I had never heard of Vi Hart before. [...]
Thank you for broadening my horizons.
My pleasure! I love her videos, along with SmarterEveryDay.
Visualize an analog 12-hour clock with hour hand, minute hand, second hand. We know that at 12:00:00 all three hands are in perfect mathematically defined alignment. At what other times if any (such as five minutes and a few seconds after one, or maybe 21 or 22 minutes and a few seconds after four) are all three hands in perfect alignment. Prove (or disprove) it.
The simple way to do it is to get an analog clock, and wind the hour hand around the entire face, taking note of the times when they overlap.
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