Produce by hand a proof of  ((A→B)→C) → (A→(B→C)) How many lines does the most natural box proof contain?

Let us first find the truth table of $(A\to B)\to C$ and $A\to (B\to C)$.

For $\mathbf{(}\mathit{A}\mathbf{\to }\mathit{B}\mathbf{)}\mathbf{\to }\mathit{C}$:

And for $\mathit{A}\mathbf{\to }\mathbf{(}\mathit{B}\mathbf{\to }\mathit{C}\mathbf{)}$:

It can be observed that $\mathbf{(}\mathit{A}\mathbf{\to }\mathit{B}\mathbf{)}\mathbf{\to }\mathit{C}$ and $\mathit{A}\mathbf{\to }\mathbf{(}\mathit{B}\mathbf{\to }\mathit{C}\mathbf{)}$ are not equivalent. So, the proof by truth table will be more convenient.