Has anyone ever published Gosper's algorithms for adding and multiplying continued fractions? You can find his write-up on the web at http://www.tweedledum.com/rwg/cfup.htm The one-sentence abstract is a classic: "Contrary to everybody, this self contained paper will show that continued fractions are not only perfectly amenable to arithmetic, they are amenable to perfect arithmetic." Also: Is there a way to combine two rational tangles (as in Conway's "square dancing" game) so that the corresponding fractions add (in the ordinary sense or in the harmonic sense)? The obvious ways of doing this (called addition and multiplication of tangles --- see e.g. Figure 3 in http://users.ntua.gr/sofial/ratl.ps) do not have this property; e.g., if A and B are rational tangles, A+B need not be a rational tangle, nor must A+B be the same as B+A. Jim Propp