Source: MacPOW Problem of the Week 1240 Take this math puzzle: The number 545 has the curious property that upon replacing the digit in any single position by an arbitrary digit (from 0 to 9; it can be a leading 0 or just the same digit), the result is not divisible by 11. Is there a positive in...

This puzzle is part of the Puzzling Stack Exchange Advent Calendar 2023. The accepted answer to this question will be awarded a bounty worth 50 reputation.< Previous Door Next Door > Hadn't heard from the Oracle in a while until this text came in... What is the Oracle trying to tell me?

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Source: MacPOW 1141 "MacPOW 1141: Capturing 5 integers" states: For which positive real numbers $x$ are there exactly 5 integers $n$ such that the following statement is true?$$x\lt n\lt x^2$$ I got that How I got this: although I'm not sure if my solution is correct. So my question is: Is my...