Q: For a calendar, with two six sided dice make all possible monthly dates using two digits (from 01 to 31) by writing one digit (from 0, 1, 2, …, 8, 9) on each face of the two dice.

A: It does not seem like it can be done — but…the Eureka part of the answer is:

the 9 is the 6 upside and vice versa!

the 9 is the 6 upside and vice versa!

Dice 1: 0 1 2 4 5 7 and Dice 2: 0 1 2 3 8 9 (9 is also 6)

or

Dice 1 : 0 1 2 3 4 5 and Dice 2 : 0 1 2 6 7 8 (6 is also 9)

or

Dice 1 : 0 1 2 3 4 5 and Dice 2 : 0 1 2 6 7 8 (6 is also 9)