Why is it that no matter what arrangement or inversion of notes are presented to me, I can quickly find the root? Is it training or is there a mathematical answer? If you play C E G or E G C or G C E, I always find the C quickly because it sounds "right". What explains this?
In each of these voicings, C is the resultant. For instance, C4–E4–G4 can be seen as the 4th, 5th and 6th overtone of C2. For, G3–C4–E4, they're the 3rd, 4th and 5th overtones of the same C2, whereas E2–G2–C3 are the 5th, 6th and 8th overtones of C1. (In 12-edo temperament, they're not exactly the overtones, but pretty close.)
So for major chords, it's really not surprising that we hear it this way. For minor chords it is a bit surprising however, because here the fundamental is not a (readily audible) resultant of all three notes, regardless of inversion. It is however (obviously) still a resultant of the fundamental and fifth, below which the minor third can then be construed as a dropped major third.