Every Number is the Sum of Three Palindromes - Numberphile

  • Published on Sep 17, 2018
  • Featuring James Grime... Check out Brilliant (and get 20% off their premium service): brilliant.org/numberphile (sponsor)
    More links & stuff in full description below ↓↓↓
    More James Grime videos on Numberphile: bit.ly/grimevideos
    Main paper he discusses: arxiv.org/abs/1602.06208v2
    And the base 2, 3, 4 paper: arxiv.org/abs/1706.10206
    Website by Christian Lawson-Perfect to run the algorithm: somethingorotherwhatever.com/sum-of-3-palindromes/
    Applet for any base 5-10 by paper author Lewis Baxter: www.rnta.eu/cgi-bin/three_palindromes/pal3.py
    Lychrel numbers: ru-clip.com/video/bN8PE3eljdA/video.html
    Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
    We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. www.simonsfoundation.org/outreach/science-sandbox/
    And support from Math For America - www.mathforamerica.org/
    Website: www.numberphile.com/
    Numberphile on Facebook: numberphile
    Numberphile tweets: numberphile
    Subscribe: bit.ly/Numberphile_Sub
    Videos by Brady Haran
    Patreon: www.patreon.com/numberphile
    Numberphile T-Shirts: teespring.com/stores/numberphile
    Brady's videos subreddit: www.reddit.com/r/BradyHaran/
    Brady's latest videos across all channels: www.bradyharanblog.com/
    Sign up for (occasional) emails: eepurl.com/YdjL9
  • Science & TechnologyScience & Technology

Comments • 1 047

  • mienzillaz
    mienzillaz Day ago

    James, you should make custom font with your handwriting, i would be first to download..:)

  • Amanda qeqqaqig
    Amanda qeqqaqig 14 days ago


  • Chris Doherty
    Chris Doherty Month ago

    There's absolutely loads of ways of deconstructing every number. This is the 100th way I found for 3141592 with a brute force type algorithm I wrote.

    (3088803, 47674, 5115)

  • Brian Dennehy
    Brian Dennehy 2 months ago +1

    This is amazing

  • Mr. H
    Mr. H 2 months ago

    And these are unique palindrome numbers?

  • Navodit Chaudhary
    Navodit Chaudhary 2 months ago

    Setters at Google used the paper to create a Google code jam final problem

    RASHO IE'TOLAN 3 months ago

    Beautiful as *fooooooook*

  • Rabbit the One
    Rabbit the One 3 months ago

    Not looking so well in this video Grimes. Have you had enough sleep lately?

  • Ahmed Elian
    Ahmed Elian 4 months ago

    But are the three palindromes a unique solution?

  • Henrik Soderstrom
    Henrik Soderstrom 4 months ago

    Is the solution unique or not?

  • Maxon Mendel
    Maxon Mendel 4 months ago

    James needs a new phone

  • Zachary Perkins
    Zachary Perkins 4 months ago

    i wrote a program that just goes through all the combinations of 3 palindrome numbers. the palindrome numbers are just generated up to the number you want, and then checking if they're the same backwards. They have a lot of possible combinations. 3141592 has 3112 (which is funnily also a palindrome) 3 number palindrome sums

  • DaveDonnie
    DaveDonnie 4 months ago


  • Thiago Soares Leite
    Thiago Soares Leite 4 months ago

    Why Just positives?
    If you multiply every palindrome by -1. You ter "ALL negatives integers" or Did I missing something?

  • tully balcom
    tully balcom 4 months ago

    The carries are also palindromic.

  • Raoul Gerrits
    Raoul Gerrits 4 months ago

    9:00 even the carry count is a palindrome, neat

  • Christian Herenz
    Christian Herenz 5 months ago

    Would be fun if you have the authors on your show ... Seems they've been working on Palindromes also in different context.

  • Brandon Brimer
    Brandon Brimer 6 months ago

    It's probably been called out already but the carried numbers form a palindrome as well. Math is amazing!!!

  • rate eightx
    rate eightx 6 months ago

    But What Three Palindromes Do You Use For, Say, 1? Or 2?

  • kaushik mitra
    kaushik mitra 6 months ago

    Is he wearing a skull cap

  • Jivan Scarano
    Jivan Scarano 6 months ago

    Is one of the palindromes allowed to be 0? If not, how do you solve for 1 or 2?

  • Jivan Scarano
    Jivan Scarano 6 months ago

    Gauss has the same birthday as Kirsten Dunst… and me.

  • Nevaeh Anglique
    Nevaeh Anglique 6 months ago

    I had a question that asked what three palindromes made up I think it was 2014?? on an old math competition problem set we were using for practice

  • debblez
    debblez 6 months ago +1

    or you could just...
    + 171
    + 8
    “small” numbers don’t require algorithms...

    • nunyabiznasty
      nunyabiznasty 6 months ago

      Your solution indeed works as a possible solution in Base 10, but the algorithm described in the paper also has to work in _literally every other base, no matter what._ It therefore has to respect quite a lot of special cases, especially when we are talking about palindromic numbers in odd bases such as 3 or 5, as mathematical operations involving digit manipulation (such as palindrome addition) can become quite funky if you're not careful and there are in many cases no "small" number solutions for these bases. (On a side note, this property of smaller bases is, if I understood the paper correctly, one of the reasons why there was this one rule mentioned by James in his demonstration that the categories are partially based on what the first digit of the target number is). Due to these cases, the resulting algorithm becomes more complicated, until it eventually becomes convoluted enough that it produces vastly less intuitive results even if it is, as you have proven, possible to generate a much smaller, quicker valid solution (at least in base 10).

  • Tony Coleby
    Tony Coleby 7 months ago

    One of the more amazing videos.

  • Chick3nWings 345
    Chick3nWings 345 7 months ago

    Any 1 digit number is easy

  • Mayor Video
    Mayor Video 7 months ago

    11 is a prime number which means I can't split equally, I am a palindrome, two ones to call my ownnnnn

  • hgam
    hgam 7 months ago

    i can think of an easier way than the methods in the video: first you identify the type of the number and fill in the first and last digits, then you subtract the 3 numbers you get from this step. For example, for 3141592 in the video, you subtract it by 2000002, 900009 and 10001 and get 231580. you then delete the zero from the end and get 23158. this number will follow the same property as it can be put into 3 palindromes, which are the middle part of the answer for 3141592. then you can identify the type of number this is and carry on.

  • KamuiSeph
    KamuiSeph 7 months ago

    Guys... It's SuDOHku.

    • KasabianFan44
      KasabianFan44 5 months ago

      Pipe down, James corrected himself a few seconds later.

  • Kiki Kikison
    Kiki Kikison 8 months ago

    The "carry" line was also a palindrome.

  • TeserT
    TeserT 8 months ago

    Pi is there 3141592

  • שחר א.
    שחר א. 8 months ago

    0322223 +
    MY BIRTHDAY -> ME!!!

  • brendan edmonds
    brendan edmonds 8 months ago

    What about 1?

  • Rodrigo Watzl
    Rodrigo Watzl 8 months ago

    About sudoku: consider a 9x9 square board of sudoku. It contains nine 3x3 boards inside. How many possible ways are there to complete the intire board?

  • Jack Scrivens
    Jack Scrivens 8 months ago

    but you do it with 1?

  • TruthNerds
    TruthNerds 8 months ago

    The website rejects any input >2^128-2^104+2^74. Don't ask me how I know that.

  • Jako1987
    Jako1987 8 months ago

    Innostunut sonni
    Saat Atte satasetta taas
    Palindromes are cool

  • Mujahid Syed
    Mujahid Syed 9 months ago

    5:55 "Soduko"?! It's pronounced "Sudoku". Plz, it really irks me when people do that. Nice vid btw.

  • Ben Heideveld
    Ben Heideveld 9 months ago

    How many different ways are there to write numbers as sums of three palindrome numbers? (on average, minimum, maximum)

  • zebracherub
    zebracherub 9 months ago +1

    For a given number, we could write down all the possible palindromes it is made of then, right?

  • J K
    J K 9 months ago

    I’m upset

  • Ian Moore
    Ian Moore 9 months ago +1

    does this algorithm generate the smallest possible combination of palindromes, or is there anything fancy and interesting about their distribution? Highly intriguing!

  • C21H22 N2O2
    C21H22 N2O2 9 months ago +1

    Is it possible that every number is the sum of three palindromes, while the carrier numbers together are a palindrome too?

  • Luke Gabriel Balgan
    Luke Gabriel Balgan 10 months ago +5

    But how do you memorize so many algorithms for each individual special case??

    sees rubik's cube

  • mirage in the desert
    mirage in the desert 10 months ago


  • mirage in the desert
    mirage in the desert 10 months ago

    So every palindrome can be written as the sun of three palindromes

  • Damien Mitchell
    Damien Mitchell 10 months ago

    The website says that 12345654321 would equal 12345654321+0+0. I'm unsure if that's considered adding 3 palindromes together.

  • Glen Gerod
    Glen Gerod 10 months ago

    I didn't realize that Johnny Lydon did maths.

  • venkat347
    venkat347 10 months ago

    Sum of palindromes for 1 please

  • Ratzler
    Ratzler 11 months ago

    Thank you for providing me with a new and fun way to fill up my spare time, I'll just go print out those *40 pages*

  • Pinto
    Pinto 11 months ago +1

    Virgin level 200

  • RLB
    RLB 11 months ago

    My name is Reinier which is a palindrome. I thought I should mention that. Bob, Anna and me sum up to Brady's birthday.

  • Billy Bob Thorton
    Billy Bob Thorton 11 months ago +2

    Some guy: Man I with their was a way I could figure out the three palindromes that summed produce this number.

    Some other guy: there's an app for that... Also why?

  • Sam Parfitt
    Sam Parfitt 11 months ago

    I bet the website now knows all of our birth dates..

  • macleadg
    macleadg 11 months ago

    Is this video the same when viewed backwards?

  • Jarom Allen
    Jarom Allen Year ago

    What if your integer is less than 3? How can, say, 2 be a sum of 3 palindromes?

  • Raluca Buta
    Raluca Buta Year ago

    why do we have only 9 numbers really ? Who invented the number system ?

  • Joe
    Joe Year ago

    here i was thinking i was gonna blow my math teachers mind....

  • ionisator idk
    ionisator idk Year ago +1

    7 times eleven equals 77.
    7 plus 7 equals 14.
    14:2 = 7

  • Stephan Jancar
    Stephan Jancar Year ago

    What about 10420275