It would be fun to have a puzzle "Find the only 10-digit number having each digit once AND property xyz", but offhand I don't know any "simple" xyz. "Is a square" gives far too many results, other powers don't work (up to 10). "Is prime" of course is an epic fail. Do you know any puzzle of this type? "Smallest/Largest such" is cheating :-)
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Here is an example of what you are looking for
Find the only 10-digit number having each digit once such that the number formed by the first $n$ digits is divisible by $n$, for each $n$ from $1$ to $10$.
I'll leave it to the reader to find it for themselves.
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$\begingroup$ This is the example I immediately thought of, too. $\endgroup$ Mar 23 at 23:28
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$\begingroup$ Now that you say, I immediately remember it :-) $\endgroup$ Mar 24 at 11:39
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1$\begingroup$ Meta-Problem: Does this only work in base 10? I hacked together a Python code, no or more than 1 solution in all bases<10 (except unary or binary) or 11. 12 is already too slow for me. $\endgroup$ Mar 24 at 19:59
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1$\begingroup$ @HaukeReddmann Some conditions on possible solutions in other bases b. b must be even for any solutions to exist. Odd digits must go in odd places and even digits in even places. The b/2th place is always b/2. $\endgroup$ Mar 25 at 0:31