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A friend asked me for the time. When I looked at my analog clock (which only has a minute hand and an hour hand, no second hand), I thought of telling her the time with the hour and minute hands swapped, and then have her work out the actual time. Then I realized that not all clock-hand configurations make sense when the hands are swapped.

Let $T$ denote the set of all clock-hand configurations on an analog clock that occur over twelve hours.

Question: What is the subset of times $S\subseteq T$ such that when the hands are swapped, the time still makes sense? (In other words, $S$ is the maximal subset of $T$ closed under the operation of swapping the hands.)

Comments: $S\neq T$, since for example $3:00\notin S$. But $S$ is non-empty, since $12:00\in S$.

geometricK
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  • I am assuming that the clock has no second hand, only an hour hand and a minute hand. Assuming so, have you done any work on the problem that you could share? – user2661923 May 12 '21 at 12:55
  • Thanks, I clarified the question. @JMoravitz: indeed, this is a repeat, I should probably close it! Thanks. – geometricK May 12 '21 at 13:04

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