Perfect Squares - Iterating Squares

For a given $i$, which perfect squares should you try? All $k^2$ where $k^2 \leq i$.

Solving $k^2 \leq i$ gives $k \leq \sqrt{i}$. So loop $k$ from $1$ to $\lfloor \sqrt{i} \rfloor$.

This comes from algebra: isolating the variable in an inequality. Recognizing the sequence structure (perfect squares) guides your loop design.