Saturday: probability

So I have six dice. How many throws will it take to get a 1-2-3-4-5-6? Any die can be any number. The first roll all six dice are rolled and for subsequent rolls only the "double numbers" are rolled.

For example, the first roll ends up with a 1-1-4-5-6-6. Two dice are rerolled, getting 1-3-4-5-5-6. One die is rerolled, getting 1-1-3-4-5-6. One die is rerolled, getting 1-2-3-4-5-6, for a total of four throws.

I tried figuring out how I would do this mathematically, but couldn't figure it out. So I did it the empirical way:

median: 7.5

mode: 4

total repetitions: 184