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I have a digital clock that shows the date and time like this: $$ \mathsf{YYYY-(M)M-(D)D\qquad (H)H:MM \; [:SS]} $$ That is, the seconds display is optional, and if the month or day or hour is single-digit, it won't display a leading zero. This is a nice year, $2013$, because all the digits are different... that hasn't happened since $1987$. What's more, if I turn off the seconds display, there are some times coming up this year that use nine of the ten digits with no repeats.

That's pretty good, but it looks like I'll have to wait a long time to see all ten digits with no repeats, and I wanted to do even better. So I custom-ordered a clock that uses base eleven (i.e., it shows the same year, month, day, hour, minute, and second as the other clock, but using base eleven -- it doesn't use $66$-minute hours or anything like that). I'm waiting patiently for that one to show all eleven digits with no repeats.

  • When's the next time (this year) we'll see nine distinct digits?
  • When's the next time we'll see all ten distinct digits?
  • When's the next time the base-eleven clock will show all eleven distinct digits?
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