jmc

The largest power of $13$ which is smaller than $222$ is $13^2=169$. The largest multiple of $169$ less than $222$ is $1\cdot 169$, and $222-1\cdot 169=53$. The largest power of $13$ which is smaller than $53$ is $13$, and the largest multiple of $13$ less than $53$ is $4\cdot 13$. $53-4\cdot 13=1$. Thus, $222=1\cdot 13^2+4\cdot 13^1+1\cdot 13^0$. So $222$ in base $13$ is $\boxed{141_{13}}$.