Now I feel silly again for having such an ugly solution.
I do feel like we can probably generalize the super ugly solutions: it seems like we have plenty of degrees of freedom and so the challenge is about how to compactly describe a strategy rather than coping with any fundamental difficulty.
(But of course there may be some fundamental obstruction that just isn’t obvious yet.)
By the way, as a bit of trivia about the problem: I don’t know who came up with it but the person who gave the problem to me said that there is a nice solution which works for all n≥3 and only fails when n is even and everyone’s starting positions are collinear. I haven’t made much headway into finding such a solution yet.
I’m wondering how helpful it is to try cases with small n and hope a general pattern is going to emerge instead of trying to find a solution from scratch that at least works for all sufficiently large n. It’s difficult to decide which you want to spend your time doing.
Oh, I think I just found that solution in parallel (or at least I found a solution with exactly the same failure condition). It was directly inspired by PatrikN’s solution. (Although I didn’t realize until after writing it that it is literally a generalization of PatrikN’s solution.)
Now I feel silly again for having such an ugly solution.
I do feel like we can probably generalize the super ugly solutions: it seems like we have plenty of degrees of freedom and so the challenge is about how to compactly describe a strategy rather than coping with any fundamental difficulty.
(But of course there may be some fundamental obstruction that just isn’t obvious yet.)
By the way, as a bit of trivia about the problem: I don’t know who came up with it but the person who gave the problem to me said that there is a nice solution which works for all n≥3 and only fails when n is even and everyone’s starting positions are collinear. I haven’t made much headway into finding such a solution yet.
I’m wondering how helpful it is to try cases with small n and hope a general pattern is going to emerge instead of trying to find a solution from scratch that at least works for all sufficiently large n. It’s difficult to decide which you want to spend your time doing.
Oh, I think I just found that solution in parallel (or at least I found a solution with exactly the same failure condition). It was directly inspired by PatrikN’s solution. (Although I didn’t realize until after writing it that it is literally a generalization of PatrikN’s solution.)