I want to know what dice to roll to get a given probability distribution(mainly normal distributions but exponential distribution would also be helpful).
I want a function $f$ so that
$$N(\mu,\sigma)\approx f(n_2, n_4, n_6,n_8,n_{10},...,n_s) $$
where $n_s$ are the number of dice with $s$ sides. I know that the central limit theorem says that the sum of stochastic variables from the same distribution approximates a normal distribution. And the sum of stochastic variables from different normal distributions gives a new normal distribution with $\mu_{sum}=\mu_1+\mu_2$ and $\sigma_{sum}²=\sigma_1²+\sigma_2²$. And with some basic statistics from this answer it gives
$$\mu=\sum\limits_{i=1}^s n_i\frac{i+1}{2}$$ and $$\sigma²=\sum\limits_{i=1}^s n_i\frac{i²-1}{12}$$.
This is how far I've gotten, I don't know how to invert this. Like for example, what function would I use to choose dice to approximate $N(60,15)$? The mean value can allways be adjusted with just adding and subtracting, the main issue is the standard deviation.
For context: I'm thinking about an idea for a pen and paper roleplaying game system where the probabilities of success are given in terms of the distribution and then the method of obtaining a stochastic variable from that distribution is up to the players. The function $f$ would give one group of methods.