Gaussian factory problem: given an iid sequence $x_i \sim \mathcal{N}(\mu,\sigma^2)$, $i=1,2,\dots$, with $\mu$ and $\sigma^2$ both unknown, construct a realization $y \sim \mathcal{N}(0,1)$.
Let $z_1 = x_1  x_2$, and $z_2 = x_3  x_4$, then $z_1, z_2 \sim \mathcal{N}(0,2\sigma^2)$.
Let $z = z_1 / z_2$. By results about ratios of Normal variates $z \sim \textrm{Cauchy}()$, i.e. is a standard Cauchy random variable.
Let $u = \frac{1}{\pi} \textrm{arctan}(z) + \frac{1}{2}$, the CDF of the standard Cauchy. Therefore $u \sim \textrm{U}([0,1])$.
Let $y = \sqrt{2} \,\textrm{erf}^{1}(2u  1)$, using the quantile function of the standard Normal.
We have $y \sim \mathcal{N}(0,1)$.

1$\begingroup$ You can even make do with only $x_1, x_2, x_3$, since $(x_1x_2)/(x_1x_3)$ already has a nontrivial distribution which does not depend on $\mu$ and $\sigma$. $\endgroup$ – James Martin Dec 19 '18 at 11:19