jmc

Call $x$ the number of days Sam works and $y$ the number of days he does not. We can set up the following system of equations to represent the given information: $$\begin{array}{rl}x+y& =20\\ 60x-30y& =660\end{array}$$ The first equation represents the total number of days Sam works, and the second equation represents his total profit. Solving for $x$ in the first equation yields $x=20-y$. Substituting into the second equation gives $60(20-y)-30y=660$. Canceling a factor of $10$ and multiplying out gives $120-6y-3y=66$. This simplifies to $-9y=-54$, or $y=6$. Thus, Sam did not work for $\boxed{6}$ days.