A person who left home between $4$ p.m. and $5$ p.m. returned between $5$ p.m. and $6$ p.m. and found that the hands of his watch has exactly changed places. When did he go out?
My attempt: The dial of a clock is divided into $60$ equal divisions. In one hour, the minute hand makes one complete revolution, i.e., it moves through $60$ divisions and the hour hand moves through $5$ divisions. Suppose, when the man went out, the hour hand was $x$ divisions ahead of the point labeled $12$ on the dial, where $20 < x < 25$ (as he went out between $4$ p.m. and $5$ p.m.). Also suppose, when the man returned, the hour hand was $y$ divisions ahead of zero mark and $25 < y < 30$. Since the minute hand and hour hand exactly interchanged places during the interval that the man was out, the minute hand was at y when he went out and at $x$ when he returned. Since the minute hand moves $12$ times as fast as the hour hand, we can say that the angle that the minute hand sweeps will be $12$ times that swept by the hour hand. But, I am not able to express this in terms of $x$ and $y$.
Any constructive hint is appreciated.