Which of the following  rods of circular cross section (given radius r and length l) each made of the same material and whose ends are maintained at the same temperature will conduct most heat

The thermal resistance is given by $\mathrm{R}=\frac{\mathrm{l}}{\mathrm{KA}}=\frac{\mathrm{l}}{{\mathrm{K\pi r}}^2}$
The heat flow will be maximum when thermal resistance is minimum.
Consider the following cases :
(a) ${\mathrm{R}}_1=\frac{{\mathrm{l}}_0}{{\mathrm{K\pi r}}_0^2}$
(b) ${\mathrm{R}}_2=\frac{{\mathrm{l}}_0}{\mathrm{K\pi }{\left(2{\mathrm{r}}_0\right)}^2}=\frac{{\mathrm{l}}_0}{4{\mathrm{K\pi r}}_0^2}$
(c)  ${\mathrm{R}}_3=\frac{2{\mathrm{l}}_0}{{\mathrm{K\pi r}}_0^2}$
(d) ${\mathrm{R}}_4=\frac{2{\mathrm{l}}_0}{\mathrm{K\pi }{\left(2{\mathrm{r}}_0\right)}^2}=\frac{2{\mathrm{l}}_0}{4{\mathrm{K\pi r}}_0^2}$
In case (b), the resistance is minimum and hence heat flow will be maximum.