Chris Mercer informed me that in Germany there are calendars made of wooden blocks for very bright kids.  There are two six sided dice, and by choosing which face is displayed for the two of these, all possible monthly dates (from 01, 02, …, to 29, 30, 31) can be displayed; note the single digit dates must start with a 0.

Your job is to write one digit (from 0, 1, 2, …, 8, 9) on each face of the two dice so that this can be done. Good luck!