Self-Describing Numbers – 3
The number 1210 is a Self-Describing Number because each digit in order describes how many digits of that type there are – there is 1 0, 2 1’s, 1 2, and 0 3’s. Similarly, 2020 is a Self-Describing Number because it has 2 0’s, 0 1’s, 2 2’s, and 0 3’s.
THE CHALLENGE
Find a Self-Describing Number that has ten digits.
EXPLORATION
Do you see a pattern in your answer for ten digits that will help you find other examples with a large number of digits?
Notes
THE CHALLENGE
A playful, disorganized approach to this is fine and should be encouraged.
Here are some useful results from the Notes on Self-Describing Numbers – 1 and – 2:
Result 1: The sum of the digits is the number of digits.
Result 2: The sum of the products is the number of digits.
Result 3: The rightmost, low-order digit, the ones digit, is 0.
Result 4 & 5: The high-order digit is at least 2 for numbers with at least five digits.
Result 6: The high-order digit is at least 3 for numbers with at least six digits.
Result 7: For numbers with at least six digits, the number of nonzero entries must be more than 3.
It is probably helpful to have the answer for 7 in front of us: 3211000. Looking at this, it is tempting to think that the number of nonzero entries is four for numbers with at least six digits. From Result 7, we know that the number of nonzero entries cannot be less than four. Can it be more than four?
We immediately know two of the nonzero entries – the high order digit that gives the number of 0’s (which is a number that is at least 3), and the place that corresponds to that digit – if that place has a value greater than 1, then that causes there to be that many 0’s and that many of some other number, and things quickly spiral out of control. So, there is at least one 1. That gives us three nonzero entries. It is now impossible to have a single 1 (how would we fill in the number of 1’s), so the number of 1’s must be at least two. So we get an additional 1 corresponding to the place that has the number of 1’s.
And that’s it. Any additional entries would once again cause a snowball effect that would cause the numbers to get too big. At last, we arrive at the answer.
The answer for 10 is 6210001000!
EXPLORATION
The answer for 7 was 3211000 and the answer for 10 was 6210001000. The pattern X210….01000 looks promising. Let’s look at how we can apply it.
It looks like, as we go up from 7, we can increase the number of 0’s by 1 each time, and put that new 0 between the two 1’s. If you check it out, it always works. Well, it works until we get to 13 with 9210000001000. After 13, the first digit becomes too large to be a single digit, but it was fun while it lasted.