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.