JA> For anyone who wants a little puzzle to start the week: if I set z_0 = i, and z_(n+1) = 0.5 (z_n + |z_n|), what is the limit of z_n as n tends to infinity? -- James Tougher: Suppose z_(n+1) = z_n + |z_n| - sqrt(z_n |z_n|). Why is z_∞ = 1/3? What is it for general z_0? Spoiler below. JA> Suppose we have a uniform spherical planet, of density p and radius R, and we drill a straight hole from one point to its antipode (neglecting any changes in the gravitational force that this will entail). Then, we drop a point mass from one of the ends, and it will travel through the hole. It is simple enough to show that the particle will oscillate with SHM, and the half-period of this motion will be sqrt(3pi/4Gp). Has anyone pointed out that this is the ground-level orbital period, so a zero-altitude satellite in polar orbit will synchronize latitudes with the point mass? --rwg Spoiler: (2Re(z)+|z|)/3. This HAD to be on a Putnam.