Let the die be symmetric and have n sides. As you start and keep rolling it you progress step by step, from having seen k-1 top sides to having seen k top sides, k running from 1 to n. The step from k-1 to k takes, on the average, n/(n-k+1) rolls. The total number of rolls required to having seen all the n top sides is, on the average, nx[1/n+1/(n-1)+1/(n-2)+...+1)].
For n=20 the sum is almost exactly 72. Hence the die is a regular icosahedron!