A corporate businessman has two cubes on his office desk. Every day he arranges both cubes so that the front faces show the current day of the month.

Note: You can't represent the day "7" with a single cube with a side that says 7 on it. You have to use both cubes all the time. So the 7th day would be "07".

Solution:

Cube One has the following numbers: 0, 1, 2, 3, 4, 5

Cube two has the following numbers: 0, 1, 2, 6, 7, 8

The 6 doubles as a 9 when turned the other way around.

There is no day 00, but you still need the 0 on both cubes in order to make all the numbers between 01 and 09.

Alternate solutions are also possible e.g.

Cube One: 1, 2, 4, 0, 5, 6

Cube Two: 3, 1, 2, 7, 8, 0