Self proclaimed islands

$\begin{cases} \lfloor \sqrt{n} \rfloor =1 &\mbox{if } 1 \leq n < 4 \text{ for a total of 3 cases} \\ \lfloor \sqrt{n} \rfloor = 2 & \mbox{if } 4\leq n < 9 \text{ for a total of 5 cases} \\ \lfloor \sqrt{n} \rfloor = 3 & \mbox{if } 9\leq n < 16 \text{ for a total of 7 cases} \end{cases}$

...and so on up until $n=100$ where have only $1$ case and $\lfloor \sqrt{n} \rfloor = 10$ .

Therefore, we can rewrite $\sum_{n=1}^{100} \lfloor \sqrt n \rfloor$ as $10+\sum_{n=1}^{9} (2n+1)n = \\ 10+2\sum_{n=1}^{9} n^2+\sum_{n=1}^{9} n = \\ 10+570+45 = \\ \boxed{625}$ .