Think numbers not dots!

Problem: (from eon)

A simple little puzzle. Suppose you are given two ordinary 6 sided dice. Is it possible to put the numbers 0-9 (with repetition) onto the faces of both dice, such that using both dice you can display all the days of the month i.e. 01 – 31 .

My Response:

Can’t be done.

My reasoning is as follows. Each of the dice must have a 1 and 2, in order to denote 11 and 22. This leave 4 spaces on each die, which in order to denote 01 – 09, we must put a 0 on each of the dice. This leaves 3 space on each die, which if we only but the remaining numbers 4 – 9 (7 total), will not fit in the remaining six places.

Is there a more elegant proof?

I am wrong. It can be done. So can anyone figure out why I am wrong? No cheating by checking out the response by tpc.

*Related*

When the six is upside down, it is a nine. The cubes are 012345 and 012678. I’ve had a calendar for over 30 years that represents the day with two cubes so I knew it was possible.

I love it when I find a problem that I can still understand, even if it is full of cubes.

Not to mention that I couldn’t resist a Model reunion. ðŸ˜‰