Suppose there are $n$ extremely paranoid, vulnerable mental patients at a hospital.
Each day at the lunch hour, they move around like frictionless billiard balls of radius $\rho$ metres in a square, padded cell (of side length $s$ metres), and each day, at the start of the hour, they each resume their wandering from last time as if nothing happened.
At the first instance of this strange behaviour, the patients were positioned in a specific pattern and each began traveling at a specific velocity.
Each patient is severely short-sighted and can only see $r$ metres ahead of them, all within an angle $\theta$ radians either side of their noses (with the centres in the middle of their respective bodies).
They know about each other's vision and always point their noses where they're going.
The nurses at the hospital are hopelessly inept. Whenever a patient sees a total of $m$ other patients looking at them in a given lunch hour, that patient will commit suicide that night (and thus can no longer resume their wandering the next day and are removed).
For what configurations of $(n, \rho, s, r, \theta, m)$ and positions of patients & velocities on the first day can we prevent the suicides?
This is a mathematics problem I made up in 2013 when overthinking the blue-eyed islanders puzzle. I put my question up as a Facebook status and worked on it for about a week until my undergraduate degree commitments made me leave it, so I forgot about it until I saw it in the "On This Day" feature of Facebook today.
I hope it won't be answered by appealing to how silly it is. Feel free to make simplifying assumptions.
The answer is clear if $n=1$ with $m>0$, say, or if $m\le n$, $\theta=\pi$, $r=s$, etc. I think there could be some more interesting results however.
I don't think it is something I can answer myself in any depth.
NB: I take suicide and mental health issues seriously. Please seek help if you're suffering; for instance, call Samaritans for free on 116 123 via any phone if you're in the UK. I mean no offense.