In my recent response to Allan, I should of course have said "not just doubt, but moral certainty of failure". After some more reflection, I've come to the conclusion that Allan's "Tightness Conjecture" probably WILL fail in 3 dimensions as well as in 4, and that Jud can probably redeem himself (so to speak) by establishing this. Namely, among the many packing configurations that Sloane et al have produced with the "Gosset" program, there are probably several records that aren't (currently) achieved by any "tight" configuration. Now if Jud were to run his program for spherical caps of the appropriate radius (minus, for safety, a very small epsilon), it should prove this. It would also be of interest to run it for the case of 4-dimensional unit spheres, to see just how far short of 24 it falls. Let me just conjecture here that it won't even find 23. I'm sure the program would also find a few unforeseen things that would amply justify the trouble of writing it. How about it, Jud? JHC