There are 24*60 minutes in a day (ignoring the imperfections of the natural world, the Earth and Sun). So there are 24*60 valid 24 hour times (excluding seconds) on a digital clock.
Each of these can be rotated 4 ways, by 0,1,2 or 3 places. For example :
In which case only 1 and 2 are valid 24 hour times.
How many of these 4*24*60 rotations are also valid minutes? (What is the fastest way to find this out?)
Also, what is the minimum generating set for all 24*60 valid 24 hours times. So can I find the minimal set of Lyndon words (the earliest 24 hour times) that generate all 24 hours times? How many of these minutes are there? So out of 24*60 minutes in a day, 1 day is actually just generated by x of these significant minutes.
(This is not a homework question)