I've been having some email problems, but despite them I've been following the discussion of kissing numbers and how hard it is to establish them. In particular, I'm wondering whether we may not be missing some possible low-hanging fruit that might result from the sort of greedy packing procedure that Jud McCranie sketched, and about which John Conway expressed skepticism. The central problem is the size of the configuration space that must be explored: 24 or 25 spheres jostling around, each contributing 3 degrees of freedom to the configuration space. JM proposed, if I read him right, to prune this space by considering only configurations of a certain special kind, in which the configuration may be built by adding each new sphere in the 'socket' formed by three previously-placed spheres. (This is hard to visualize, but in four-space, you need three other spheres to form a well-defined socket with no degrees of freedom.) Enumerating such configurations would be much easier than searching an enormous 75-dimensional configuration space. JHC expresses doubt that this program could work, because of the possibility of losing the baby with the bathwater. It might be possible for 25 non-overlapping 4-spheres to kiss a central one, but only in configurations that are _not_ in the restricted class JM proposes. Let us call these JM-type configurations _tight_ configurations; forgive me if I do not give a rigorous definition -- I'll be willing to ante up later, but for the moment I think we all know what we're talking about. Obviously, the JM program depends on an unstated conjecture, one that JHC believes to be false. If forced to place bets at this point, I'd wager that JHC is right ... unless JHC offered me attractive odds. Because I'm not yet morally certain he's right. The conjecture, which I will call the Tightening Conjecture, is: if there exists ANY configuration of n non-overlapping radius-R k-spheres kissing a given unit k-sphere, then there exists a TIGHT such configuration. JHC puts forward what I believe is intended to be a counterexample to this conjecture, but I claim it isn't really a counterexample. In the case he presents, k=3, n=8, and R=(sqrt3+1)/2. He points out that it works to position each of the 8 peripheral spheres at the corners of a cube of side 2R, whose center is the center of the unit sphere. This configuration is _not_ tight, in the sense given above, and JHC implies by his use of the phrase "the correct configuration" that it is the only one. If this were true, it would certainly follow that a search confined to tight configurations would fail to find one, and if we believed the Tightening Conjecture we would incorrectly conclude that the 8 peripheral spheres simply would not fit. However, the configuration described by JHC is _not_ the only one that works. Begin with the cubical configuration described by JHC. Run a plane through the center of the central sphere, so that all eight peripheral spheres are tangent to that plane. Now it is clear that four peripheral spheres on one side of the plane (call it the north side) can be rotated independently of the other four, and after they have been rotated 45 degrees, they can move south a little bit, across the equator, and settle into the four northward-facing sockets offered between the four southern spheres. The resulting crownlike configuration still has one degree of freedom, but I am reasonably sure that this extra flex can be expunged by puckering the square formed by the four southern spheres. Eventually something will bump, at which point the configuration will be tight. The Tightening Conjecture may still be false, but JHC's cubical example does not kill it outright. My guess is that JHC has a stronger counterexample in his arsenal, that _will_ disprove the TC; my only real point in this message is that the offered example doesn't do the job. -A