 Solution

Assume no hook has length two. The last row has length one, since otherwise its penultimate box has hook length two. If two adjacent row lengths are equal, take the last two rows in the maximal block of that length; the last box in the upper of those two rows has hook length two. Thus all row lengths are distinct. If a gap ${\lambda }_i-{\lambda }_{i+1}$ were at least two, the penultimate box in row $i$ would again have hook length two, since there would be no boxes below it. Therefore every successive gap equals one and the final part equals one, proving (2) implies (3).