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Self-Referential number


What is the smallest (by value) self referential number? Obviously it can't be 0 or 1.

See for the rules of self-referential numbers.

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Obviously it can't start with a 0.

If it starts with a 1, it must contain a 0, but its second digit can not be 0 because it has at least one 1. Also its second digit can't be 1. If its second digit is >2, then it has at least 5 digits (at least three 1s, its second digit which is >2, and at least one 0), and so it is >1210. If its second digit is 2, it needs one more 1 and one more 0, meaning at least 4 digits total. But it can't be 1201, so 1210 is smallest.

If it starts with a 2, it must contain two 0s, but its third digit can't be 0 and so it has at least 4 digits, making it >1210.

If it starts with >2, it must contain at least three 0s, also making it >1210.

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I like these self-refential number questions. They're interesting.

Since I'm guessing that there are only a finite number of self-referential numbers, another good question would be "what is the highest?"

In fact, an even better question would be "what are all of the self-referential numbers, and is there a pattern?"

BMAD, can I look forward to a new thread being opened up for these discussions?

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