  The \emph{anti-van der Waerden number}, denoted by $aw([n],k)$, is the
smallest $r$ such that every exact $r$-coloring of $[n]$ contains a rainbow
$k$-term arithmetic progression. Butler et. al. showed that $\lceil \log_3 n
\rceil + 2 \le aw([n],3) \le \lceil \log_2 n \rceil + 1$, and conjectured that
there exists a constant $C$ such that $aw([n],3) \le \lceil \log_3 n \rceil +
C$. In this paper, we show this conjecture is true by determining $aw([n],3)$
for all $n$. We prove that for $7\cdot 3^{m-2}+1 \leq n \leq 21 \cdot 3^{m-2}$,
\[ aw([n],3)=\left\{\begin{array}{ll} m+2, & \mbox{if $n=3^m$}\\ m+3, &
\mbox{otherwise}.
  \end{array}\right.\]
