It is 5:55 - is that all?

I woke up last night and noticed that, at that moment, it was 5:55. As I went back to sleep I wondered how many other times there were with all the same digits. (My digital bedside clock does not show leading zeroes. 05:55 is the ISO specified abbreviation for times, but that would not work with the rest of this question.)

How many such, valid, identical-digit times are there?

The most obvious dozy plan was to think forwards along the lines of "how about 6 something". I was awake enough to recall that 6:66 is not a recognized time, although it met the requirement. I quickly dozed off, but not before working backwards, and helped by the satisfion that I had answered the question, finding a small number of possible answers.

In the morning I realized that I had not fully answered the question. I briefly thought of several other answers, but decided life was to busy with other things to spend time finding the answer. I could ask you, dear reader, to do it for me. So ~ in summary ~

Using a typical three or four digit clock-time format how many times during the day are there with all digits are the same?

Last night my real thought was "What is the statistical chance, to the nearest minute, of waking up at such a identi-digit time? My next thought was "Yuk, stasitics, bound to be difficult. Go to sleep." Now, in the light of day, I realize that even with my elementary knowledge of stats it was too trivial a question to bother asking. Or is it? Try this ~

Working to the nearest minute what are the chances, in one day, of looking at a clock to find that all digits are the same?

 

 


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