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.
- 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.
- 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.
- 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.
- 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.
- That means, other than the first digit, the set of all other non-zero digits consists of several ones and 1 two.
- 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), 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.
Share:
n:
i need a biography for the number 272
31 March 2009, 4:32 pmAndrew:
Why can’t 9210000001000 be autobiographical?
5 August 2013, 1:54 amit has:
9×0
2×1
1×2
0x3
0x4
0x5
0x6
0x7
0x8
1×9
0x10, 11, 12 🙂
Tanya Khovanova:
Andrew,
By definition, they can’t have more than 10 digits: https://en.wikipedia.org/wiki/Autobiographical_number
If you allow more digits and they all happen to be the same, you create an ambiguity.
5 August 2013, 12:57 pmMandeep Kumar:
whats wrong with
23 August 2018, 1:11 pm8000000010
0 – 8 times
8 – 1 time
tanyakh:
Mandeep:
In addition, it means 1,2,3,4,5,6,7,9 should be present zero times. Which is not the case.
23 August 2018, 1:35 pmMe:
This is cool 😀
23 August 2018, 11:21 pmOctotroph:
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!
24 August 2018, 1:41 amRaine Riny:
Is 221100 autobiographical?
24 August 2018, 8:06 amXeuron:
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.
25 August 2018, 12:37 pmDragon:
Raine Riny
25 August 2018, 12:40 pmNo.
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
Zehra:
When we gather digits of an autobiographical number it is always equal to numbers of all digits in number.
25 August 2018, 6:29 pmFor 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?
Zehra:
Uh! I thought a litle bit and I found out:)
25 August 2018, 6:32 pmKurabaya:
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.
26 August 2018, 10:16 amHero:
There is no autobiographical number for 1,2,3 and 6 digits
27 August 2018, 2:48 pmRakesh Kumar:
Summing up the digits should give out 10
28 August 2018, 1:38 am# manideep Kumar
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
28 August 2018, 12:07 pmAkshay Anil Shaha:
800000010
28 August 2018, 3:41 pmIs not an autobiographical number because 1 is present but 1’s position is having 0
Anirban:
Hey Mandeep, u have missed the number of 1s=1, so yours is not an autobiographical number
29 August 2018, 1:02 amPurav Patel:
Is 01, a autobiographical number?
29 August 2018, 3:50 pmAnkit:
Purav, 01 is not. No. Of 0s cannot be 0.
1 September 2018, 12:48 pmIts a paradox.
Holla:
Nice. Proved to be a very nice page. And by the way, nice question paurav
3 September 2018, 7:11 amBhaskar Biswas:
its really confusing
4 September 2018, 11:09 amNatalie:
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?
4 September 2018, 1:18 pmJackson:
What about 9000000010? Ten numbers, =10 and violates no rule you stated.
13 September 2018, 6:17 pmGeorge Anastasiou:
9000000010 has a “0” at 1’s place, even though it has a “1”
17 September 2018, 6:26 pmAlan 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 September 2018, 10:35 amhms south:
2020 is an autobiographical number because
0s: 2
8 October 2018, 4:54 pm1s: 0
2s: 2
3s: 0
HMS South:
2020 is autobiographical: 2 0s, 0 1s, 2 2s, and 0 3s
8 October 2018, 4:55 pmFor andrew:
for andrew BECAUSE THE VALUE OF ALL THE NUMBERS HAS TO EQUAL THE NUMBER OF DIGITS
11 October 2018, 6:31 pmShihab SN:
Aren’t 72100001000, 821000001000 and 9210000001000 auto-biographical numbers?
16 December 2018, 10:42 amtanyakh:
Shihab, it is usually assumed that an autobiographical number has no more than 10 digits.
16 December 2018, 4:29 pmHet patel:
I came up with this method to find autobiographical numbers:
For finding ‘n’ digit number
((n-4)*(10^(n-1)))+(2*(10^(n-2)))+(10^(n-3))+(10^3)
18 February 2019, 6:02 amHet patel:
Hi
18 February 2019, 6:04 amHet patel:
Is my method useful
18 February 2019, 6:08 amJeremy Mannikko:
Andrew,
It could be, but only if you’re not using a base-10 counting system. Since your number is 12 digits long, it must allow for the possibility of at least 1 “10” “11” or “12” which, in standard base-10 counting, are 2 digits, not 1. for example, with a 12 digit number, you could have “11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0” However, as you can see, for this to work, you need a 2-digit number.
If you’re using something like hexadecimal counting (where 10, 11, 12, 13, 14, and 15 are written as a single digit using letters of the alphabet A=10, B=11, ect) you could make a autobiographical number of more than 10 digits. In fact both “B000000000010000” and “9210000001000” would be valid autobiographical numbers for a base-16 (hexidecimal) or base-12 counting system, respectively.
Does that make sense? I’m not sure if I explained that well.
6 May 2019, 3:33 amAbhinav Upadhyaya:
Are 9000000000 and 8100000000 are autobiographical numbers?
14 May 2019, 5:04 amtanyakh:
Abhinav, No these are not biographical numbers. The first number has one digit 9.
14 May 2019, 11:44 amMary Audrey Brown:
To Abhinav, those two are NOT autobiographical numbers. True numbers like this that had ten digits must add up to 10.
26 July 2019, 12:02 pmPeter S Christopherson:
I see that 9 billion is not autobiographical but would it be self-describing?
17 August 2019, 10:57 pmE Koh:
Find a 8-digit number where the first digit is how many zeros in the number, the second digit is how many 1s in the number etc. until the eighth digit which is how many 7s in the number.
15 September 2019, 3:57 amHenry:
Hey, I think 81000000100 is a autobiographical number, right?
5 January 2020, 1:36 amHenry L.:
I think 90000000010 an autobiographical number, right?
5 January 2020, 9:01 amtanyakh:
Henry, no, both your examples do not count the number of ones correctly.
5 January 2020, 2:23 pmperson:
90000000100 is though
13 January 2020, 7:02 pmperson:
oops wait its not
13 January 2020, 7:09 pmbut 90000000010 is
OliveTree:
2020 is an autonomous number =D great year!!
19 January 2020, 6:34 pmAndy:
why is 9000000000 not an autobiographical number
12 March 2020, 9:56 amRay:
I think 10 is the smallest autobiographical number
26 March 2020, 10:06 pmRISHI:
whats the use of autobiographical numbers .
9 April 2020, 9:38 amRISHI:
no RAY 10 IS NOT autobiographical number
9 April 2020, 9:40 amVICTOR:
WOW COOL SEQUENCE
For Those Still Not Getting It, See It This Way
You Have A Number “N” With Ten Digits “ABCDEFGHIJ” Where,
A=Number Of Zeros In “N”
B=Number Of Ones In “N”
C=Number Of Twos In “N”
D=Number Of Threes In “N”
E=Number Of Fours In “N”
F=Number Of Fives In “N”
G=Number Of Six In “N”
H=Number Of Seven In “N”
I=Number Of Eight In “N”
J=Number Of Nines In “N”
A + B + C + D + E + F + G + H + I + J = 10
10 June 2020, 5:23 pm2020 Had a Silver Lining for Math Geeks - Carelyst:
[…] I pointed out last year, 2,020 is an example of what mathematicians call an autobiographical number: Its first digit indicates how many of its digits are 0s (there are two); its second digit […]
31 December 2020, 8:15 pmWaqar:
Fascinating
27 March 2023, 5:52 pmJack Taylor:
Anyone know a 15 digit autographical number?
28 February 2024, 4:10 pm