My 4-D intuition is certainly no good, so I was wondering if what seems would work in 3D would carry over to 4D (about whether or not it is sufficiently general for a ball to touch the maximum number of other balls). But let me ask this about the "continuum", and I'll use the 3D analogy.

Suppose you've got a few balls touching the central one and you're finding the places where you can add another one. The simplest approach would be to find all of the places where it touches the central ball and two others, but that isn't sufficiently general.

No, it isn't. Your problem is not geometry, it's logic. You can't assume that the new ball will touch more than one of the previously placed balls (and you can't even assume _that_ just by saying so, you have to prove it--see below).

Consider the new ball touching the center ball and is in contact with ball A.

You can't assume that. It might be in touch with just one of the previously-placed balls, and you can't even guarantee which one.

Swing it one way (staying in contact with ball A and the central ball) until it touches ball B.

But your "swinging" it is not general, because it might interfere with the further placement of other balls. If you interfere with future placements, your solution is not general, even in two dimensions. The way you can guarantee that each ball touches at least two others, one of them previously placed, is this way: Hypothesize a complete kissing arrangement of N>2 spheres. I will prove there is a connected kissing arrangement in which every ball touches at least two others, by modifying the given arrangement: 1. If the arrangement is not connected, you can move one connected component as a unit until it touches another ball. 2. If ball A touches only ball B, then ball B must touch some ball C (otherwise case 1 would hold). You can swing ball A around B until A hits some other ball-- perhaps not C, but some ball. This modification increases the number of adjacencies by 1, so if you repeat it as long as there is a ball touching fewer than two others, it will terminate in at most (n-1)(n-2)/2 + 2 steps, because with that many adjacencies every ball must touch at least two others, QED. This _allows_ you to search for arrangements by placing each new ball so it touches one previous ball somewhere. If you further restrict your choice. you invalidate the entire effort, since you might not find an arrangement even if one exists. I looked at trying to modify this for four dimensions, and I find that you can prove that every ball must touch at least two others. Same in K dimensions: every ball must touch at least two others. You might have all N balls in a loop, so you have to guess each ball but the first, second, and last with K-1 degrees of freedom. You can't do better with "geometric intuition". John was right, of course. Dan