The following question is motivated by the spraypaint tool in Microsoft Paint. In a sense, I'm asking how many clicks on average are necessary to fill each pixel within the range of the tool if we leave the cursor fixed. The following is a more precise statement of the question:
Consider $n$ dice, each with $k$ sides. A roll consists of rolling all $n$ dice. Determine $E(n,k)$, the expected number of rolls needed so that all numbers $1,\ldots,k$ have appeared on some roll.
The translation to spraypaint can be seen by assuming the range of the spraypaint tool is $k$ pixels, and each click paints $n$ random pixels.
I remember looking at this problem years ago, and only partially solving it. I believe I worked on the case $n=1$:
$$E(1,k)=kH_k=k\sum_{i=1}^k\frac{1}{i}$$
Here $H_k$ is the $k$th harmonic number. I'm not entirely convinced of this result. For $n=2$ I could only find a closed form for asymptotic behavior as $k$ got large. My memory of the solution is a bit fuzzy though.
For $n \gt 1$ you can imagine the die rolls in a batch of $n$ are sequential. Except for the last roll, the situation is the same, so $E(n,k) \approx \frac 1nE(1,k)$. It will actually be a little higher than this, as you will (in effect) keep rolling up to the next multiple of $n$, when with $n=1$ you would quit. The heuristic answer would be that on average you would waste half the rolls, so $E(n,k) \approx \frac 1nE(1,k)+\frac 12$. To do better, I think you would have to assess the probability of completion at each multiple of $n$.