= 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 \b[every successive gap equals one and the final part equals one], proving (2) implies (3).
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