1. How many primes are in a clock if you regard the numbers on its face as a continuous clockwise string of digits; not to exceed 15 digits i.e. one full round?
2.Same question for a counterclockwise direction.
3.Same question for a digital watch ((HH(0-23)MM(0-59)) - in ascending order).
Probably you must make a sharp distinction between "numbers" and "digits". As Ady has stated it "if you regard the numbers on its face as a continuous clockwise [also counterclockwise] STRING OF DIGITS". Those who do not have Mathematica and/or an infinite amount of time, may wish to compare Sloane A036342 (clockwise) and A036343 (counterclockwise). Those, however regard the '10','11' and '12' at those hour points each as a single NUMBER, even though each has two DIGITS. The puzzle saying "not to exceed 15 digits, i.e. one full round" again puts the emphasis on digits rather than numbers (i.e. there are twelve numbers, but fifteen digits).
I discovered the disparity when I started making a list of the primes, and treated 10, 11, 12 as two digits each; this clearly allowed 101 (a three-digit prime) using the '10' from 10, and the first digit of 11. The Sloane lists did NOT include 101 in either of the primes list, and I noticed that they explicitly stated that 10, 11, and 12, were treated by them as numbers, not as pairs of digits.
Those who try to solve this one without lookup, may want to consider which interpretation to take. Perhaps other online sources follow the "as digits" and give lists with that criterion. For Sloane, there were only about a dozen of each direction.
I did not search for the "digital watch", since that poses a somewhat different question , depending also on how that is viewed. The puzzle may suggest a 24-hour watch since HH is given as "0-23", and the MM as "0-59". But probably these should read as "00-23" and "00-59" (the digital watch I am now wearing, though a 12-hour view with separate "AM"/"PM", shows minutes and seconds with explicit leading zeros, if we are looking for digits). The third test, using the digitals, does not indicate how many numbers/digits are the max length to be in the search strings (24 * 60 ?). Even Pythagoras would demur. As the old joke says "Time flies -- you can't do it."