Consider a cubical room of edge $l$ with light bulbs which can be placed on any of the faces (imagine they are point size). For a given light bulb, the lumens produced decreases with inverse square of the distance from the bulb . Given that each bulb produces $x$ lumens, how many bulbs are required to light each point in the room to a minimum of $y$ lumen.
Source: Something I came up with
My thoughts:
If we have $n$ bulbs, and consider a point $z$ inside the cube where distances from each bulb is $\{r_i \}$ then we find that the total lumens at $z$ is :
$$L(z) = \sum_{i=1}^n \frac{x}{r_i^2}$$
I think the optimal place to place a bulb would be the center of a face, and that if $ y \leq \frac{x}{l^2}$, then obviously just one bulb is enough. But, the problem becomes a lot more complicated when $y$ exceeds $\frac{x}{l^2}$.