Autobiographical Numbers

Do you know that 1210 is the smallest autobiographical number? You probably do not know what an autobiographical number is. You are right if you think that such a number should be a pompous self-centered number whose only purpose in life is to describe itself.

Here is the formal definition. An autobiographical number is a number N such that the first digit of N counts how many zeroes are in N, the second digit counts how many ones are in N and so on. In our example, 1210 has 1 zero, 2 ones, 1 two and 0 threes.

Let us find all autobiographical numbers using the “zoom-in” method.

  1. By definition, the autobiographies can’t have more than 10 digits. It is nice to know that these egotistical numbers can’t be too grand.
  2. The sum of the digits in an autobiography equals the number of the digits. Consequently, the sum of the digits will not be more than 10.
  3. The first digit is the number of zeroes. As you know, self-respecting integers do not start with a zero. Hence, the number of zeroes is not a zero.
  4. Subtracting statement “c” from statement “b” above, we get a resulting statement that the sum of all the digits, except for the first one, is equal to the number of non-zero digits plus 1.
  5. That means, other than the first digit, the set of all other non-zero digits consists of several ones and 1 two.
  6. Furthermore, the number of ones is either 0, 1 or 2.

Now we continue zooming in in three different directions depending on the number of ones. In this blog entry, I will consider only the case in which there are no ones; I leave the other two cases to the reader.

  • If the number of ones is zero, then the only non-zero non-first digit of such a number is 2.
  • This 2 should be included in the autobiography; since the third digit of the number is not zero, it must be 2.
  • The number has 2 twos.
  • It must be 2020.

Here is the full set of autobiographical numbers: 1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000.

This is the sequence A104786 in the Online Encyclopedia of Integer Sequences (OEIS). The OEIS, where I first encountered the autobiographical numbers.

Autobiographical numbers are very cute numbers. But there is a problem with their name. If there is a notion of an autobiography of a number, then it would be logical to expect that there is a notion of a biography of a number. What would be the logical candidate for a biography of a number? Let us say that given a number N, its biography is another number M such that the first digit of M is the number of zeroes in N, the second digit of M is the number of ones in N and so on.

Of course, for a number to have a biography, we need to assume that none of its digit is present more than nine times. Still there are several problems with the definition of a biography.

The first problem is that if N doesn’t have zeroes, its biography starts with a zero. As numbers don’t start with 0, that biography is not a number! Furthermore, if N starts with 0, it can have a biography but N is not a number. Luckily for this article, a digit string starting with zeroes can’t be an autobiographical string, because the number of zeroes is not a zero. It is a relief that those illegitimate strings that are trying to pretend to be numbers can’t actually be autobiographical.

The second problem with biographies is that a number can have many biographies. Indeed, if a number doesn’t have nines, you can remove or add zeroes at the end of a biography to get another biography of the same number. Since mathematicians like to define things uniquely, we might consider it a problem if a number has several biographies. In real life it is possible to have many biographies of a person. So the second problem is not a big problem. I will call the shortest possible biography of a number the curriculum vitae and the longest possible biography the complete life story.

The third problem is that numbers with the same digits in different permutations have the same biographies. So in a sense a biography follows the life not of a number, but rather the set of its digits.

Suppose for now we allow a biography to start with 0. Also, let us choose the curriculum vitae — the shortest biography in case there could be several. Let us build a sequence of CVs. As an example, we start with 0. Zero’s CV is 1, one’s CV is 01, continuing that we get the following sequence: 0, 1, 01, 11, 02, 101, 12, 011, 12, 011, 12, …. You can see that the CVs’ sequence fell into a cycle in this case. I tried sequences of CVs starting with many numbers. I found that they fall into two cycles. One cycle is described above and another one is: 22, 002, 201, 111, 03, 1001, 22. Can you find another cycle or, alternatively, can you prove that all the numbers that allow the sequence of CVs converge to only these two cycles?

Let us build the sequence of complete biographies, that is, life stories, starting with 0: 0, 1000000000, 9100000000, 8100000001, 7200000010, 7110000100, 6300000100, 7101001000, 6300000100, …. We see that this sequence falls into a cycle of length two. The members of this cycle are legitimate numbers. These numbers are too shy to advertise themselves. But Alice praises Bob, because Bob praises Alice. It’s a very advantageous flattery pattern! I will call such a pair a mutually-praising pair. We’ve already seen mutually-praising strings: 12 and 001. Two other examples of number pairs thriving on each others’ compliments are, first, 130 and 1101, and second, 2210 and 11200.

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29 Comments

  1. n:

    i need a biography for the number 272

  2. Andrew:

    Why can’t 9210000001000 be autobiographical?
    it has:
    9×0
    2×1
    1×2
    0x3
    0x4
    0x5
    0x6
    0x7
    0x8
    1×9
    0x10, 11, 12 🙂

  3. Tanya Khovanova:

    Andrew,

    By definition, they can’t have more than 10 digits: http://en.wikipedia.org/wiki/Autobiographical_number

    If you allow more digits and they all happen to be the same, you create an ambiguity.

  4. Mandeep Kumar:

    whats wrong with
    8000000010
    0 – 8 times
    8 – 1 time

  5. tanyakh:

    Mandeep:

    In addition, it means 1,2,3,4,5,6,7,9 should be present zero times. Which is not the case.

  6. Me:

    This is cool 😀

  7. Octotroph:

    Wait, so there’s no 6-digit autobiographical number? I mean, I’ve been sat here with a pencil and paper for 20 minutes trying to figure it out and I’m no closer to it than when I started, but I would not consider myself to be by any means an intellectual. I’m sure either I’m just missing something or my high school concession of “math is weird and somewhat dumb” is correct. WHAT EVER HAPPENED TO MATH NEVER BRING WRONG? Next you’re going to tell me that up is down, the sky is pancake, and we’re eating blues for breakfast!

  8. Raine Riny:

    Is 221100 autobiographical?

  9. Xeuron:

    Hey octotroph. Same life bro. Tried to find an autobiography of a random number of digits. Chose 6 and got stuck. Until i just found out that it doesn’t exist. GG.

  10. Dragon:

    Raine Riny
    No.
    It has 2 zeros correct.
    It has 2 1s correct.
    It has 1 2s correct.
    It hasn’t 1 3. Wrong.
    Since it doesnt have 1 3. We lost a 1 meaning the second digit is wrong. Meaning there is no 2 anymore so the 3rd digit is wrong. And so on. There is no autobiographical numbers for 6 digit. Farewell GG

  11. Zehra:

    When we gather digits of an autobiographical number it is always equal to numbers of all digits in number.
    For example;
    1+2+1+0=4 (there are 4 digit in 1210)
    6+2+1+0+0+0+1+0+0+0=10 (there are 10 digits in 6210001000).
    Why?

  12. Zehra:

    Uh! I thought a litle bit and I found out:)

  13. Kurabaya:

    I just created a code to solve this autobiographical number (actually just for 10 digits number since I found out about this kind of number from TED-Ed’s riddle). But it turns out that there is no autobiographical number for some digits like 2, 3, 5, and 6. I don’t know, it might be I did mistake with my code or it just never exists.

  14. Hero:

    There is no autobiographical number for 1,2,3 and 6 digits

  15. Rakesh Kumar:

    Summing up the digits should give out 10
    # manideep Kumar

  16. Mahir:

    I saw in your youtube puzzle of Leonardo Da Vinci valut that there is only one to digit autobiographical number.

    Can you tell me if the following is autobiographical number or not?

    8000000010

  17. Akshay Anil Shaha:

    800000010
    Is not an autobiographical number because 1 is present but 1’s position is having 0

  18. Anirban:

    Hey Mandeep, u have missed the number of 1s=1, so yours is not an autobiographical number

  19. Purav Patel:

    Is 01, a autobiographical number?

  20. Ankit:

    Purav, 01 is not. No. Of 0s cannot be 0.
    Its a paradox.

  21. Holla:

    Nice. Proved to be a very nice page. And by the way, nice question paurav

  22. Bhaskar Biswas:

    its really confusing

  23. Natalie:

    I am doing a project on autobiographical numbers and I can’t really find a whole lot of information… Do you know anything about who created them and why?

  24. Jackson:

    What about 9000000010? Ten numbers, =10 and violates no rule you stated.

  25. George Anastasiou:

    9000000010 has a “0” at 1’s place, even though it has a “1”

  26. Alan S:

    Octotroph and xeuron:

    I found the best way to “understand” the pattern is to change step f) in the original post and instead notice that the 2 must be the second or third digit.

    A) When the 2 is the second digit, you just iterate through choice of the first digit, and the rest of each number is forced:

    1210
    22… impossible
    3211000
    42101000
    521001000
    etc

    And you can informally see why the 6 case does not exist (you now need two 1s and they “clash”). There’s also a jump over the 5 case due to the extra 1.

    B) If you consider the 2 as third digit, that means there are two 2s so the first digit must be 2, which means you only have to enumerate the choices of second digit:

    2020
    21200

    This is why there’s the “extra” 4 case, and the fill-in for the 5.

    Not sure if that helps, but it worked for me:-)

  27. hms south:

    2020 is an autobiographical number because

    0s: 2
    1s: 0
    2s: 2
    3s: 0

  28. HMS South:

    2020 is autobiographical: 2 0s, 0 1s, 2 2s, and 0 3s

  29. For andrew:

    for andrew BECAUSE THE VALUE OF ALL THE NUMBERS HAS TO EQUAL THE NUMBER OF DIGITS

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