This is a problem I encountered on an exam. I understand the problem and what needs to be done but I cannot figure out a way to solve it systematically other than brute forcing. It deals with physics but the fundamental problem is mathematical.
Consider four point charges $2q, 2\sqrt{2}q, 4q, 4q$ and this grid:
The task is to place all four charges in the intersection points in such a way so that the force on the central charge $Q$ is $\vec{0}$. There can only be one charge per intersection point. The force between two charges are given by Coloumb’s law: $$F = \frac{q_1q_2}{4\pi\epsilon_0}\frac{\hat{r}}{r^2},$$ where $\hat{r}$ is the normalized direction vector and $r$ the distance between the charges. Thus the problem simplifies to finding $r_i, \hat{r}_i, \ i=1..4$ such that $$\frac{Qq}{4\pi\epsilon_0} \left( \frac{2\hat{r}_1}{r_1^2}+ \frac{2\sqrt{2}\hat{r}_2}{r_2^2}+ \frac{4\hat{r}_3}{r_3^2}+ \frac{4\hat{r}_4}{r_4^2} \right) = 0 \iff \frac{2\hat{r}_1}{r_1^2}+ \frac{2\sqrt{2}\hat{r}_2}{r_2^2}+ \frac{4\hat{r}_3}{r_3^2}+ \frac{4\hat{r}_4}{r_4^2} = 0. $$
How would I go about systematically solving this problem, i.e. not just testing different combinations (which I tried but failed)?