2048 bit key would be over 10^600.
Yes but what percentage of those have been used already? I'd say very little.
and this is a number a human thought of. Who manually creates a 2048 bit encryption key? maybe spies who treat it as a one time pad
But they are all products of two primes. There are plenty of 3+ prime products between them. A 1% guessable encryption key would be terribly bad.
“Think about” vs “encounter”, two very different things. Very few people just “think” about numbers purely as numbers. However, I routinely encounter them, some large:, most small.
I'm pretty sure "think about" is a subset of "encounter"
I, for one, love the puzzle! Between phone numbers, ID identifications, credit cards... thinking what's a number that no one has thought of is definitely intriguing! We're even accounting for computers here. Great video and puzzle!
IPv4 addresses are 32-bit (about 4 10^9) numbers. So, many people have been using, seeing and thinking about 10 digits all the time without knowing them in decimal form.
There are the GUID that have been used for over 20 years. They are 128-bit (about 10^40) random numbers. As a result, many people have been using, seeing and thinking about such big numbers as well.
But those are laughably small to the 224 bit number 10^67...
The fact that there is a 99 percent chance that nobody has ever thought of a bigger number doesn't matter. According to his assumptions if the humanity has thought of 10^17 numbers then the chance you think of a unique number is huge very quickly. Let's say you pick an 18 digit number. Even if every number ever thought of had 18 digits that would still be only 1 tenth of all the 18 digit numbers so you already have at least 90% chance. If you choose 19 digits than it's at least 99%. In reality it's likely much higher as they mentioned as well most numbers thought of are tiny.
It mustn't be a tiny number one think of. Most likely someone would think of an unspecific number like 4.8 billion$.
Probability doesn't work like that
@@kisslab Just because it does need to be big doesn't mean it needs to be *bigger* than every number everyone has thought of, it just needs to not be identical to any of the previous numbers thought of. He should be choosing with replacement basically.
@@Amethyst_Friend you're right, but their math isn't far off. Let's keep the assumption that there are 10^17 numbers that have ever been thought of, and we'll pick a random number with 10^18 digits. Now, let's assume that all 10^17 numbers ever thought of were all random and unique numbers with exactly 10^18 digits. Assuming no leading zeros, that leaves a total of 9*10^17 numbers that we're picking from. In this absolute worst case scenario, where nobody has ever thought of the number 2, the odds of picking a number with 10^18 digits that has been thought of before is (10^17)/(9*10^17)=1/9.
In other words, picking a random number with 10^18 digits should give no worse than 88.88...% odds of having randomly picked a number that has never been thought of before.
This seems a bit more clever, and it's a more satisfying and believable answer too.
Then let's consider that each number thought of begins at 1 and there's no gaps, ie, every number imagined has been unique (slightly more realistic but still a massive overestimate). Then the probability just changes by a factor of 10 for any numbers above 10^17, so 17 digit numbers would be 88% and so on
Note on life expectancy for people in the past: People actually lived to a decent age (60+) as long as they made it out of childhood. Why you always hear people in the past died at 30 is because they are taking an average which includes kids that died. But again, as long as you made it out of childhood, you'd live a decently long life.
I was thinking about a similar problem the other day. I was wondering what the smallest integer no human has ever seen is. Obviously if you were to find it then it would stop having that designation, but people really don't see large numbers written out very often so my gut instinct is that it's somewhere around the point where people usually switch over to scientific notation. That would put it somewhere on the order of 10^15, because once you pass the trillions into the quadrillions that's too many digits for most people to easily wrap their heads around. Though I have no idea where I'd find data to make a more informed estimate!
A human? maybe.
But a machine ? definitively not.
Just the collatz conjecture alone was verified for all numbers up to 2^68 ... hum weird it's quite close to 1.76*10^67
I got in an internet argument on a similar topic just a few days ago. Person 1 said the well-ordering principle says there is a smallest number no human has seen or used before. Person 2 claimed you have to define the set the number is being pulled from, and "numbers no human has seen or used before" is not expressible in first-order ZFC therefore is not a definable set. Other commenters had already pointed out this is just the difference between two sets. I said it would only be paradoxical if you used the number; there's no problem saying it exists, but also Person 2's same form of argumentation could be used to (erroneously) argue Uncomputable numbers are not a definable set because (by definition) you cannot write a first-order ZFC predicate to recognize an uncomputable number.
@@ballom29 you can be clever and dismiss most numbers outright when checking the collatz conjecture so a bunch of integers between 1 and 2^68 have been missed. Also 2^68 and 1.76*10^67 are not even close to each other.
For most of human history, nobody ever thought of a number greater than 1000. There was no way in the language to express such a number, so there was no way to think about it.
That’s clearly BS
Another way someone can think of huge numbers Is in incremental games like cookie clicker, at the start sure its thousands and then millions and billions, but as you go on you go bigger and bigger. And if you look at it once while it's running, and think of the amount, then someone has thought of that number
Especially if we consider how they defined "using a really big number", every number I'm those game would jave technically be "used"
The framerate of the PC is only about 60 numbers per second, so that's an upper limit on the number of digits you can see. However, I think "thought of" should have a higher barrier than "see" because we know from neuroscience that the vast majority of visual information never filters into your conscious thought.
Yeah - I'm thinking in terms of a number never interacted with human consciousness or computed. You can have a computer generating numbers within the space of 1.76x10^67. An society of observers are watching the screen, and registering them as it provides with a new one. There's no distinction at that point besides labels. At this point it's like organizing subsets of sand on the beach, by holding them in your hand and watching them fall. You aren't interacting with a written number (conscious besides "reading"). I guess you can make it an unconscious reading: shape the sand particles in to a '1' shape and this represents 1. Blow on it, "read it", this obtuse configuration of sand particles, represents the character for the number of sand particles still present. Learn about pieces of it. But you cannot seem to hold it. It is too complex.
What Tony's argument actually concludes is that if you pick a random 67-digit integer, it's likely to be *bigger* than any number ever thought of before. Which is, of course, a ludicrous conclusion, but it's supported by his assumptions.
Yes. The assumptions are probably completely wrong (depending on how you define "think of a number"), but the probability he calculates does not correspond to the question he asks.
I just thought of his number plus one. Oops
Not only that it is supported by this You Tube Channel because it was posted.
Let's turn this around. The largest number considered is always: the last largest number + 1. Argument closed. End point.
@@clahey then you didn't think about the number...
Disagree as the number is still far less than a googol (for instance) and many other large numbers, which we know have been thought of.
A 1KB file in your computer can be interpreted as a 1024-digit number in base 256 or a 8192-digit number in base 2, which is way bigger then 10^67, and is a number you've "used" but not thought of.
This is actually an important question in computer science when we want to assign unique IDs to things without knowing what other IDs have been taken. There's a specification called UUID that programmers use all the time to "think" of random numbers while being fairly confident that they'll never randomly generate a duplicate UUID. The space people use is 2^128 or around 3.4 * 10^38, or just over the square root of Tony's number!
So many years later.. Numberphile is sticking to their original angle.. talking individual numbers! Excellent!
The set of numbers that no one has ever or will ever think of has some interesting properties. Any statement "X ∈ {that set}" where X is a numeric literal must always be false, since reading it would exclude that number by definition. Unless the statement is generated by a computer and saved or printed somewhere and never shown to a human.
This statement about self-reference always leading to contradictions is false.
@@Bennici This sentence is true. Therefore, so was yours.
Is it 258? I just discovered it yesterday and it feels like a hidden treasure.
I agree with Brady, I think Tony is off by an order of magnitude in the exponent. Matt Parker's ten-billion-human-second-century seems to suggest the same.
wish this vid focused more on how intentional under- or overestimation can be used to generate legitimate results. Say you want to show that statistic X is greater than value Y. You can compute an estimate for X using assumptions which will definitely result in an estimate less than the actual value of X. If this definitely-less-than-X value is greater than Y, you have proven that the actual value of X is also greater than Y. I think there would be a lot less controversy over your assumptions if you emphasized this as the goal of your assumptions -- it doesn't matter that the assumptions are realistic. It matters that they definitively result in an underestimate or overestimate of the value in question -- whichever matches the point you're trying to make.
I agree
Phone numbers are 10 digits in the United States. If you randomly pick a 10 digit number it might happen to have been a real phone number at some point.
According to a quick search on the web, there are 336 area codes in the US. And I suspect not all the possible exchange numbers are used either.
(Area code 8x10x10) x (Exchange 8x10x10) x (local 10x10x10x10) = US phone numbers, except that the 5xx, 7xx and 8xx area codes use substantially less than their capacity.
@@quintrankid8045 india has 10 digit numbers too... the rest of the world exists...
@@footballbranthan2396 Well then we ought to add the country codes in front of each number to get as complete a set of numbers as possible. But I suspect each country will have some exceptions, for example emergency numbers.
4:16 the uni-brow caveman thinking of a Calabi Yau Manifold is so cute !
So is it just coincidence that it comes out at a very similar number to 52 factorial, the number of possible orders of a deck of cards, and we are always told that a well shuffled deck will be in an order that has never existed before.
to keep the axioms in check for distribution, I would compare your results against assuming all numbers thought of are unique and see how much bigger that number is
I would think that "thought of" would mean seen or conceptualized all of the digits in that number in a given base.
The chance that all numbers anyone ever thought of are less than Graham's number is by definition 0. After all, Graham thought of a number equal to Graham's number.
There are some videos about large numbers and they thought much further than Graham's number. Of course for every number n>1 you can always use n^n for example to find a larger number.
we know for a fact that people in our universe have thought of Graham's number, but the math question is "if you randomly selected an amount of numbers equal to the amount of numbers we're estimating have been thought of in our universe, what's the probability that all those randomly selected numbers would be less than n?"
Noone can write/think or spell down exact Graham number, not only because there is not enough material in the universe, but tbh no one knows what digits it's consists of.
@@MrWorldOfQuests We also only know a finite number of digits of pi. So it is not possible to think of all digits of pi. Of course Graham's number is quite abstract and nobody will probably understand how large it is.
@@skyscraperfan I do agree, and pi even worse because it has infinite number of digits, so it is impossible to know precise value. while Graham number is finite and in theory it is possible to know exact number, but practically we have no means to do so.
4:10 "in the iron age people were only living to about 20 or so"
the life expectancy at birth is weighted down by infant deaths. if they survived to 15, their life expectancy would be to live to around 28-36.
indeed! and many people lived into the 70s and beyond, if they had people to help them.
He needed to more precisely define what “think of” means. Somebody doesn’t think UP their phone number but I guess they do think OF their phone number. Social Security numbers in the US are nine digits. Say you had a ticket with a long ID number on it, is that a number that someone has thought up?
I was also thinking that when a computer thinks up a number, that’s a person thinking up a number indirectly. Shouldn’t we be coming up with numbers that no computer has ever thought up either?
He answered that, he took the distribution of numbers from the number occurences on the internet.
I just assumed "think of" meant any number your brain has processed. So if you held the number as an object of thought, typed it in, wrote it down, looked at it, were a mathematician working with it, regardless would all count as thinking of the number. AFAIK, he never said you have to have created the number. I think it's safe to assume that all (or nearly all?) 64-bit numbers have been thought of in this way.
@@MetroAndroid We all "use" big numbers every time we use encryption for anything. I don't think that qualifies as "thought of".
@@burnfire4617 While a computer using a number internally without ever printing it or putting it on the Internet or in a database file or anything probably does not qualify as someone thinking of a number, I still feel like the goal of the whole exercise should be to come up with a number that simply has never been produced in a human brain or by a human in a machine or on a piece of paper or in a database or file. It is still simple, it just comes down to estimating how many digits do you need for such a thing.
I'm gonna be a teacher and make an entire exam on this number alone. Take that
I'm not a mathematician (i studied law), however my question is whether any consideration was given to those who were born, but were unable to think of numbers? For example, stillborns, or those who died before the concept of numbers had developed.
Not a criticism - i enjoyed the video, but i couldn't shift this thought from my mind throughout and wonder if this would have an impact on the conclusion.
I love the fact you can have your "own" number that no other person would likely ever think of for eternity.
Arthur C Clarke once began a story with "There are thirty ghosts for every living human". I imagine that proportion has considerably reduced since that was written (50-60 years ago?) 117 billion implies that it's about halved.
I think xkcd did make a graph of alive vs total humans in time
something about fey being created by first smile of a child
Just about. According to the World Economic Forum, with us approaching 8 billion people in 2022, that means 7% of all humans that have ever lived are alive right now. So, "There are fourteen ghosts for every living human" would be more correct today.
Thank you for sharing this. I'd never heard this line. I appreciate that he brought it down to smaller numbers, so we could comprehend it without going glassy-eyed.
That only works if there is not reincarnation.
Anyone who's played idle/incremental games for an extended period of time are accustomed to seeing exponents over 100. Javascript can natively handle up to around e308. So if we're saying "thinking about" as in "have some sort of comprehension of", then you'd have to go higher than that. Depending on the game, *much* higher.
Agreed but how many of those numbers that players of those games think about are 3 or 4 or 5 significant figures followed by a bunch of zeros? assuming as many as 10 digits with trailing 0s that's still only ever looking at 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000001% of numbers above in the e100 range that those players think about
Some incrementals (namely AD prestige tree several more long web incrementals) even use additional libraries like break_infinity to go beyond e308 (10^308) some mods of these games use libraries like expanta_num start using eee which represents 10^10^10 and goes into F and G notation which are compressions of 10^10 stacks to give an idea how high these sorts of numbers go to.
Yeah, I believe that's because e308 is close to 2^1024 or 2^2^10.
I have seen numbers as big as eee100 in some games 😂
The fact that he thought of Graham's number while doing this already proves that people have thought of numbers greater than 10^68 with probability 1, disproving his reasoning
are there any really big numbers? let's googol it
He said essentially 1, because it is. As a proportion it's effectively zero even if you thought of a trillion numbers the same number of digits as grahams number you'd still have pretty much all of them to choose from. The probability of choosing a fresh number would be 1-(n/grahams number) where n is the number of graham number sized numbers that have already been picked. Grahams number is so huge even something like 10^100/grahams number is effectively 0.
@@nekogod ah yes, "essentially" that works in this case.
Except... "thinking of" numbers isn't an actual random sampling process. Doesn't it kinda get us into the "smallest uninteresting number" paradox?
This makes me think of a question I had in my sophomore-level statistics class. I didn't understand how a point could have no mass. I asked, "Let's say I'm going to choose a random number. The probability that I'll choose pi + 3 sqrt(2) is zero, right? ... Well, I chose pi + 3 sqrt(2). That means I did the impossible!" (Actually, I chose something more like 7 + 3 sqrt(2), but I got a little more creative still when I was watching this video. And I didn't even consider complex numbers.)
10 to12 digit is way too low depending on how you define "thought of". People that program see big numbers very often if you allow different display methods than base 10, people that work in national economic analysis work with trillions and more . Then there are all the phone numbers, bank numbers, IP Adresses, etc. Maybe if we go near 20 to 30 digits you can have some reasonable amount of confidence.
Yeah, I'd say that 10 digits is low. I'd venture that 2^128 (~ 2 * 10^38) is a very safe lower bound:
- The longest phone number is 15 digits, so it takes care of phone numbers.
- It also takes care of programmers being ticked off due to 64-bit memory errors.
- Takes care of Unix timestamps, along with IPv4 and IPv6 addresses.
- Computer searches for counterexamples to conjectures (e.g. Goldbach) haven't gone up that far.
- Value of goods + services humanity has produced is estimated in quadrillions of USD, or roughly 10^21 Iranian rials (lowest exchange rate against USD).
The only numbers I've seen above 2^128 are estimates regarding the number of atoms in Earth and whatnot, but those are usually given in scientific notation, so effectively ending with a bunch of zeroes. Just writing out 38+ _non-zero_ digits would be a near-guarantee that your number hasn't been thought of before.
@@bane2201 also, GUIDs are 128 bits (random ones are 124 bits), and is designed to be realistically unique.
...unless your default router password is a 20-30 digit number that you read off regularly when connecting a new device to your wifi.
the US debt is 14 digits
@@Bennici I set my password to be 'incorrect'. That way, if I forget it, I can just type in something random and the computer will prompt me with the right password, Your password is incorrect.
Don't we need to better define what "thought of" implies before we start with these assumptions? Does leveraging a computer to work with numbers get included?
_Don't we need to better define what "thought of" implies...?_
Yes, that question was asked near the start of the video but was never properly answered.
That's true
Mathematicians do tend to think about very big numbers but above a certain size they rarely specify numbers exactly. There are interesting exceptions, for example 16 digit numbers - lots of people will think about their credit card number.
If we take for example 19 digit numbers, 19 digit primes have a reasonable chance of having been thought of but most other irregular numbers (not powers of 2 or 10 or all digits the same) will tend to be "thought of" as 3 significant figures and a power of 10.
The question goes way beyond mathematics into human biases and what we mean by "thought of".
I can't help but adore Tony Padilla. I love his take on numberphile: "Let's get Brady thinking, damn the torpedoes, etc."
I wonder what's the *smallest* number nobody has ever thought of.
I also smoke pot.
I would say 50% of the numbers I think of are from 1-10, 25% from 10-100, 12,5% from 100-1000, etc
Maybe calculate using this?
I think numbers with short expressions like 2^31 - 1 will be far far more likely to be thought of than numbers expressed in their full length. There is usually no reason whatsoever to think of a number larger than a few billion perhaps in its full length. I would even argue that larger numbers with dozens and more digits only appear in any thought if they have a short expression.
Thanks Brady for questioning the assumptions. That really helped.
A phone number is 9 or 10 digits so Brady's first guess (which he kept pushing) is really impossible
That's not the same. Let's say your PIN is 4293. That's four two nine three. Not four thousand two hundred and ninety three.
Same with phone numbers. Even when you know and use them, you do not treat them as if they are numbers in their own right. Instead they are a string of digits.
@@silkwesir1444 That seems like a _really_ technical argument, when mathematicians literally notate big numbers by "2 and then 12 zeros" commonly...
He means "integers", doesn't he? Because I would believe that a number that has 76 decimal places is equally if not even more probable that someone has never thought of it.
or one with an imaginary component.
Using integers is a simplification, I'm sure. While 314 and 3.14 aren't the same number, they are the same string of digits. Likewise -3 and 3 are identical except for the minus sign. Factoring in all these modifications would complicate the math and the model a lot.
No. He means (natural) numbers.
@@pjbrady47 I'm genuinely interested in seeing just how many digits less are necessary when factoring in those modifications!
As soon as he was talking about the number of people that "will ever live" I knew that the Doomsday Argument was coming... I think saying that it "has its critics" is being very charitable. The argument inherently requires you to perform an experiment with a sample size too small to have confidence in its conclusion.
Problem with one of your problems: The actual odds that a number you choose will never be thought of by anyone ever is zero because as soon as you think of it, someone has thought of it.
Underrated comment.
People have been thinking about 42 at least since the Golgafrinchans arrived 2.5 million years ago.
Actually prior to that, since the Magratheans had already constructed the Earth some millions of years earlier.
Oh yes, the native humans were hardwired to think of 42 but they were outcompeted by the telephone sanitisers and management consultants.
I can easily think of the collection of all numbers nobody ever thought of. All numbers nobody ever thought of are in there. In a way, I thought of them all now.
and that set is also now the empty set.
A number that nobody's thought of yet doesn't yet exist. But as soon as it is thought of, it will exist.
I believe that by pinpointing the social media follower and like/interaction count ranges common people witness daily and placing a large portion of the recent population's numbers in that range, we can get a much higher confidence interval. Great video as always 👍
I Googled the number "1.76e67" and unfortunately I found it has search results..
Is it possible that since we use scientific notation we can reach much higher numbers easily?
A very large proportion of the 117 billion people who have ever lived did not inhabit a world where number systems even existed. That means any estimate based on the whole 117 billion people is way too big.
You are right. For most of human history and for most people now scientific notation is not a 'thing'. The vast majority of all humans have either thought of no numbers, very small numbers or numbers that represent things they encounter in normal life. Almost all of the numbers most people encounter in normal life are at most no more bigger than a trillion. The most likely way for most people to encounter any number in the trillions is probably a news story about government budgets and a country's debt. Until recent decades almost nobody would ever really have had to consider any numbers bigger than 1 billion.
His estimate is way too high
He said the total number of numbers ever thought of is 1.5x10^18
So even if all those numbers were unique, if you chose a number smaller than 1.5x10^20 it would have to be a new number with a probability of 99% or above
I think an easy way to do this to reduce the entropy of storing a 67-digit number is to pick a 6-to-12 digit number and then assign a random set of operations on it. Say 0-sqrt() 1-*pi 2-()^2 3-()^3 4-cos() 5-sin() 6-exp() 7-ln() 8-factorial/gamma 9-1/(), and you pick a random sequence of 5 digits assigning an operation and successive operations to that number. At a couple operations, there is rarely going to be a reason for anyone to do that specific sequence of operations on that specific number, so after about 5 different operations it's profoundly unlikely anyone would have thought of that specific number.
The probability that all numbers which have ever been thought of are LESS than your number is different than the probability that all numbers that have ever been thought of are DIFFERENT than your number.
and we all know that once you've had a new idea, all around the planet people have the same idea.
thinking of a number programs it into the collective consciousness and bingo, it is not longer unique
To be honest if you pick any transcendental number between 0 and 1 it will have over 99% probably that it's unique. you don't need the number itself to be large. You just need it to be difficult to specify from a dense set.
I think it is fairly obvious from the rest of the video that the number in this context is restricted to positive integers.
But that would not actually be a number but a function. I don't think that qualifies.
Gonna use this in college and look 'all powerful'
my favorite part of Padilla is how is always up to some kind of mischief
It's a common misconception that people didn't live as long way back like the bronze age... but life span hasn't changed too much... the life expectancy was so low back then because of infant deaths driving the average down.
His assumptions are wrong. He's making estimate that "what is the propability that the whole sample size is less than something", but he also knows the counter example exist, he knows there are mathematicians who have thought of larger numbers and even with this experiment we are regularly thinking about numbes around that size.
He's not saying that no one has ever thought of a number greater than N, obviously people have. What he is saying is that if you pick a random number greater than N then it is likely that no one has thought of that specific number.
No, that doesn't work. You say 1% of the thought-of numbers are above 1.7e67, but that doesn't mean that if I think of a number above that I have a 1% of it not being a "new" number", or that if I think of a number below that, that probability is 99%. The proportion of thought numbers at each side of this barrier is 99:1, but the probability is not, because the amount of numbers in those two pods is way different.
So, in addition of the gross overestimation to reach the 1.7e67 barrier, you are also putting it yet artificially higher than necessary by ignoring the size of the amount of numbers there are.
Good to see Tony back again!
I feel like the function 1/n^1.3 isn’t steep enough at far enough values. How many people have thought of 20 digit numbers for example?
If you count digital music as a numbers, then people have thought of some pretty big numbers !
What's the biggest number someone has counted up to? That would eliminate a lot of lower digit counts.
I'm going to guess the very low millions. I conjecture that some people somewhere have actually counted to one million, but that very few of them, if any, have gotten significantly further, given how much effort is required to count to one million in the first place.
I really appreciate that leap years were considered! I think they make a difference!!
2:37 I love how there's a very obvious and jarring cut in the texture of the ocean on that sphere :D
I already did this thought experiments several months ago, and 67 is way too high. The conclusion I came to is that you have an extremely reasonable chance of making a number that nobody on earth has ever come up with before with 28 digits.
Instead of something arbitrary like "thought", let's go with something more concrete like "written". So, if you are writing a natural number with length (n) using random digits, how long does the number have to be before you can be reasonably certain this has never turned up in written form anywhere in the history of the world?
First, assume all numbers are natural numbers. This only reduces the space in which a number can be depicted, so that means if you expanded the space to include all real numbers the final count would be less, but the video assumes only natural numbers so we'll go with that. Decimals and other modifiers to natural numbers are ignored.
Assume every number generated by humans is itself random and never repeated. Now, this is completely at odds with the video, but I'm actually deliberately making this adversarial; we're actually assuming that all human endeavor, ever, is being used to make a guess at (n). In reality, most numbers generated will be low, but why don't we just concatenate all of those into guesses?
The probability (p) of guessing (n) is = 1/10^n
The number of attempts (N) until the correct guess is expected to turn up is 1/p
Thus, N = 10^n total attempts must be made before a *specific* number of length n is expected to turn up. So, given these assumptions, how big can we expect (N) to be?
Let's use Matt Parker's 10-billion human-second century (10BHSC). This assumes that 10 billion humans are taking 1 action every second for a whole century, creating an upper bound of the total number of guesses that could possibly happen in the scope of our lifetimes. So, let's assume that every human writes 1 guess at (n) every second. (we could make them write one digit every second, but this won't be significant until later). So the total number of written guesses from a 10BHSC is 3x10^19.
Since this is our N, you only need to make a number with a digit total higher than the total number of guesses made, with each additional digit reducing the chances for an early guess quite rapidly. So (n) only needs to be 20 digits long to have been unlikely to turn up in the 10BHSC.
Now since we're going through all of history, it may be more appropriate to use a 100 Billion Human-Second Century (which assumes 100 billion humans have each lived 1 century making guesses at (n) every second). The result for this is just one digit higher, 3x10^20.
Now let's assume all guesses from the 100BHSC instead create just ONE number, and (n) cannot appear anywhere inside this number. So let's make out our 3x10^20 guesses into a single number that is 300 quintillion digits long.
A number that is x digits long contains roughly the same number of strings of a portion of its length. So a number that is 10^20 digits long contains almost 10^20 20-digit numbers--which just about covers the total number of strings with 10^20 digits. So the probability of finding a specific 21-digit string in this massive number (of which about only one-tenth appear in our massive string) is 10%, a 22-digit string is 1%, and so on. To make the chances of this number containing (n) to be 1 in 1 billion, (n) only needs to be 28 digits long.
Now I'm not a mathematician so something I did here might be off, but I'm pretty sure this shows the number is significantly lower than the video claims. If I'm off-by-one somewhere, or you think that more digits should be appended to the end of our final number, remember that this only makes a significant difference by orders of magnitude. You need to add 10x some factor in order to increase this number by ONE more digit.
Exactly. The video is rubbish. (Well technically they claimed 10^68 was sufficient. It is. So is 10^300.)
This is in some ways correct, but it ignores that some very big number ranges are used a lot more often by people than others. In some cases, smaller numbers might be rarer for humans to ever see than big ones.
For example, 67 digit numbers are probably seen less often than 68 digit numbers, because 68 digit numbers represent different playing card shuffles spread open on the table and 10, 11, 12, 13 digit numbers represent telephone numbers which are probably well "thought about at least by one person" across the entire range.
@@blumoogle2901 I still feel like this underestimates the number of numbers. Like, when talking about a sixty-eight digit number, Even if, culturally, 68-digit numbers come up more often than 67-digit numbers, the number of 68-digit numbers is still 10^68. A single person going through all 68-digit numbers, one per second, will still take over a hundred octodecillion years. So even if it's culturally significant, the number of card shuffle spreads that have occurred in the history of humanity is statistically insignificant in making a dent in all possible 68-digit numbers.
@@rickpgriffin perhaps that's true, but I feel like just thinking "what could the next card in the draw pile be?/how is this pack shuffled?" constitutes "thinking about" several 10^20s worth of possibilities for numbers all at once.
A lot of correctness got lost in the simplification. It's not correct that there's a 99% chance of picking a number never used before picking one above 10^68. The intuition about 66 digit numbers is closer to correct. There's 9*10^65 of those, and very few of those have ever been thought of, so there's way higher than 99% chance a random number of those has been thought of.
IPv6 addresses are 128 bits in length. An example from the IPv6 Wikipedia article looks like this:
IPv6 addresses are represented as eight groups of four hexadecimal digits each, separated by colons. The full representation may be shortened; for example, 2001:0db8:0000:0000:0000:8a2e:0370:7334 becomes 2001:db8::8a2e:370:7334.
That is 39 digits if you do the decimal conversion. IPv6 data packets consist of source address and destination address plus a host other fields. There are people supporting the Internet that work with these numbers daily.
If you think of a number using the formula: x*10^y then the probability of someone having though of it before, is much greater then if you choose random digits,
but if you think of a number with 67 digits, can you confidently say that you are thinking of the whole number at the same time?
Awhile ago my friend and I made a sequence that went on the OEIS, for fun I calculated the 100,000th term. It has 100,000 digits and I thought that I was probably the first person to ever see that number. Turns out I didn't need to go that far into thr sequence to find my own number.
Either I'm missing something, or his working out answers a very different question than the one he claims. Calculating the probability that there is a 99% chance of all the random numbers selected are under a threshold does not mean that you have to guess over that threshold to have a 99% chance to find a number no one has thought of. In fact his own working out works against him. His threshold is 49 orders of magnitude greater than his estimate for how many numbers have been thought of. Even if every number thought of was unique (and we know that it wasn't), picking a truly random number under that threshold must have at most a 0.00......1 chance of picking a duplicate.
I mean, I'm not really convinced by most his assumptions for what he does calculate, but even if I grant all of them, his threshold is ridiculously high.
I usually love the content here but this was just pointless.
You are correct that he was working out the answer to a very different question than the one he claims. It doesn't have to do with uniqueness, though. His final working out does not assume that everyone's always thinking of unique numbers (you can see this in that he works out a non-zero answer to the probability for "all thoughts of numbers are of numbers < 10" question). The question he's working out the answer to is: how big a number do I have to pick so that there's a 99% chance no one has ever thought of my number *or anything bigger*. That "or anything bigger" makes a huge difference.
@@intrepidca80 "anything bigger" doesn't work either because of course people have thought of numbers higher than this.
52! for example is 8*10^67
10^100 even has a name.
this episode was just bad
@@dmtc6913 Right, but that's not because of anything to do with the math. That's because of his assumption that we're disregarding special-case numbers. He stated that assumption (which entails the caveat you described) up-front, though, so I don't have a problem with that.
@@intrepidca80 I know he didn't assume uniqueness, and I wasn't claiming that uniqueness was part of the problem. I brought it up, because assuming uniqueness gives you the largest probability of accidental duplication, and that probability is abysmally small even with that assumption. It was to underline how far off beam the calculation is.
this reminds me of when I was at school and debates were settled by who could think of the biggest number ("bagsy the back seat times a million") which would undoubtedly lead to "bagsy the back seat times one more than you can ever say".
"Nobody in the history of humanity will think of this number", prints the number on the screen with millions of viewers.
An average human thinks about a number every 2 minutes...
Sure, go ahead.
Can I say - that's a lovely home made Fathers' day card :-) Top job Jessica!
Numberphile out here answering my childhood questions
52! = 8 * 10^67. And it is usually assumed that the chance of a random shuffle of cards repeating is negligible (much less than 1%)