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For an $n$-sided die, the number of rolls needed, on average $n\log n$ for large n. For one die the question is here: "A Collection of Dice Problems" by Matthew M. Conroy.

What about $r$ dice?

swe
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    Is there any difference between rolling one die $r$ times and rolling $r$ dice? – Ted Shifrin May 27 '14 at 13:00
  • No essentially not, but then rolling one die r times will count as 1 throw. @TedShifrin – swe May 27 '14 at 13:03
  • @TedShifrin: If the answer for one die is $k$, I think the answer for $r$ dices will be within $[k/r,k/r+1)$, which gives $\lceil \frac{k}{r} \rceil$. – user21820 May 29 '14 at 04:16
  • The expected number $E$ of throws with one die is not an integer anyway. The answer is just $E/k$. – Ted Shifrin May 29 '14 at 04:20

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