jmc

We focus on the portion of the floor that is left unpainted. After one day, $\frac{1}{2}$ of the floor is left unpainted. After two days, $\frac{1}{2^2}$ of the floor is left unpainted, and so on. After $n$ days, $\frac{1}{2^n}$ of the floor is unpainted. The floor has area $15^2=225$ square feet, so we look for the least number of days $n$ when at most $\frac{1}{225}$ of the floor is left unpainted: $$\begin{array}{rl}\frac{1}{2^n}& \le \frac{1}{225}\\ \Rightarrow 2^n& \ge 225\\ \Rightarrow n& \ge 8.\end{array}$$ Thus, Zeno took $\boxed{8}$ days to paint the floor.