Complexity of nested loops
I'm not convinced the approach translates well. In particular, the last statement would become something like "the square root of the number of boxes [...]" and that doesn't work out well. A more traditional way to approach this is:

The inner loop has complexity $\Theta(i\sqrt{j}))$. This loop is executed for $1\leq j \leq i$ (middle loop) which gives a complexity of

$\Sigma_{j=1}^i i\sqrt{j}=i\Sigma_{j=1}^i \sqrt{j}$

Evaluating $\Sigma_{j=1}^i \sqrt{j}$ is tricky. There is no closed-form formula, but we can obtain an approximation by changing the sum to an integral. Note that

$\int_0^i x^{1/2}=\frac{2}{3}i^{3/2}$

is a lower bound, while

$\int_1^{i+1} x^{1/2}=\frac{2}{3}{i+1}^{3/2}-\frac{3}{2}$

is an upper bound. We may conclude that

$\Sigma_{j=1}^i \sqrt{j}=\Theta(i^{3/2})$.

Hence the middle loop has complexity $\Theta(i^{5/2})$. For the outer loop, we have to evaluate

$\Sigma_{i=1}^n i^{5/2}$

for which there is once again no closed-form formula. By applying the same integral technique, we find that the complexity is

$\Theta(n^{7/2})$.

However, finding a closed-form formula for the exact complexity (without asymptotic notation) is impossible.