»  Solutions to puzzles in my VDARE.com monthly Diary

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•  Solution.

To get to a solution I've chosen to work my way up to "self-descriptive" from merely "descriptive."

I start with a random eight-digit number. Here's one from an actual random-number generator (RNG): 53479811. I'll call it U. Obviously U is not self-descriptive. Its right-most digit is 1, so if U were self-descriptive there would be one zero in it, but … there isn't. U is not self-descriptive.

OK; now generate a second number — I'll call it U₂ — that describes that number U in the way laid out by the problem statement.

There are no zeros in U, so the right-most digit of U₂ is 0.

There are two ones in U, so the second digit from the right in U₂ is 2.

There are no twos in U, so the third digit from the right in U₂ is another 0.

There is one three in U, so the fourth digit from the right in U₂ is 1.

There is one four in U, so the fifth digit from the right in U₂ is also 1.

There is one five in U, so the sixth digit from the right in U₂ is yet another 1.

There are no sixes in U, so the seventh digit from the right in U₂ is another 0.

There are seven different digits in U, so the left-most digit of U₂ is 7.

So U₂ looks like this:  70111020. Self-descriptive? Nope. The right-most digit is 0, so if U₂ were self-descriptive there would be no zeroes in it, but in fact there are three. U₂ is not self-descriptive.

In the same way I can get another eight-digit number, which of course I shall call U₃, that describes U₂. It works out to 40000133 because U₂ has four different digits, none of them 6, 5, 4, or 3, one of them 2, three of them 1, three of them 0. U₃ isn't self-descriptive, either. (Although it's a tad closer to being self-descriptive. That leading 4 does indeed tell us the number of different digits in itself:  0, 1, 3, and 4.)

Lather,rinse, repeat. U₄, U₅, U₆, U₇, and U₈ work out to be, in order:  40012014, 40020123, 50011213, 50101132, 50101132, … Whoa! The descriptive numbers seem to have got stuck on 50101132 — which is self-descriptive! (Five different digits, no sixes, one five, no fours, one three, one two, three ones, two zeros.)

So yes: self-descriptive numbers do exist and I've found one: 50101132.

If I start with a different random number U and apply the descriptive process, perhaps I'll find another. Let's try it.

The next number out of the RNG is U = 11593061. That stabilizes — again after six steps, U to U₇ — at … 50101132!

Will any random number get me to 50101132 after six steps? I cranked the handle on the RNG.

• 22175952:  50101132 after six steps.
  
  
• 64023840:  50101132 after only four steps.
  
  
• 57739351:  50101132 after five steps.

So it's 50101132 every time, but the number of steps varies. (At the lower limit: If the RNG had cranked out 50101132, the number of steps would have been zero.)

How about some seriously non-random number? I tried 99999999. I landed on 50101132 after seven steps.

Conclusion: 50101132 is the only self-descriptive number you'll find using my process. Quite likely it's the only one there is. Starting with any eight-digit number at all, there is some upper bound to how many steps it will take you to reach 50101132, and the upper bound is not very high.

There are of course ways to expand the inquiry. Are there other self-descriptive numbers not captured by my process? What is that upper bound to the number of steps? How about numbers with more than eight digits, or fewer than eight? What about different bases for our notation — hexadecimal, for example?

And so on. I couldn't find much relevant literature on the internet, but you may be more diligent than I am …