$\mathrm{\rho }-$Charge density of the cube

${\mathrm{V}}_{\mathcal{l}}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}}-$ Potential at the corner of a cube of side $\mathcal{l}$

${\mathrm{V}}_{\mathcal{l}}^{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{e}}-$ Potential at the centre of a cube of side $\mathcal{l}$

${\mathrm{V}}_{\mathcal{l}/2}^{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{e}}-$ Potential at the center of a cube of side $\frac{\mathcal{l}}{2}$

${\mathrm{V}}_{\mathcal{l}/2}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}}-$ Potential at the corner of a cube of side $\frac{\mathcal{l}}{2}$

By dimensional analysis

${\mathrm{V}}_{\mathcal{l}}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}}\propto \frac{\mathrm{Q}}{\mathcal{l}}=\mathrm{\rho }{\mathcal{l}}^2$

${\mathrm{V}}_{\mathcal{l}}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}}=4\mathrm{ }{\mathrm{V}}_{\mathcal{l}/2}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}}$

But by superposition ${\mathrm{V}}_{\mathcal{l}}^{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{e}}=8{\mathrm{V}}_{\mathcal{l}/2}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}}$

Because of the centre of the larger cube lies at a corner of the eight smaller cubes of which it is made

Therefore, $\frac{{\mathrm{V}}_{\mathcal{l}}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}}}{{\mathrm{V}}_{\mathcal{l}}^{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{e}}}=\frac{4{\mathrm{V}}_{\mathcal{l}/2}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}}}{8{\mathrm{V}}_{\mathcal{l}/2}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}}}=\frac{1}{2}$