thermodynamics - Why can the entropy of an isolated system increase?

From the second law of thermodynamics:

The second law of thermodynamics states that the entropy of an isolated system never decreases, because isolated systems always evolve toward thermodynamic equilibrium, a state with maximum entropy.

Now I understand why the entropy can't decrease, but I fail to understand why the entropy tends to increase as the system reach the thermodynamic equilibrium. Since an isolated system can't exchange work and heat with the external environment, and the entropy of a system is the difference of heat divided for the temperature, since the total heat of a system will always be the same for it doesn't receive heat from the external environment, it's natural for me to think that difference of entropy for an isolated system is always zero. Could someone explain me why I am wrong?

PS: There are many questions with a similar title, but they're not asking the same thing.

Answer

Take a room and an ice cube as an example. Let's say that the room is the isolated system. The ice will melt and the total entropy inside the room will increase. This may seem like a special case, but it's not. All what I'm really saying is that the room as whole is not at equilibrium meaning that the system is exchanging heat, etc. inside itself increasing entropy. That means that the subsystems of the whole system are increasing their entropy by exchanging heat with each other and since entropy is extensive the system as whole is increasing entropy. The cube and the room will exchange, at any infinitesimal moment, heat $Q$, so the cube will gain entropy $\frac{Q}{T_1}$, where $T_1$ is the temperature of the cube because it gained heat $Q$, and the room will loose entropy $\frac{Q}{T_2}$, where $T_2$ is the temperature of the room because it lost heat $Q$. Since $\frac{1}{T_1}>\frac{1}{T_2}$ the total change in entropy will be positive. This exchange will continue until the temperatures are equal meaning that we have reached equilibrium. If the system is at equilibrium it already has maximum entropy.