Why 381,654,729 is awesome - Numberphile

Discussing pandigital numbers with Dr James Grime (and a special guest).
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[@TheEnergeticPanda]

James came to my school. He asked a question and i put my hand up. He asked me, i had to say "first of all i love numberphile and i'm jealous that you get to meet Brady Haren" and a big smile went on his face

[@iconredesign]

He's just always so happy!

[@kezzyhko]

I thought he could came to my school too but then I remember that I from Russia D:

[@thefremddingeguy6058]

+Сергей Семушин Same here but I live in America. There's no way he's going to America, especially since (from experience) most people aren't really interested in maths anyways. Everyone would be sleeping if he came to my school and did a speech, except for me....

[@a1b3a3c14nbcv]

and then the smile clapped

[@soupisfornoobs4081]

@@a1b3a3c14nbcv this time it's actually pretty likely. He's a professor that goes from school to school all the time, or at least did when the world wasn't bricks, so it's not impossible he'd meet numberphile viewers. I'd call it inevitable

[@sussurus]

Lara: "I'm doing good" ... Brady: "H-how are you doing?"

[@steinscamus8037]

+Sussurus XD

[@steeevealbright]

Sussurus *polite laugh*

[@notstarboard]

ONE OF US

[@ramind10001]

Sussurus Lars sees the future :O

[@NathanY0ung]

Awkwaaaaaaaaaaaaaaaaard

[@hiitakasai1264]

Try each row, column, and box in a finished Sudoku, You'll find pandigital numbers there too.

[@velocity173]

Hiita Kasai well duh

[@SomebodyLikeXeo]

Mind Blown!

[@harrisluo5818]

lol it doesnt work when i do my sudokus

[@musicappreciate]

Knowing the digits will add up to 45 helps me greatly and thinking through a sudoku

[@ApplicationBot]

420 is special because if you add the digits from left to right you get the same as adding them up from right to left.

[@Victor4X]

+ApplicationBot but.. that's true for any number I know that this is the joke, but a lot of people don't know

[@ApplicationBot]

+Victor4X lol

[@katnos4609]

That is the same for any number

[@jakeypearce]

339. 3+3+9=15 9+3+3=15 How is that number special?

[@SilentMath161]

+jakeypearce its a joke, newbie

[@keltzar1]

You fools! You think you've hidden Lara's number but you've really only reduced it to down to 9 factorial possibilities. It's only a matter of time now before the numberphile fans crack the code!

[@spitfire83]

you have to guess the area code as well

[@potato-hj9nm]

keltzar1 atually its 9*9! but won't take long to try them all.

[@Stevobulfer]

Potato 123 no, he said it didn't include a zero which means it's only a 9!

[@Nulono]

Times area codes.

***** I think that's the joke.

[@5hak3itup]

I really like james. He seems like a cool guy to hang out with

[@SirCutRy]

James' favorite pandigital number also has the following property: if you add the triplets, you get an arithmetic sequence of length 3 with a0 = 12 and d = 3: 381 654 729 12 15 18

[@MuzikBike]

All six of the numbers are also divisible by 3.

[@loocheenah]

And the largest one-digit divisors of those triplets are, respectively: 3, 6, 9

[@stefstefstef6789]

If you were to enter 381,654,729 on a phone... You enter the digits symmetrically to the center too.

[@nicolasfpauly]

woooooow

[@rx1589]

For those who don't know why there is no pandigital prime without redundant digits, here's why. No matter how you replace the digits, the sum of 1+2+3+4+5...+8+9 will always be 45, and it means this number is divisible by 3 and 9. Thus, it's not a prime

[@musicappreciate]

😊

[@TomHasVideo]

polite laugh

[@Wumbolo]

isn't it the most polite laugh you've ever seen on Numberphile?

[@MuzikBike]

Wouldn't want to laugh disgustingly ruldely like everyone else does would you?

[@izhaanahmed3038]

+Muzik Bike Why do I keep finding you on the internet? This is like the 5th youtuber I swear

[@Peter_1986]

More like "sexy laugh".

[@Laser0rick]

Numberphile comment sections are literally the only comment sections that actually make me feel stupid compared to others.

[@bestnocture]

Spoke has been Truthen.

[@xerus7425]

lel

[@OriginalPiMan]

James McTrollington Try the comments of 'PBS Space Time'.

[@KrisPBacon69]

look at guava juice comment sections... trust me, you'll feel like a GENIUS

[@desia.brimou]

Same.

[@louismelahn1805]

I would prefer to call them “pandactyl” numbers or “omnidigital.” “Pan” comes from Greek, and “digitus” from Latin.

[@avecus]

Louis Melahn did Greeks call digits as dactyl? In that case, pandactylic. (I know they called fingers but idk if the idea of digit existed)

[@banger107]

"tele" comes from greek (τῆλε) and "vision" comes from latin (visione) and we still call it "television"

[@jwcfive7999]

@@banger107 visione, as far as I know, is not Latin. Video, videre means see, though.

[@duncanhw]

​@@jwcfive7999It's the ablative of visio

[@derf4886]

"polite laugh"? Who wrote that?

[@arunabhganodwale1022]

And I see this another almost same comment just above this, spooky indeed, even more see if I am replying after 7 years.

[@AaronJLong]

The sudoku flashbacks are real

[@stargazerequiem]

James, James everywhere Numberphile cannot be as entertaining as I is without our friendly neighborhood James

[@fazlulhowladerbibi5377]

JAMES, THE TYPE OF GUY WHO WAS A FAVOURITE PANDIGITAL NUMBER

[@johnnycochicken]

can relate, I was also a favorite pandigital number

[@bulgaria9003]

19113519tm

[@pinomial9888]

Random search, got this: "An ambigram (from Latin: ambi both + gram = letter) is a word or words that can be read in more than one direction, where the word reads the same when upside down or turn over to form an entirely new word."

[@MuzikBike]

i said pans sued pies ! ^There's an example I just made up

[@deliriumredbonnie7188]

Inexorability cool

[@Jivvi]

NOON

[@Jivvi]

Also "chump" is an ambigram if you write it in cursive.

[@michaelmarkidis5291]

I wrote a program to check all pan-digitals from length 1 to 9. I call it a "perfect pan-digital" if it shares the same unique property as the number 381654729 in this video. Checking all 1 permutations of: 1 ------------------------------------- Perfect pan-digital found: 1 Checking all 2 permutations of: 12 ------------------------------------- Perfect pan-digital found: 12 Checking all 6 permutations of: 123 ------------------------------------- Perfect pan-digital found: 123 Perfect pan-digital found: 321 Checking all 24 permutations of: 1234 ------------------------------------- Checking all 120 permutations of: 12345 ------------------------------------- Checking all 720 permutations of: 123456 ------------------------------------- Perfect pan-digital found: 123654 Perfect pan-digital found: 321654 Checking all 5040 permutations of: 1234567 ------------------------------------- Checking all 40320 permutations of: 12345678 ------------------------------------- Perfect pan-digital found: 38165472 Checking all 362880 permutations of: 123456789 ------------------------------------- Perfect pan-digital found: 381654729

[@flashpointwhite]

examples please?

[@georgeslade1753]

A strobogrammatic number is a number that looks the same upside down

[@RichieCarter2002]

69

[@georgeslade1753]

+Richie Carter xDD

[@Carlos244]

George Slade Wow omg

[@aaronclark2599]

is that when you turn it upside down via reflection or rotation?

[@dhnyl]

rotation

[@adamoore8073]

I zoned out for like 5 seconds and came back to numbers that made no sense. its like algebra all over again

[@numberphile]

a strobogrammatic prime sounds amazing! :)

[@h-Films]

a numberphile comment with 1 like

[@lecombattant1000]

The smallest one shouldn't be "-9876543210" ?

[@jacoboblandonpineda]

+Theo De Castro Pinto Maybe they just counted the smallest positive pandigital number.

[@ifn151]

Yeah probably

[@MasterHackerChannel]

"No I didn't particularly like that" xD

[@TranslatorCarminum]

4:11 - 4:17 Vertical digital palindrome?

[@parthivshah202]

Yes

[@Skimbleshanks73]

He sounds like he's getting an orgasm just thinking about it

[@damienhodsall9645]

strobogrammatic is what it's called

[@D4arkSh1nes]

"Polite laugh"

[@Wumbolo]

isn't it the most polite laugh you've ever seen on Numberphile?

[@Jivvi]

_sensible chuckle_

[@DeepanshuKush]

How cool is that! Actually, around 5 years ago, in my 7th grade, one of my teachers asked me to find a pandigital number in which the nth digit is divisible by n. It took me 3-4 days of really hard thinking (and a lot of trial and error!) but I finally arrived at 381,654,729! I had completely forgotten about that until I saw the video. Thanks!

[@oskarlindelof9685]

If you should use every didgit once and once only. Then the only pandidgital number in base 2 is 10. And if you dont include 0 then the only pandidgital number is 1. In general, the number of pandidgital numbers in base b is (b-1)! And if you include 0 as well it is (b-1)!(b-1)

[@hershellayton4566]

Wouldn't the smallest pandigital number be -1234567890 or .0123456789?

[@weeboogaijin]

Hershel Layton positive integers

[@jamesgarrett7340]

Hershel Layton if

[@goldenbeast2673]

At 7:30, you spelt 'really' incorrectly...

[@Henry-ih3jh]

A simple mistake like that is realy quite shocking.

[@goldenbeast2673]

Lol, I see what you did there.

[@ncamen68]

+Henry Bownes I SEE WHAT YOU DID THAR

[@Xyvir]

Ambigrams are numbers or words that look the same upside-down and right-side up. It can also refer to numbers or words that are mirrored at any particular angle.

[@JeSuisWeedle]

hahahaha two years ago I got the following problem on this math olympiad: a nine digit number has the following properties. 'the first n digits of the numbers forms a new number (in the same order obviously) which is divisible by n, where 9>=n>=1. Find the nine digit number' After a while I proved that 381654729 was the only possible number (: now I'm randomly youtubing, and I find this video! What's even more interesting is that I live in the Netherlands :O such coincidence :D

[@doorhanger9317]

I do love things like this about numberphile because it makes me feel so connected to pretty much everyone in the world where as in real life even another county seems so... distant. Great video anyway, keep up the good work and keep lifting spirits after hard days!

[@MauricioFernandezF]

Evert riow and column in a sudoku is going to be a pandigital number without zero. :)

[@AngryMrFixit]

Anytime there’s a puzzle cube in a numberphile video, I want to reach into my phone and solve it.

[@GretgorPooper]

She's so pretty \*---**\*

[@syt6179]

also if you look good you can add 2 numbers and get 10 until the number 5 in the middle .. 381,654,729 example: 3+7, 8+2, 1+9, 6+4, and then 5

[@ramdassbhaiya9287]

any pan-digital #'s that are in the Fibonacci sequence?

[@ramdassbhaiya9287]

That Guy I recognize it is the fibonacci sequence, but is 1123581321345589144233377610 a fibonacci #? I could figure it out, but I'm lazy.

[@rroggr]

well, its not a fibonacci number, but if you want a pan-digital fibonacci number then: 1304969544928657 is the first one I found, maybe I missed a few that is, or maybe I didn't.

[@robertopatino3632]

Without repeating, no. Repeating, the first one is 2504730781961, which is the 61th Fibonacci Number

[@jwmmath]

...one could use a "brute-force" computer program to select series of 'seed' numbers, then generate Fibonacci-like sequences until the least 10-digit number to appear is indeed pandigital.

[@Username93611]

"I'm doing good..." "How are you doing?" I chuckled. I do that sometimes.

[@torgo_]

"something you _need_ to be doing".. that seemed a bit aggressive. 2:37

[@Vulcapyro]

Usually we say 1 is a small [positive] number, -1 is a small negative number, and -(10^100) is a large negative number.

[@JetBoxSuper]

Wow, just watching this video, I found out I have a pandigital phone number. O.o

[@anders160196]

Well, we use commas, dots, and spaces when we wan't to write a long number correctly. Commas we use to seperate decimals and whole numbers, as an example One and a half becomes "1,5" - Dots are used instead of commas, which means that "1,000" becomes "1.000" - Spaces are placed at every third decimal, which means Pi becomes "3,141 592 653 59..."

[@SilentHollow69]

How can he smile so hard?

[@Whoeverheis11]

Using the strictest definition of a pandigital number, Whole number, Requiring zero, not allowing repeats, there are 10! or 3,628,800 pandigital numbers. not requiring zero, I believe there are 10! + 9! (3,991,680) of them, since those with zero are encompassed within that definition. Expanding to rational numbers, we get 3.6288*10^16, or 3.991680*10^16, depending on whether 0 is required. And for any of those, allowing repeats gets you infinite pandigital numbers.

[@sambolini7160]

there are NO prime pandigital number. they all can be divided by 3 !!

[@zx_taaaaaaaaaaaaaaaaaa]

unless you repeat a digit again

[@sewbernard]

"polite laugh" IT WAS NOT NECESSARY TO SUBTITLE THAT, BRADY

[@therealshard]

"Polite laughter" Hahahahahahaha whuuuuutttt

[@wafelsen]

As other commenters have pointed out, the 2nd, 4th, 6th and 8th digits must be even to be divisible by 2, 4, 6 and 8 respectively. The other digits must then be odd as there are no even numbers left (without repeating).

[@socool2197]

0. an infinite number of 0s 123456789 would surely be the smallest one, 123456789 being the one without 0.

[@JackJack-cf7le]

it dosent have the n-th place dividing thingy

[@Azzucips]

Do the numbers in the decimal place still make the number pandigital?

[@Anonymous71475]

No, it's the closest to zero, but the smallest pandigital number with 0 is: -987654321000.... or, to make it easier, -9.87654321*10^∞.

[@joealias2594]

If you add up a number's digits, if that sum is equal to a multiple of 9, the original number is divisible by 9. 1 + 2 + ... + 9 = 45, so any number that has digits 1..9 as its only digits will always be divisible by 9 and therefore not prime.

[@heartsden2901]

"Polite laugh" OMFG WOW

[@ananyablonko1249]

pi at 1:53 misses an 8 on the 4th line after "05" just before the 2. also the digits "45923078164" contain the first two pandigital numbers within pi

[@evalkyrie]

"polite laugh"

[@iambic-kilometer]

My favorite pan digital number is 44445x54444=2,419,763,580.

[@NotAJollyPotato]

Ambigram, for a pic that is the same upside down. Not sure if it would be the same.for a number

[@cecasiahaan6468]

689

[@liesdamnlies3372]

+Ceca Siahaan Nice. Though it does assume rotation 180º and not a mirrored image.

[@kopopo23]

+Dylan Le Lerre ambidigital numbers? lol idk

[@NotAJollyPotato]

kopopo23 sounds relatively correct. That is there name then!

[@valentinemay4766]

+Dylan Le Lerre It's actually 'strobogrammatic'

[@EdoTimmermans]

Some other interesting pandigital numbers: 6398410752 has the highest number of primes in the prime factorization (25, 6398410752=2^21*3^3*113. A pandigital number can have at most 9 different primes in the factorization, there are 17 such numbers, 5948623170=2*3^2*5*7*17*19*23*31*41 has the lowest prime number in the factorization (41), 2148736590=2*3^2*5*7*11*13*17*23*61 is the lowest and also contains the 7 lowest primes and so does 3628194570=2*3^2*5*7*11*13*17*23*103. When you multiply the last number with 2 then you get the third pandigital number containing the 7 lowest primes (7256389140). There are 9 pandigital numbers with a maximum of 7 different primes in the factorization that together are pandigital as well, the smallest is 2841659730=2*3^2*5*7*13*43*8069, the one with the lowest prime among these 7 is 9538764210=2*3^2*5*47*83*101*269.

[@MJPOAS]

Does anyone notice that the value of each term is the number of places you have to go through to get to that term? IE the first term is 3, so when you go in three places you get 1. The forth term is 6, so when you go in six places you get 4.

[@CapriSjonnie]

Like I said, it is because all the telephone numbers in the Netherlands contain of 10 digits. Mobile or landline, it's all 10. Except for commercial numbers, they can contain less. So in that case, her pandigital number will consist of the base10 sequence of 0-9.

[@dnclvr]

Each grouping has two numbers that add up to nine and each grouping adds up to 12,15,18 respectively so separated by three and each divisible by 3

[@cocoa1996]

Numberphile always cheers me up

[@Gewath]

I mean scientific notation. Where all numbers are expressed as a*10^b, where b is a whole number. 381,654,729, for instance, is 3.81654729*10^9, and will be rounded off anywhere down to 4*10^9, depending on how accurate you want to be. In standard notation, it'd be 0. ...(9,876,542 0's)... 2.

[@N4voru]

My favorite pandigital number is 9,753,086,421, because you get it if you take 9876543210 and subtract it with 123456789. And if you left out the zero, take 987654321 and subtract it with 123456789, you'll also get a pandigital number (864,197,532).

[@RincewindIsMyHero]

We call "," a comma whether or not it is used in math or punctuation. :) We refer to "dots" as periods when we use them as punctuation or points in math. So we'd say 3.14 as three point one four. We ignore commas though and don't say them. So "1,427" is read as one thousand four hundred and twenty seven.

[@Boomshicleafaunda]

The original question is why 0! is 1. You can't just state that and be happy with it. There's always a reason behind why things are the way the are. 0! is 1 because of the convention with Empty Products (Example: prod({2,3}) = prod({2})*3 = prod({})*2*3 = 1*2*3) Factorials can be written as products. 1 is the Multiplicative Identity, so when you take the product of nothing, it has to be 1, because otherwise, when we start the problem, where we start with nothing, products wouldn't make sense.

[@alan2here]

All the binary pandigitals, using the definition that doesn't permit 0 or duplicates: 1 And all the base 3 ones: 12 (5) 21 (7) (+2) And all the base 4 ones: 123 (27) 132 (30) (+3) 213 (39) (+9) 231 (45) (+6) 312 (54) (+9) 321 (57) (+3) Some interesting maths there I think :)

[@Cornampoo]

For those of you that find Lara very attractive: in the Netherlands all cellphone numbers start with 06, so you only have 8! possibilities left. Happy calling :P

[@EnergyCuddles]

'polite laugh' That made me laugh. x)

[@MinecraftWarriors358]

Look at 1:53. I was just randomly reciting pi (I've memorized the first 60 or so) and I saw an error in there. In the 4th line it says "75105209" It should actually be "75105*8*209.

[@rrestoring_faith]

***** dam lol nice find i think they would have just used a random image or something tho

[@jaapaarts1007]

Rrestoring faith it's written in the style they usually do so it's not a picture at all its just text

[@mrmoneyboygurl]

I've memorized 260 haha

[@scslre]

hallie lieberman haha

[@avecus]

Who said it's π?

[@jukmifggugghposer]

Cracking the Cryptic recently featured a sudoku on their channel that uses this very number in the solution. It's got a few special rules, but the main one is that divisibility property - there's a few strings that have to satisfy that rule around the puzzle, including one at the top that's nine digits. So this video is kind of a spoiler for the beginning of that sudoku.

[@GRAVEAUDIO]

This dude is so enthusiastic about his numbers its great

[@ScorpioPK]

The reason why there can't be prime numbers without redundant digits (with or without the 0) is because the sum of the digits will always be 45, which is divisible by 3, making the entire number divisible by 3.

[@asdfghyter]

"Pandigital phone number, how cool is that?" :)

[@0x2710-b]

The notation for this is (13)(28)(46)(5)(7)(9)

[@ygalel]

Not only this number has all digits, from the left counting up it's divisible by all digits and it's unique is what makes it insanely crazy

[@JWQweqOPDH]

Yes. It's fits his description of them all being (2^n)-1. 2^0-1=0.

[@alan2here]

I thought this might be surprisingly easy: Pick a digit, lets say 5, it's divisible by 1. Append an as of yet unused digit that makes this number divisible by 2, for example 4 to make 54. Repeat for divisibility by 3, 4, etc... 5436 But now I see, it's easy to get stuck before getting to 9, for example there is only one choice for 5 and its a digit that I've already used. Surprisingly 123456 gets up to 6 digits! 3216549 :)

[@icanhazsourcecode]

"Hi, I'm doing good." "How are you doing?" That wasn't awkward.

[@Tompazi]

i is defined as a number with one property: i² = -1. It is not defined as the square root of -1. Just that when it's squared it's -1.

[@thefurryfile999]

James said his favorite pandigital number was the only one were leading N digits together were divisible by N. But he then stuck a zero on the end to shift it a place over, but doesn't this mean his fave pandigital number is not the only one that does this, since adding a zero to the end makes it a different number?

[@Vulcapyro]

Yes, that's correct, but that isn't -just- dependent on the n-digit numbers being divisible by n; it has to follow from the number being pandigital. Saying n divides each n-digit number implies odd-even-odd is false. Nor is the odd-even-odd pattern necessary for the divisibility, as you claimed above.

[@innertubez]

I think not because it has only 3 (as a natural number component). But that's just my guess as an ignorant layman. Perhaps in a system under the less strict rules they count decimals. Also Google "Is it possible for an irrational number to never have '1' as a digit?" for a cool discussion.

[@pausebeforeviewtube]

I'd call numbers that look the same when you flip them on the vertical axis, vertically palindromic numbers. However, with most fonts, I don't think any vertically palindromic numbers exist except for 0. 99066 would have to be _mirrored_ upside down (i.e., flipped both vertically and horizontally) in order to look the same as it does right-side up; if you just flipped it vertically, you'd have backwards sixes and nines. Normal palindromic numbers are numbers that look the same when reverse ordered, like 121. Horizontally palindromic numbers would be numbers that look the same when you flip them on the horizontal axis, like 808. Some fonts (and some entire typefaces) render eights and ones so that they are symmetrical both vertically and horizontally, so numbers like 808 and 101 would be palindromic both vertically and horizontally if rendered in such fonts.

[@StereoBucket]

Sum of all digits is 45 which is dividable by 3. :D Easiest trick to find out if you can divide the number with 3 or 9. Just add up all the digits and find out.

[@Lugmillord]

oh, I noticed our mistake. The definition of those numbers doesn't include that divisible by n thing, that only worked for the first number. So that 9814... number was just the biggest square number to feature each digit once. You can see at 1023456789 that the divisible by n doesn't work that well either.^^

[@ragenFOX]

i had a pandigital phone number once. i used to always say that * and # are the only ones left so i would use all of the buttons on the board.

[@martintuma9974]

2:23 Chuck Norris CAN divide by zero.

[@blackieblack]

987,654,321 - 123,456,789 = 864,197,532. Subtract one pandigital number from another, and you get a third pandigital number. I can't believe this wasn't included in the video.

[@iOSMinecraft120]

numbers that are read the same upside-down are called ambigrams (also words). See wikipedia for more info

[@CDolph296]

There was an AMC problem on those types of numbers. They were called "cute numbers". The question was to find the number of 6-digit cute numbers.

[@ThomasdenHollander]

There are 5^5 (3125) different combinations of phone numbers excluding the area part, and 5! (120) of them are pandigital, so 3,84% of all dutch phone numbers are pandigital, which is 614.400 people.

[@anticorncob6]

If pandigital means that it has to contain all digits 0 - 9, but you are allowed repeats, then from that "Three is everywhere" video, virtually all numbers are pandigital numbers. They are only rare when you're dealing with relatively small numbers.

[@Jivvi]

2013 is the first year with four different digits since 1987. The last date with all different digits was 25/06/1987, and the next one won't be until 17/06/2345. If you don't use leading zeroes, the next would have been 5/4/2013, but that's cheating.

[@JustinHallPlus]

Well it's much cheaper if you mainly make local calls. Since most cable companies bundle the service, people end up paying for a phone line whether they use it or not. It may be much different in Finland though, and I don't personally have a land line, but there is a reason for some people to have one.

[@subterreanhighrise]

Brady has anybody ever told you that you sometimes sound like Murray from the Flight of the Conchords series? ... well ... this only adds another awesome to an already awesome Video series :) Really love it!

[@tarikhusein]

These guys are so polite!

[@ragnkja]

You're right. A riddle for you: One evening when I was taking the bus home, I noticed that the mirror image of the clock showed the time from twenty minutes earlier. What time was it?

[@singalo123456789]

2:22 "of course we can't divide it by zero :O"

[@SkinnyCow.]

its always more interesting when mr.grimes is the video as he looks like he is 15 and knows so much !

[@wolfelkan8183]

“It’s impossible to have a prime number without redundant digits”. In base 10, that is. But there are four in base 3 and one in base 2.

[@lexicoll1]

Of the extant Germanic languages, Scots (i.e. Lallans, not Scottish-English or Gaelic) is the closest to English, followed by Frisian. Then Dutch, Low Saxon (Plattdeutsch), German, Swiss German, Danish, Norwegian, Swedish, and finally Icelandic. I've probably missed a couple out: Faroese springs to mind.