I'm trying to figure out the question in the title for a project that I'm working on.

My goal is to find a configuration of five integer points on the plane, where we can overlap any pair of them without it covering the other three points.

For $n=2$ it's trivial, for $n=3$ I believe any acute triangle works.

$n=4$ is slightly trickier, but still easily solvable:

Unfortunately $n=5$ is exactly where I'm stuck on. From some doodling around it seems impossible, but maybe I'm missing something. It feels like any configuration of four will be equivalent to the one above, and I don't think there's any way to add another point to it, but I can't tell for sure.

If it *is* possible, what would be the "smallest" configuration that satisfies this restriction? In other words, the one with the smallest total bounding box.

In case it is not possible, a proof would be nice for closure, but if there's some "dumb" way to brute-force this it'd also be ok.