  We show that for any set of n distinct points in the complex plane, there
exists a polynomial p of degree at most n+1 so that the corresponding Newton
map, or even the relaxed Newton map, for p has the given points as a
super-attracting cycle. This improves the result due to Plaza and Romero
(2011), which shows how to find such a polynomial of degree 2n. Moreover we
show that in general one cannot improve upon degree n+1. Our methods allow us
to give a simple, constructive proof of the known result that for each cycle
length n at least 2 and degree d at least 3, there exists a polynomial of
degree d whose Newton map has a super-attracting cycle of length n.
