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I assume you mean a digital clock and that it only displays hours and minutes, not seconds. I'll also assume it's set to standard AM/PM time, not 24 hour time, which would be a bit different.

The times that would display exactly three 1s are:

  • 1:11
  • 10:11
  • 11:01
  • 11:10
  • 11:12
  • 11:13
  • 11:14
  • 11:15
  • 11:16
  • 11:17
  • 11:18
  • 11:19
  • 11:21
  • 11:31
  • 11:41
  • 11:51
  • 12:11

That's 17, and they would be displayed 2x per day each, so 34 minutes in total.

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Q: How many times a day does a clock display exactly three 1's?
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