I am trying to group numbers based on the count of digits in the square they produce, and below is the list containing the grouping that I've done so far.
1 - 3 // Numbers producing 1 digit when squared ...
4 - 9 // Numbers producing 2 digits ...
10 - 31 // Numbers producing 3 digits ...
32 - 99 // and so on ...
100 - 316
317 - 999
1000 - 3162
3163 - 9999
10000 - 31622
31623 - 99999
100000 - 316227
316228 - 999999
As one can observe, every range that produces odd digit count has a lower bound of $10^{i-1}$, while its upper bound has an observable pattern. I want to know if this can be represented by a formula or if the upper bound is a known constant such that we can use it as $C \cdot 10^{i-1}$.
The numbers producing $1$ digit when squared has squares between $1$ and $10$, which means that the numbers themselves lie between $\sqrt1$ and $\sqrt{10}$. The numbers producing two digits when squared has squares between $10$ and $100$, so they lie between $\sqrt{10}$ and $\sqrt{100}$.
Continuing this way, we see that numbers that leave $n$ digits when squared has squares between $10^{n-1}$ and $10^{n}$, so they must lie between $\sqrt{10^{n-1}}$ and $\sqrt{10^{n}}$. This is the pattern that you have picked up.