-1
A 3 or 4 digit number.
I am pretty sure you can figure this out on your own. Raise different numbers to the square, until you get a 4-digit result. Similary, calculate the cube of different numbers, until you get a 4-digit number. If you want the SAME number to be both a perfect square and a perfect cube, then it must be a power of 6. In that case, just experiment raising different numbers to the sixth power, until you get a 4-digit number.
There are 28706 such combinations. 5456 of these comprise three 2-digit numbers, 19008 comprise two 2-digit numbers and two 1-digit numbers, 4158 comprise one 2-digit number and four 1-digit numbers and 84 comprise six 1-digit numbers.
There are 9 1-digit numbers and 16-2 digit numbers. So a 5 digit combination is obtained as:Five 1-digit numbers and no 2-digit numbers: 126 combinationsThree 1-digit numbers and one 2-digit number: 1344 combinationsOne 1-digit numbers and two 2-digit numbers: 1080 combinationsThat makes a total of 2550 combinations. This scheme does not differentiate between {13, 24, 5} and {1, 2, 3, 4, 5}. Adjusting for that would complicate the calculation considerably and reduce the number of combinations.
It doesn't matter if you know that the largest three-digit prime number is 997 or that its cube root is 9.98998998. If you include zero in the whole numbers, the answer will be zero.
If the numbers on the cube are one through six, you have a one out of six chance of rolling any of them. If they are double-digit numbers, your chance is zero.
There is no such ratio that applies for all single-digit and double-digit integers.
-2
There are two single-digit numbers less than 10 which are cubes of whole numbers. These are 13 (one cubed = 1) and 23 (two cubed = 8).
72. (with the range of two digit numbers being from 10 to 99).
The answer will depend on how many digits there are in each of the 30 numbers. If the 30 numbers are all 6-digit numbers then the answer is NONE! If the 30 numbers are the first 30 counting numbers then there are 126 combinations of five 1-digit numbers, 1764 combinations of three 1-digit numbers and one 2-digit number, and 1710 combinations of one 1-digit number and two 2-digit numbers. That makes a total of 3600 5-digit combinations.