Berry-Esseen Bound for a Sum of Uniform Random Variables

Let $U_1, \ldots, U_n$ be i.i.d.\ $\mathrm{Uniform}(0,1)$ and $S_n = \sum_{i=1}^n U_i$. The Berry-Esseen theorem states $$\sup_x \left|P\!\left(\frac{S_n - n/2}{\sigma\sqrt{n}} \le x\right) - \Phi(x)\right| \le \frac{C\,\rho}{\sigma^3 \sqrt{n}},$$ where $\sigma^2 = \mathrm{Var}(U_i)$, $\rho = E[|U_i - 1/2|^3]$, and $C \le 0.4748$.

(a) Compute $\rho = E[|U_i - 1/2|^3]$ exactly.

(b) Evaluate the Berry-Esseen bound for $n = 50$.

(c) How large must $n$ be for the bound to guarantee the CLT error is below $0.01$?