IMPORTANT At 1:02 I said that, in the first 1000 digits of pi, there is a 100% chance that we would see the same digit 3 in a row. That is false. Assuming the sequence is random, there is always a chance that we woudn't see the same digit 3 times in a row. The actual probability is not that easy to calculate. It's approximately 99.99%. Calculating the probability of getting 6 digits in a row also isn't straightforward. I said that that it's 0.1%. It's approximately equal to 0.93%. Thanks for all the comments pointing this out and sorry for the mistake, hope you enjoyed the rest of the video.
Take square-root of 1111....11(n times) in a high precision calculator. Increase n from 1 to infinity and look at the decimal expansion of the square-root.
Correction about pi: the chance of getting 6 of a SPECIFIC digit in a row in the first 1000 is 0.1%, but the chance of getting 6 of ANY digit in a row is 1% as it can be any of the digits 0 to 9. This is a super common mistake.
correction: the chance of getting 6 of the same digit within the first 1000 digits of pi is 100%. The digits of pi are not random, it's a constant, that 999999 is always guaranteed to be there.
Honestly, not that crazy. Ramanujan had an amazing intuition for numbers. He might have noticed his birthday had this property of summing to a prime when divided into two-digit numbers and decided to try if he could expand it into a bigger configuration.
@@tuures.5167 actually, indeed, it's that crazy. Think about the probabilities that a math genius had born exaclty this square describes this birth day
@@SBImNotWritingMyNameHere A bit of both. It started as being used to describe features of how things seem to work. If you have one apple, and another apple, then putting them together gives two apples. There are a lot of properties of math that are actually physical like that, which are then described using rules. But then those rules can also be used for other things, taking us into the realm of 'pure mathematics' which seems disconnected from the natural. But it is all still based in those rules that describe how natural things work. The thing is that occasionally the 'pure mathematics' is later discovered to actually apply to something real, after the math was developed. As an example imaginary numbers were found to be useful in mathematics hundreds of years before they showed up in electrical engineering and quantum mechanics. So it seems in some way that the natural world really does have math at its heart, and we are really just discovering it more than inventing it.
I like how most of these are actually coincidences, it's just so many chances for something "exceptional" to happen it's almost inevitable something will.
@@brightblackhole2442 Let's categorize all the numbers into 2 groups, interesting and uninteresting. Interesting numbers have a unique property about them, for example 2 is interesting because it is the only even prime number. Out of all these numbers, there are an infinite amount of uninteresting numbers. One of these is the smallest uninteresting number, which imo is pretty interesting, so it's no longer uninteresting. But wait! holy smokes its a pArAdOx!! (taken from jan misali's paradox video)
9 | 99 9 + 9 = 18 ≠ 9 The real property is that all multiples of 9 have digits which add up to another multiple of 9, but not necessarily 9 itself. a LOT of these are "literally not a coincidence", yes, 360 included (in fact, the whole point of still using 1/360th of a turn as a degree is bc 360 is a highly composite number, so it divides neatly by a bunch of factors. No surprises there). Still, sum of digits of ANY multiple of 9 isn't always 9 so this property isn't especially more or less coincidental than other entries in the video imo
BEAUTIFUL I love statistics and how in math there isn't really a "coincidence" the unexpected is expected, every number will theoretically have infinite "special" values and coincidences which will fascinate us, it is expected.
For some of them it's true, but all of the patterns of numbers repeating in irrational numbers are coincidences, because they exist only in a base 10 counting system, which is human made. Maths works regardless of how many digits we use to form our numbers, we could write pi only with 0s and 1s if we wanted to, and for any number of digits we use for a counting system, there will be different patterns, so yes. Those are actually all coincidences.
@@theterron7857 While it's not entirely wrong to call them coincidences due to how obvious the patterns are in base 10, looking at the representations in other bases for long enough is bound to lead to the discovery of interesting patterns, simply due to the sheer number of possible patterns one could find. Since the fact that patterns can be found is essentially guaranteed, what the patterns are is irrelevant and calling them coincidences feels a bit disingenuous.
i feel like you dont understand probabilty, you wouldnt have a 100% probability of getting three digits in a row even if you were considering the first quadrillion digits.
What he means is that it is not rare that there is three digits, because the probabilities of it happening were already met, is like being suprised of winning a 1% prize at your 100 attempt, it still is just 1%, but it had to appear at some point, because you already met the 100% probability, so if it didn't pop off, then it would start being bad luck
@@matitello4167 nah. I don't think there is such a thing as meeting percent change at some point, from which point things become more likely or surprising. A 1% event need not happen within the first 100 trials. It need not come every hundred trials. It does not even have to come within the first 1000 trials, or every 1000 trials. The idea that it must, is the gamblers fallacy: the idea that certain outcomes become 'statistically due' to happen if they haven't come in a while, as if the amount of trials, and their outcomes, have some kind of influence on the next one in order to force statistics to balance out. Trials are only independent if such influence does not exist. So while you expect a 1% event every 100 times, there might not be one for 100000 trials and then, suddenly, there could be 1010 in close succession, and the stats would still work.
This video's thumbnail and title are almost identical to the ones of the kuvina saydaki's vid. Is this just an another weird coincidence or it has some explanation?
I made a video on this in January. My video actually explains what is and isn't a coincidence (a lot of these are not). Also, intentional or not, you totally ripped off my thumbnail. Edit: thank you for changing the thumbnail to something more original!
It probably is just because they were doing random stuff. Mathematicians do enjoy maths (surprising, I know!), and we do enjoy to just doodle with numbers and ideas. Some might have been discovered by computers programmed to find stuff like that, but there has been a mind behind it, that probably accidently came across something and wanted to check if it happened again any other time.
@@Faroshkasas a math student (i like to study math a lot but i can’t really consider myself as a mathematician) i thought there was some more complex process behind it. i guess i overlooked it. 😅 thanks for the answer anyway!
@@sevenpenceLOLZ I guess there could be. But, in my experience, when it is something that has no real use, it's just people having fun lol. But maybe there was some deeper reasoning. Ramanujan's square, for example, definitely needed a lot of thought, but I doubt he was trying to solve a real world problem
4:00 if a number is divisible by 9 the sum of its digits is also divisible by 9. When you divide by 2 over and over again you dont change the fact that the number ks dkvisible by 9. The fact that it is 9 instead of something like 18 is coinsidence, but there were few possibilities to begin with
3:55 this works for every number that is initially divisible by 9. im pretty sure everyone knows that you can figure out a number is divisble by 9 if its digits' sum is divisible by 9
I want to call 360 as "anti-prime". It's divisible by: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 45, 60, 90, 120, 180. By adding them up you get 638, which is bigger, than 360(not including the 1 and 360 itself as divisors).
Also did you knew, that 2^n is equal to all the previous 2^n + 2(not including 2^0)? For example, 2^10=2^9+2^8+2^7+2^6+2^5+2^4+2^3+2^2+2^1+2. You can check it
@@ІсаєнкоАртем0 has infinite factors adding up to infinity making it the better anti prime, infact 0 isn't a composite number because it has infinite factors so let's just call it that
0:50 zero might appear unooften at the start, but maybe millions of magnitudes of digits into pi there is a ton of zeros, actualy, it has to happen at some point as pi is irrational and goes on forrever
It is. That was my first thought too. I think he means that, for every 100 digits or whatever, each number will appear ten times. It’s a dumb, non-real assumption, but a lot of these things are ridiculous.
@@RaiRajeswori someone claiming to be a genius and making math videos would know very little things are 100% certain. It's a massive mistake and should be called out as such
@@tmplOS First I want to address that as far as I saw his videos, only his username is digitalgenius. Secondly, I agree with you on the fact that this big misconception should be discussed on a bigger level than comments
4:15 The result is the original number mod 9 (assuming it's natural and a version of mod where 9 mod 9 is 9, but the usual numeral system is used). So, you can just 1*2 = 2 2*2 = 4 4*2 = 8 8*2 = 16 = 7 7*2 = 14 = 5 5*2 = 10 = 1 (all mod 9)
congrat you found what was behind this "coincidence". Now you can do that for everything he said in his video (except for the approximation, these are just scams)
@@midahe5548I remember making a separate comment about another one For the first one, I had some thoughts then, but I finally figured it out now. The second digit is the arithmetic mean of the other two. So, it's 111111(the second digit) ± (100001 - 1100)(the difference). Both are divisible by 37 (111111 = 91*1221 = 3003*37, 98901 = 81*1221 = 2673*37. In fact, all these numbers are divisible by 1221
I figured out something amazing about squares. Here's the sequence: 1 4 9 16 25 26 49 etc. At 25^2 (625) is what I have called 'the splitting point'. Here's some more of the sequence: 484 529 576 625
5:01 this makes sense since 10! is 8! * 90, from 8!, multiplying by 60 will convert to seconds, and multiplying by 1.5 will convert 4 weeks into 6. 60 * 1.5 = 90
5:10 look what I found for 4 digit numbers: 1420^3+5170^3+1000^3 = 142,051,701,000 2 digits have several solutions as well, like: 16^3+50^3+33^3 = 165033 22^3+18^3+59^3 = 221859 34^3+10^3+67^3 = 341067 44^3+46^3+64^3 = 444664 48^3+72^3+15^3 = 487215 98^3+28^3+27^3 = 982827 98^3+32^3+21^3 = 983221 After that I checked for two 3-digit numbers and 2nd powers, and found only this: 990^2+100^2 = 990100 But I guess these results are not that beautiful because of how we group digits in triples. I'll look for other powers then.
@@studyonly7888 yeah, I'm fine. At the moment I'm searching for 12-digit numbers. The closest I got was 531^4+174^4+170^4+819^4=531,174,170,818. One off =(
actually this is only true if pi is a normal number (roughly meaning all strings of digits are equally likely to be found in the decimal expansion). even though we know almost all numbers are normal, we still don't know if pi is or not.
37*3 = 111. that's why all "repeating digit" numbers are in some way related to 37. for exemple 111, 222, 333, 444, 555,..., 121212, 131313, 141414, ... 134513451345, ... are divisible by 37. I made the proof of why anumber in a form abccba is divisible by 37, with c = b + i and b = a + i with i being the offset (for exemple 123321 have an offset of 1, whereas 135531 have an offset of 2). these numbers divided by 37 are equal to a*3003 + i*330 with a being the lowest digit
These numbers are of the form abccba = 100001a + 10010b + 1100c. In 123321, a=1, b=2 and c=3. 100001/37 gives remainder 27 10010/37 gives remainder 20 1100/37 gives remainder 27 abccba/37 gives remainder 27a + 20b + 27c = 27(a+c) + 20b When b is the median of a and c, this is = 27(a+c) + 20(a+c)/2 = 27(a+c) + 10(a+c) = 37(a+c) divisible by 37 But b on the keyboard is always in the middle of a and c, and is also always their median, so it always holds.
Just to check the 6 weeks = 10! actually makes perfect sense. A week is 7 days and there’s 6 of them, so that handles the 6 and 7 in 10!. A day has 24 hours, which is 8*3, so that takes care of those factors. An hour has 60 minutes, which is 2*3*10, taking care of the 2 and 10. Since 9 is 3*3, we can split it into 2 factors of 3, and have this take care of one of them. A minute has 60 seconds, which is 3*4*5, taking care of the 4, the 5, and the other 3 leftover from the 9. And of course 1 times anything is itself. You could say it’s somewhat coincidental, but inevitably we’d math time with numbers divisible by 2s, 3s, and 10s, and that handles most of the factors of 10!, then getting lucky with 7 day weeks gets us the hardest to get factor, leaving just one last factor of 6 to add in. Going from 6 weeks to 4 weeks for 8! minutes also makes sense. You’re swapping which factors apply to seconds and minutes in the above scenario, and by removing seconds removing a factor 10, and one of the factors of 3 from the 9. You’d be losing a factor of 2 as well, but by changing it to 4 weeks from 6 you effectively gain it back for losing the other factor of 3 that makes up the 9, getting 8!.
1:28 I don't really like using probability for the decimals of known numbers. Like no, the probability of getting the same digit 6 times in a row in the first 1000 digits of pi is 100%, not 0.1%. No matter how many times you bring up the digits of pi in base 10, it will always have those 6 9's in there in the exact same spot. You can say this is assuming the digits are random, but that isn't really fair, is it? The digits of pi aren't random, they're pretty much set in stone with formulas and infinite series. this was all very cool tho
@@TriglycerideBeware The idea is that it works off the assumption that the digits of pi really are random. If they aren't then it implies there has to be some reason as to why these digits are appearing in these kinds of interesting orders.
@@TriglycerideBeware Yes but as I said, if it is not random then it implies there is probably a reason for the strange appearance of numbers that we haven't found yet
@@staticchimera44 I'm afraid I don't understand the point you're making. Could you say it a different way? Pi obviously isn't random--it's the same every time. The probabilities he gave were assuming that the first 1000 digits were selected randomly from a uniform discrete distribution of [0,9], and I think his script was pretty explicit about making that assumption. All I was saying was it doesn't make sense to assume the digits were generated randomly, since they aren't. I feel like we're mostly on the same page, but it sounds like you're trying to make an additional point. I would like to understand it, if you're okay with explaining it a different way
@@xian3themax311 imo 99.9% is effectively the same as 100% in statistics, but in most other parts of maths they are very different. I’m not sure what branch this is (number theory?), but it’s not statistics
@@Lege19 No, it's very different also in statistics. If an event has a probability of 99.99% it is very likely to happen but maybe it doesn't happen. WIth 100%, it is guaranteed that the event happens, which is very different
@@xian3themax311 The probability of rolling a six at least once if you roll a dice six times is around 66.5% Using probability, the calculation for this is 1-(5/6)^6, meaning the probability for everything except for not rolling a six for six rolls or something idk probability
The number 10^7.5 (or sqrt(10^15)) is almost exactly equal to the number of seconds in a leap-year; with the difference being just 6 minutes and 16 seconds (or an error of about 1 second per day).
congrat. you made me laugh with your "almost exactly equal". NB: in mathematics, "almost exactly equal" is "not equal". So your sentence is correct that way: The number 10^7.5 (or sqrt(10^15)) is not equal to the number of seconds in a leap-year. Interesting right ?
1:35 You said that the probability that six digits in a row are equal in the first thousand digits of pi is .1%, but I beg to differ. As you have demonstrated in this first few minutes, the probability of that happening is 100%, because it actually happens. I think what you intend to say is that if we consider a number whose digits are generated randomly, then the probability of getting six equal values in a row is approximately 0.1%. While don’t think that the notion of random is coherent, I will concede that it may make sense in probability calculations that the event of having six equal digits in a row in the first 1000 digits of a number, under the equally likely assumption, maybe as you claimed .1%; this is certainly very different from the claim that a number whose expansion we know through the first 1000 digits has a .1% probability of a certain string of digits in that first 1000 digits.
Im confused am i missing something? The title is "its just a coincidence" in quotes, which seems to be saying "it isnt a coincidence" and then preceeded to list a bunch of things that seem coincidental without explaining why they arent Why is 6 9s not coincidental? Or is it just not coincidental because "pi is infinitely long therefore every combination of numbers will appear" In which case thats super dumb Or are the quotes around "its just a coincidence" useless and this video is actually listing coincidences In which case this is also super dumb The description seems to support my original view so.......... why is he not explaining why they arent coincidences
the next digits of e are 45 90 and 45, the degrees in an isosceles right triangle, then 235, the first three primes, and 360, the amount of degrees in a circle
The 360° coincidence extends way beyond 360 and under 11.25, it eventually increases by integer multiples of 9, 2880 (360*8) sums to 18, and 5.625 (360/64) sums to 18 as well. At 360/1024 or 0.3515625 it sums to 27, divide by 2 again and it sums to 36.
The first fact can be generalized to (and explained by) the following statement: any number of the form abccba (note that the letters are digits, they are not being multiplied) is a multiple of 111 if a,b,c form an arithmetic sequence. Note that each line in the calculator is an arithmetic sequence and 37 divides 111, which is why the statement implies the fact. The statement is true because 111 divides 999 = 1000 - 1, so 111 divides x000 - x. With this, one can see that 111 divides abc000 - abc. Since a,b,c is an arithmetic sequence, we know a + c = 2b, so abc + cba = 100 * a + 10 * b + c + 100 * c + 10 * b + a = 100 * (a + c) + 10 * (2b) + (a + c) = 111 * 2b, which is clearly a multiple of 111. Therefore, 111 divides abc000 - abc and abc + cba, so it divides the sum, which is abccba.
That 100% from 1:05 is wrong. There is no way there is a 100 percent chance, as that is always. You could make a number that doesn't follow this simpily: 1234567890 repeated 100 times.
With continuous probability distributions, the probability of any individual event happening is infinitely small, so we say 0%, but still events happen anyway. So sometimes our intuition about what it means when something has 0% or 100% probability needs to be loosened, to not merely mean impossible/certain. ...that being said, selecting random digits is a discrete process... so I have no idea where the 100% came from either. Unless he's trying to say that pi *isn't* a random sequence, and it's always the same? But then so many of his other points are completely invalidated. Either way, there are quality issues.
@TriglycerideBeware it's not just continuous distributions, infintine number of things can sometimes be like that - we expect pi and some other trancendental numbers to be "normal", which means we think we should be able to find any finite string of digits somewhere in them with 100% probability i think there's a mistake in the video because he says "within the first 1000 digits" which is just not true...
I would argue that’s not coincidental. Mathematics was probed and researched for thousands of years before the Bible was written. The significance of certain numbers is far older than the Bible.
4:00 that is no coinceidence, as all those numbers are multiples of 9, so their sum is 9, 360=9*40, so we can divide a few times before we get to decimals
for 5:36 I actually made a program that finds numbers just like that in Lua, and there’s a few more than the ones you showed. Interestingly, both 333,667,000 and 333,667,001 have this property, along with 334,000,667.
For those, who want some statistic, probability chances, fun facts and explanations: 0:52 A little error: Statistically, theres should be 10 triple numbers on average in 1000 random digits, and the mistake was, that you counted up only 1 possible outcome, when theres 10: (000),(111),(222),(333)...(999). And the fact, that there are less than 10, is just a statistic. Also, there's NEVER a 100% on anything random with digits. Even infinite amount of random digits could consist of every number except of 1 specific, and the chances are 1×10 / Infinity. Which is not a 0, but still, very-very unlikely to ever happen. 1:28 By the statistic, we have 10 different outcomes, so we multiply the probability chance by 10 assuming, that probability of the next number to be the same - is 1/10. We get probability of "1/10,000" So, on average we get: 1000 digits of pi / 10,000 and we get a 1/10 chance of getting 6 equal digits in a row of 1000 random numbers. Not a 0.1% as mentioned in the video ;) 3:06 If you assume thay everything is random (e^pi - pi ~ 20; 2143/22 ~ pi⁴; pi⁴ + pi⁵ = e⁶; pi = √2 + √3; sin(60°) ~ e/pi; etc.) than it may look that chances of those coincidences are very slim, but, remember: 1) Math is a science, and constant at every point of space and time; 2) The ammount of different combinations with pi, e, sin, are almost endless; 3) Aldo, never forget, that those specific numbers are known, to be infinitely precise constants of universe, and have more in general, than other numbers based on what they represent. 4:00 There wont be any numbers, but instead, a fun fact: Amount of degreece can be ANY number that we want, but people have choosen 360° as a standart of circle, cuz this number can be divided by a LOT of numbers: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, (almost 16 "22.5"), 18, 20, 24, (almost 25 "14.4"), (almost 27 "13⅓"), 4:48 10! = 6 weeks; 4 weeks = 8! Heres an easier representation: 6 week (in seconds) = 6w × 7d × 24h × 60m × 60s 1h = 3600s 10! = 1 × (2×3) × (7) × (6×4) × (5×8×9×10) (5×8×9×10) = 40×9×10 = 360(circle😊) × 10 = 3600 3600 × (1×2×3×4) = 3600×24 = 79200 79200 × (6×7) 3628800 4 weeks (in minutes) = 4w × 7d × 24h × 60m 1d = 24h × 60m = 1440m 8! = 1 × 4 × 7 × (2×3×5×6×8) = 28 × (48 × 30) = 28 × 1440 = 40320 minutes
@@midahe5548 Bro, i just have no life. When i woke up i immediately checked telegram, and saw 1 guy, that typed me, and as a result i bursted out laughing about series we watch, and made a fkn 7 THOUSAND symbols long story, which had almost the same plot as a series, and worked out with HIS life in the Internet.. on a mobile (those 2 comments are written fully on mobile too)
within 1000 digits you have a 100% chance of getting six "9's" in a row because pi is an irrational number not a randomly generated number the odds of an irrational number containing six of the same digits in a row is infact 0.1% of irrational numbers
The two power thing is probably because of the modulo 9 rule. Any number has the same modulo 9 (remainder when divided by 9) as the sum of its digits. Since 2^6 = 64 which is one more than a multiple of 9, the modulo 9 keeps on repeating. It will never be divisible by 9, so the sum will never be 0 or 9, leaving 8 distinct options for each remainder, and creating a cycle. Cool video!
3:58 isn't this true for all numbers divisible by 9?(or 3). In base 10 divisibility by 9 (or 3) is testable if the sum of digits are divisible by 9 (or 3), so if a number is divisible by 9, it has prime factors of 3*3, and if you divide by any number other than those that have a factor of 3^n, and the result is a whole number, the result will still have the prime factors of 3*3
It’s not a coincidence, it’s just fascinating. Math is a series of random numbers created by us humans that start out so simply but increase in complication the further you look into it. The randomness and repeated unexpectedness is truly amazing honestly and it’s crazy how many other coincidences there are out there that we still don’t know of. How did we ever even start out with numbers?
7:40 that's cool. Ohhhhh that's even good OHHHHH MY GOOOOD HOW ARE ALL SQUARES ADD UP TO SAME PRIME *NOOOOOOOOOOOOO EVEN THE DATE OF BIRTH WHAT THE F-----*
Btw 3 raised to the power of n, such that n > 1 results in: 3 ^ 2 = 9 3 ^ 3 = 27 --> 2 + 7 = 9 3 ^ 3 ^ 3 = 81 --> 8 + 1 = 9 ... 3 ^ 3... no matter what, the sum of the digits, by repeating until we come to a single digit(18 would be 1 + 8), they will all be 9. for 4 raised to the power of n, the repeating sequence goes like 4, 7, 10, 4, 7, 10. for 5, it is undetermined. For 6, the same pattern appears just like 3. For 7, the sequence is 7,3,1,7,4,1 for 8, it is 8, 1, 8, 1. For 9, it is always 9. For 10, it is always 1. But for 11, where 11 is raised to the power of n(and add all the digits): n = 1 --> 2 n = 2 --> 4 n = 3 --> 8 n = 4 --> 16 n = 5 --> unfortunately, not 32. Cool, right!
The number of seconds in a minute minus 1 is a prime number. The same is true for the number of minutes in an hour - 1, hours in a day - 1, seconds in a day - 1, and the number of minutes in a day - 1.
The *dalton* (1⁄12 of the mass of a *¹²C* atom) is still "around" (that is determined experimentally and is known only with finite accuracy), but the Avogadro number from now on is fixed and is equal to an integer with 9 higher significant digits, the rest of them (lower 15 digits) being 0.
@@pumpkin_pants3828Agree, that way it makes perfect sense. Though drawing the audience's attention to the fact that now it is an *exact* number would have served a much better purpose.
He said it was "around 6.02·10²³" because he omitted the last 6 decimal places. What makes this property of Avogadro's number such a big coincidence is how arbitrary its definition originally was. Avogadro's number was originally defined as the number of hydrogen atoms in one gram of hydrogen. A gram was originally defined as the mass of one cubic centimeter of water. And a centimeter was originally defined (during the French Revolution) as 10⁻⁹ times the distance from the North Pole to the Equator along the meridian passing through Paris.
I really like the Ramanujan square - i mean, not just because of the identical summing, and the hidden link to his BD, one easy approach for me is, for numbers 1-25 these are some of my fav piano concerto pieces of Mozart (to name a few, I listened frequently to No.9, 23, 24, and 25), and the years 86 - 89, is the periods 1786-1789 where he wrote most of his famous master pieces. for the sum 139, well I loved sym No.39 (in addition to No.41)
speaking of the first 37 fact, all the quotients you get are divisible by 11 (as well as the dividends!) which makes all the dividends divisible by 407!
4:17 is not a coincidence at all, it's a simple consequence of modular arithmetic and works with any modulo (not just mod 9 == sum of digits of a number in base 10) and any base number other than 2 similarly, at 3:53, we start with a number whose digital sum (= the number mod 9) is 9 (which is the same as 0 modulo 9), so dividing or multiplying that number by anything would keep the digital sum 9. if you get to fractional numbers, taking their digital sum is equivalent to multiplying them by a power of 10 then taking it mod 9, which would also keep the digital sum 9
the 3x3 square is not a coincidence because 111 is divisible by 37 and 111 is the common divisor of each number. for base 5 in a 2x2 square, its common divisor is 11. for base 17 in the 4x4 square, 1111. Note that these are all perfect squares and the base system is the product of the length by the width added by 1. 11 base 5, 111 base 10, 1111 base 17. This pattern always holds true and it must hold true (arrange every number formed by least to greatest and take the derivative to make this fact more obvious). 6, 111, 5220 in base 10
The thing about numbers summing to 9 is less improbable considering that when you sum together the digits of a multiple of 9, you get a multiple of 9, and since we’re dividing by 2 all subsequent numbers will be divisible by nine.
0:00 Yeah, they're all divisible by 37 because they are all divisible by 111, by construction, and 111=3×37. Likewise, a 4×4 base 17 square would have have all numbers divisible by 1111 in base 17, which is 5220 decimal; then you could say that they're all magically divisible by 29, because 5220=2²×3²×5×29. 3:56 As others have mentioned, that happens because 360 is divisible by 9, and dividing it by 2 doesn't remove this factor. 4:13 "Digital root of x", a.k.a. mod(x,9). If you multiply any of the remainders by 2 modulo 9, you get the next remainder. Since there are finitely many different remainders, it is expected that you'll eventually reach a cycle. 6:55 Proof by induction. Done. 7:18 See wikipedia / wiki / Magic_square#Extra_constraints (RU-vid doesn't like links) and make your own Ramanujan magic square with your own birthday. (the rest is left as an exercise to the reader)
Apart from the already acknowledged "100% mistake", the digits of pi having some parts in it that are theoretically unlikely is not actually unlikely in itself. You have to keep in mind that the question is not "How likely is it that there are 6 9s in a row", the question is "How likely is something to happen that could be considered unlikely in retrospect" or in simpler terms, the question isn't "How likley is X thing to happen", the question is "How likely is something unlikely to happen" and SOME unlikely thing happening is generally actually very likely. That is also the reason why so many theoretically unlikely coincidences happen in day to day life. After all, we only notice the few coincidences that DO happen, bot the billion that COULD but DON'T. Statistical analysis of the likelihood of an event can only be measured if you FIRST define what specifically you look for, and AFTERWARDS actually look for that specific thing, not the other way around. And I looked it up. The digits of pi have been statistically analysed and the actually do appear to be completely normally distributed.
