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2. I.e.: could the gentlemen circle a rectangular table in a clockwise fashion, then rearrange themselves and continue in another fashion such that given any number of men, every man would be paired with every other man in the smallest number of iterations and without pairing two men together twice. Imagine a long table with a seat at one end and $\frac$ seats along each long side. After each round, each person moves one seat clockwise. This gets us N-1 rounds in the even case, which is optimal. It's a little complex, but essentially you split the room into 2 parts, and then have 1/2 the room meet the other 1/2.