Lighting a unit square

Formal definition

• ContainerThe unit square [0, 1]², with lower-left corner at (0, 0) and upper-right corner at (1, 1)
• SubmissionThe n light positions lights, together with a claimed minimum intensity that the verifier can rigorously verify
• ConstraintsAll lights lie inside the square; multiple lights may share a position, in which case their intensities add
• Point intensityP = {p₁,…,pₙ} is the multiset of light positions; each unit-brightness light contributes the inverse square of its distance$$I_P\left(x\right)=\sum _{i=1}^n\frac{1}{{\Vert x-p_i\Vert }^2}$$
• Darkest intensityL(P) is the continuous minimum of the intensity over the entire closed square [0, 1]²$$L\left(P\right)=\underset{x\in {\left[0,1\right]}^2}{\min }{I}_P\left(x\right)$$
• ObjectiveMaximize L(P) over all valid light configurations P; the submitted intensity must be a lower bound on L(P) that the verifier can rigorously verify$$\underset{P}{\max }L\left(P\right)$$