let $s=\{0,1,4,9,...\}$ be the set of squares. there is a unique set $r$ of nonnegative integers such that every positive integer $k$ can be written in the form $s+r (s \in s, r \in r)$ in an even number of ways. are the only even numbers in $r$ those of the form $2 n^2$? does the set $r$ have positive density? \vskip .1in \noindent the more general problem is to develop methods for describing $r$ for a wide variety of initial sets $s$. specifically, i will talk about sets $s$ that are the range of a quadratic polynomial, the thue-morse set, and random sets. i will ask more questions than i am able to provide answers for. \vskip .1in \noindent this is joint work with dennis eichhorn and joshua n. cooper.