*Originally posted by Aetherael*

**not to be a troll, but i have to disagree with the way you posed the problem. more accurately, i think you are asking us to "prove that there are finitely many integers n such that n^2 and n^3 exhaust the digits, brute force methods not allowed." then, you ask if we can find one.
**

however, as posed, your question is that of an existence proof: "is ther ...[text shortened]... are asking for a more elegant and inclusive proof that accounts for all possible solutions.

No no, you're neither a troll,nor jerk. I see you quite friendly.

The problem with a problem (?) is to formulate it correctly. A good formulation of a problem includes to define it properly. You gave me a good formulation. I chosed to formulate it as good as I could (despite my lack of English skills) and show an example how it could be done without any other help to the problem solver. So a faulty calculation of a faulty solution (42) gave a hint how I meant.

I've seen many good problems ruined by poorly formulation, with side solution, pseudo solutions, and faulty solutions due to poorly formulated problems.

So the problem #2 can be expressed: "prove that there are finitely many integers n such that n^3 and n^4 exhaust the digits, brute force methods not allowed."

Btw - What's wrong with brute force? Ah, the interesting analysis is lost. For example: Every number ending with a zero gives a double zero when squared and therfore must be excluded. Are there more numbers that can be excluded?

This problem is however a bagatelle. No deeper insights are needed. Excell can be a good tool.