South African Mathematics Olympiad

It is easy to see that for $n=1,2,\dots ,20$ there are $n$ equally probable ways to obtain a total of $n+1$. (One die shows any number $x$ between $1$ and $n$, and the other die shows $n+1-x$.) In particular, a total of $21$ can be obtained with $20$ different throws. Beyond that, the number of possibilities decreases again from $19$ down to $1$. Thus the most probable total is $21$.