The minimum number.

There is a formula you can use to calculate the probability of all faces showing on any number of dice. From that, you can easily determine the minimum number of dice you need to roll. Here is the formula in question: Just change "n" as required and it will give you the probability.

n = 10;  1 - ((6C1) *(5/6)^n - (6C2)*(4/6)^n + (6C3)*(3/6)^n - (6C4)*(2/6)^n + (6C5)*(1/6)^n).

Here are half a dozen calculations to give you an idea: 

n=10        p=0.271 812 128 5

n=11        p=0.356 206 418 6

n=12        p=0.437 815 680 6

n=13       p=0.513 858 194 0

n=14       p=0.582 845 348 9

n=15       p=0.644 212 738 6

As you can easily see that you will need to roll a minimum of 13 dice in order to get at least 50% probability that all 6 faces will show.

I'd really like a mathematician to talk through this question/answer.     

A variation on this question has come up before on this forum. The most important example is discussed here in detail: https://web2.0calc.com/questions/probability_882