Square root of a six-digit number in your head

Describe how you would find the square root of a six-digit number mentally.

The root has three digits; bracket it, then refine. Use the first three digits to find the hundreds and tens digits (e.g. $N=556{,}516$: $740^2=547{,}600$ and $750^2=562{,}500$, so the root is in the 740s). If $N$ is a perfect square, the last digit narrows the units digit to two options ($6$ means $4$ or $6$), and comparing with $745^2=555{,}025$ picks $746$. Otherwise refine with Newton's step $\sqrt N\approx a+\tfrac{N-a^2}{2a}$.