Answer:
If you don't want the leading zero, then you only need the digit 0 on one of the cubes so you can represent 10, 20, and 30. Create one cube with (1,2,3,4,5,6) and the other with (0,1,2,7,8,9) and you can represent any date from 1 to 31 (well, really 32).

If you do want the leading zero, you need to have a 0 on both cubes to represent each date from 01 through 09. However, this presents a problem: You need 0, 1, and 2 on each cube, but that only leaves room for six more digits, and there are seven left to place (3 through 9). The trick is to note that 6 and 9 look pretty much the same when turned upside down, so you can omit one of them and turn the cube over to represent either one. So, you have cubes with (0,1,2,3,4,5) and (0,1,2,6,7,8).
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