I said Partial success: A huge effort produced a tricuspid packing, but with undesirable radii. The good news is that there are three degrees of freedom in the equations. Small matter of additional effort. The main difficulty is with arnoldp(x,y,r), a 3367 term polynomial which vanishes if a circle at (x,y) of radius r is tangent to the ovoid. Unfortunately, not iff. The only way I could eliminate extraneous parameters and reduce arnoldp to three variables was to numerically approximate the coefficients. Then the elimination process (aided by Mathematica, since my copy of Macsyma was compiled for a 286 in NT) introduced huge spurious factors. The false solutions are raising hell with Newton's method, many of whose 36 equations are arnoldps. It's taking forever, and screwing up. The 3367 floating point coefficients preclude symbolically extracting the bogus factors. But DUH! With a fish dinner and a shot of chai, I just realized an obvious way for Arnold to visit Jenny Craig. If any funsters chirp up with the answer, I'll kick myself for not requesting help earlier. If not, stay tuned for more arnage. No chirpers. It so happened that when given x and y, the desired r was always the least positive real root of arnoldp(x,y,r). This is of no help to the Newton problem where x, y, and r are all unknown, but it lets you to construct a couple of hundred valid (x,y,r) triples, to which you fit successively larger polynomials until they start successfully predic(a)ting new cases. It turns out that 161 terms, of degree 10 in x and y, and 5 in r^2, do the trick. (Although I am disconcerted by the large remainder I get from dividing the old arnoldp by the new one.) The resulting hundredfold speedup in the search process led to a promising design. http://gosper.org/Dozenegger.rtf (2MB)-: But instead of running off to laser-cut fancy new Dozeneggers, I'm just making a set of pieces from thin scrap to test in the existing Arnolds, having been cooked once too often by unintended solutions. Alan Schoen will no doubt find a solution, which I pray is the unique, intended one. "This time for sure." --rwg --------------------------------- Looking for last minute shopping deals? Find them fast with Yahoo! Search.