Ace the AMC

Autobiographical Numbers

This quiz is devoted to solving a single problem:

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. What other autobiographical numbers are there? The autobiographical number 1210 is broken down below as an example.

We'll start with the 10-digit case, and then go back to the general problem.

Note if you watched the related TED-Ed video (screenshot below) which continues to the solution, this quiz takes a different solving route (and gives hints as to how to extend it) so it will be helpful even if you've seen the rest of the video before.

Autobiographical Numbers

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.

Again, our first goal will be to focus on the question: What is the autobiographical number with 10 digits?

As this problem can be difficult to answer outright, it's broken into parts. First, what is the sum of the 10 digits?

               

Autobiographical Numbers

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.

Consider the last 5 digits of a 10 digit autobiographical number (as marked above). How many 0 digits must there be in that set?

               

Autobiographical Numbers

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.

Using the previous answer, we know the latter digits must be all 0s (except for a 1 somewhere; the extra digit can't be more than 1 because that will cause the overall sum of digits to exceed the target of 10).

What about the 3s and 4s digit places (as marked above)?

               

Autobiographical Numbers

To summarize the previous results (including information from the solutions), we know the sum of the digits must be 10, and the facts from this diagram:

We're close to an answer now!

Autobiographical Numbers

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.

What is the first digit of the 10 digit autobiographical number? That is, how many digits of the digits in an 10 digit autobiographical number must be 0?

               

Autobiographical Numbers

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.

Combine the information from the previous questions, and you should now be able to answer:

What is the autobiographical number with 10 digits?

Autobiographical Numbers

Now, how to solve the general case?

Our first steps can apply in a general way. For example, when checking 8-digit autobiographical numbers, this diagram arises.

From these steps it's possible to get a general formula for the first digit for when \( 4 \leq N \leq 10 .\)

Once you have this information, you have enough constraints that the different cases fall very quickly. Note, as a word of warning:

  • At least one of the cases of \(N\) has two numbers that fit rather than one.

  • At least one of the cases of \(N\) has no numbers of that length.

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