Converse of $(A\rightarrow B)\rightarrow((B\rightarrow C)\rightarrow(A\rightarrow C))$

To see how I found the assignment that falsifies the converse, $$\Bigl((B\to C)\to (A\to C)\Bigr)\to (A\to B),$$ recall that an implication $P\to Q$ is false when $P$ is true and $Q$ is false. So to (try to) falsify this implication, we want $A\to B$ to be false, which means that we must have $A$ true and $B$ false.

Once you have $A$ true and $B$ false, the implication $(B\to C)$ will be true regardless of the truth value of $C$, so to make the antecedent true we just need $A\to C$ to be true as well, which we can achieve by letting $C$ be true. Thus, $A$ and $C$ true and $B$ false will make this proposition formula false: the consequent $A\to B$ is false; the antecedent is true because its consequent $A\to C$ is true (because its consequent $C$ is true).