pqr is a three-digit natural number such that $\mathrm{pqr}={\mathrm{p}}^3+{\mathrm{q}}^3+{\mathrm{r}}^3.$What is the value of r?

If $\mathrm{Q}=0$, then $\mathrm{R}=1$;

If $\mathrm{R}=0$, then $\mathrm{P}=\mathrm{S}$;

If $\mathrm{S}=0$, then$\mathrm{P}=1$;

In a certain system of writing of numbers, we write $(12\mathrm{\Phi }{)}_4=(124{)}_{\mathrm{\Phi }}$, where $\mathrm{\Phi }$ is the units digit in the first place and number of digits in that system in the second place.

How many values of $\mathrm{\Phi }$ are possible?