Abstract problem, any known solutions?

I don't think I have enough information to go on (mostly it's not clear what's being distributed (liquid? powder? pellets?), how quickly and easily it can flow around your cylindrical distributor (which I assume is really an annulus), what the cost is if you don't have good distribution (presumably product on the floor, a big mess, and lost raw materials -- but how soon does this happen compared to when your downstream operations run out of material?).

But if it's any consolation, from the way you frame the problem I don't know that I could do anything that's not clunky.

Do you have a good vision of how to do it manually?

Thanks Tim. That overcomplicates the problem somewhat, it's just about achieving mathematical symmetry around the circle. I've already done a reasonable job with scrappy code as I mentioned. There are some extensions if you can make a prediction of future states, for example, in the 8-case, if you have 1,5 and add one, minimum imbalance (as in moments) is achieved by adding, eg. 2, but if you know that you're going to add another one shortly, 3 would be better (followed by 7). That would make my code even more messy, I just wondered whether anyone somewhere has a 'book proof'. I know it's a long shot.

I guess I'm just not visualizing what you're trying to do, and thus can't quite figure out what you need. The reason I was asking about material is because if you're dumping in some viscous fluid, you could in theory service the #4 outlet with the #1 inlet, just with a rate limitation. But if you're dumping in sand, then you can't, period.

Without knowing the in/out relationship, I think there's not algorithm to be had. And I don't see how to see the in/out relationship without knowing the material characteristics.

But -- clunky but working is a lot better than an elegant failure!