Boundary Values & Fourier Series · Challenge problem

A rod $0 \le x \le \pi$ obeys $u_t = u_{xx}$ with both ends held at $0$ and starts at $u(x, 0) = \sin x + \sin 2x$. At first the hottest point of the rod lies left of $x = \dfrac\pi3$, and as the rod cools it drifts toward the middle. At what time $t$ is the hottest point exactly at $x = \dfrac\pi3$?