Neil Sloane, OEIS Foundation

Abstract

I'll discuss problems from geometry, number theory, and the theory of computing.

1. Poonen and Rubinstein counted the intersection points in a regular n-gon with all diagonals drawn. But what if we start with n points on a line rather than a circle? (A6561, A290447).
2. Mysterious things happen when you iterate arithmetic functions, for example n -> (φ(n)+σ(n))/2. Although it is hard to believe, the orbit of 270 seems to be integral and ever-increasing (A291789). John Conway recently lost a $1000 wager on the iteration of another arithmetic function (A195264).
3. Back in the 1930s Emil Post studied "tag systems", which in general are now known to be universal Turing machines. But Post's simple 3-shift tag system is still open, 80 years later. Is this really hard (A284116)?