Fun Math Puzzle

GeekPress found this puzzle at Car Talk:

I'm getting old and a little absent-minded, so my friends got together and bought me a stylish little desk calendar. It's a cradle for two cubes, each with one number per face.

They figured I probably had enough left in me to figure what year it was and what month it was, but the date was going to elude me. So, this little gift was going to show the date. So, for example, if it were the 21st, I'd rotate one cube until a "2" was showing, and the other would show a "1". The next day I would know to rotate one cube so, together, the two cubes would read "22".

With the two cubes, I was able to express every date. For example, if it were the 2nd of the month, it would be expressed as "02". If it were the 18th you'd put up a 1 and an 8, and so on.

Here's my question. If you were designing the cubes, what numbers would you paint on each one so you could express all the dates from "01" to "31"?

My answer (and reasoning) below the fold.

One cube: 0,1,2,3,4,5

Each cube has six sides, giving a possible 12 numerals. The only repeated digits in a day's date are 1 and 2. (i.e., the 11th and 22nd of a given month -- there's no 33rd). So each cube must have a 1 and a 2. That leaves 10 possible cube sides to list the numerals 0 through 9. I just listed them sequentially on the first cube (i.e., 0,...,3,4,5) and then continued the sequence (6,7,8,9) on the second cube. Easy.

Update: Owlish points out a flaw in my reasoning. Not so easy after all! (Never do math while drinking ale). Not only would the third, fourth, and fifth days be just 3, 4, and 5, but there's no way to express "30" under my solution. Back to the drawing board...

Update 2: I had to cheat and Googled the answer here.

The answer is: 0,1,2,3,4,5 and 0,1,2,7,8,9(6)

Yes, the "9" doubles as a six. Very clever.