Heat transferred by **conduction** an be calculated using the equation

#Q/t = (k * A * (T_"hot" - T_"cold"))/d#, where

#Q# - heat transferred;

#t# - the time needed to transfer #Q# heat;

#k# - the *thermal conductivity* of the material;

#A# - the surface of the material;

#d# - the *thickness* of the material;

#T_"hot"#, #T_"cold"# - the higher temperature and lower temperature, respectively.

So, you know that your rods are built from the same material, which means that #k# is the same for all four rods. Moreover, the temperature difference between the reservoirs is the *same* for all four rods.

Assuming that you measure the heat transferred in the same time interval for all four rods, you can write

#Q = underbrace(k * (T_"hot" - T_"cold") * t)_(color(blue)("constant")) * A/d#

Therefore,

#Qprop A/d#

So, in your case, the rods are *cylinders* with radius #r# and length #l#. This means that the area and the thickness of a rod will be

#A = pi * r^2#

#d = l#

This means that you have

#Q prop (pi * r^2)/l <=> Q prop r^2/l# #-># since #pi# is constant as well.

Therefore, the rod that has the largest #r^2/l# ratio will conduct more heat, given that all other parameters are identical for all the four rods.