Can you number the faces of a pair of dice so that they can display from 01 to 31 for the days of the month?

You use a Dodecahedron for the month, numbered 1 to 12, a Tetrahedron numbred 0 to 3 for the first digit of the month, and a Decahedron for the second digit, numbered 0 to 9.

But how can you do it for a pair of 6 sided dice?

Embedded LCD screens.

Not with a pair of dice. But with a pair of numbered cubes.

Both cubes have 0,1,2,3. one has 4,5,6, the other has 7, 8, 9.

One die has 0, 1, 2, 7, 8, & 9.
For 6, turn over the 9.
The other has 0, 1, 2, 3, 4, & 5.

Comic book stores sometimes have 30 sided dice. Called a D30. You could probably buy for under $5. Amazon might have it as well.

@filmfann got it right. The tricky part is using one face for both 6 and 9. The important thing is to have two copies of 0, 1 and 2. It does not matter where the other numbers go.

I know of two ways of finding the total number of numbers that can be represented. One method is long but easy. The other is short but a little tricky.

Method 1:
The numbers 00 to 29 count for 30.

Assume the dice are numbered as @filmfann had them. In addition to having two copies of 0,1 and 2, place 3,4,5 on one die and 6/9, 7 and 8 on the other.

For numbers starting with 3, 4 or 5 there are 7 possibilities for the second number on the other die, giving 3×7=21 entries.

For the numbers 6,7,8 and 9 there are 6 entries for the second number, giving 4×6=24 entries.
The total is therefore 30 + 21 + 24=75.

Method 2:
6 numbers on one die times 7 on the other is 6×7=42. Reversing the order of the dice gives another 42 for a total of 84. We must eliminate the repetitions due to the numbers 0, 1 and 2 on both dice. They form 3×3 =9 different numbers, each of which is repeated when the order of the dice is reversed. 84 – 9 = 75.