Dan writes ...

It has been known for some time that K(4) must be 24 or 25. An arrangement for 24 has been known for a long time. The most symmetrical such arrangment of 24 4-balls all touching a 25th is described by assuming the 25th 4-ball to be centered at the origin, and the 24 "kissing" 4-balls to be centered at the 24 points having [2 coordinates = +-sqrt(1/2) and the other 2 coordinates = 0].

This is a rotation of the 24 unit Hurwitz quaternions: The 8 usual +-{1,i,j,k} and the sixteen (+-1 +-i +-j +-k)/2. JHC remarks that he wouldn't trust a numerical geometric proof. I'll ask: Why not? It's a packing problem, and computer calculations might well provide a proof. I assume even John accepts computer searches for error correcting codes and similar packing problems as valid. --- I recently read the "popular" book Prime Obsession. It's an attempt to explain the Riemann Hypothesis to a layman. I've called it "popular" since I found it prominently displayed at Borders. I enjoyed it, particularly the historical chapters. (The author alternates history chapters with math chapters.) I haven't seen a serious attempt to explain complex variables to a non-mathematical audience in forever. I'm not in a position to say how well the explanation works, but it seems to be a good try. The graphs are good. There are a couple of typos. Only a few proof sketches. I recommend the book for the historical background. It will also serve as a very gentle introduction to RH for the interested funster, although many will find the math a bit thin. --- Jeff Lagarias has recently released a 50-page annotated bibliography for the 3N+1 probem. http://arXiv.org/abs/math/0309224 I was amazed that so much has been written about it. The conjecture has been checked to 2.5 x 10^17 by an ongoing distributed computing project at http://personal.computrain.nl/eric/wondrous. The lower bound for the size of a hypothetical new loop is now 630 million. Rich rcs@cs.arizona.edu