ERRATUM:
Using numerical methods for numerical integration of the standard equation for gravitational force (1/r^2), you'll observe particles being slingshotted way out of the gravity well of the entire system, as they have the ability to "clip into" a spot deep in the 1/r^2 curve for a single timestep, and thus be accelerated enormously. Therefore, this video was rendered using the force equation f=1/(r^2+epsilon). I expected that the epsilon I chose was sufficiently small that it wouldn't change the diagrams, but that seems not to be the case. Most notably, in the 2-planet case, it turns out that with no such epsilon term, you should expect to get two perfect half-plane attraction basins...