Punctually at midnight or noon, both hands of the clock are more than an hour 12. An hour later, the hour hand will stop on the number 1, and the minute hand will be above the number 12. When the minute hand reaches the number 1, the hour hand will move forward by 5/12 minute graduation; when the minute hand has reached this point (after 5 i 5/12 min. from the beginning of the hour), the hour hand will move farther again - you can continue this way forever.
So, actually, the minute hand, "Basically” and "theoretically" - it should not overtake or even catch up with the hour hand!
How to explain this paradox?
In this race of clues, similar to Achilles' race with the turtle, the whole thing is this, that successive shifts of the minute hand give an infinitely decreasing geometric series, namely
Since, As you know, the sum of an infinitely decreasing geometric series is given by the formula
so at an hour 1 minutes 5 i 5/11 the clues will come together for the first time on this day, counting from the south or from the north.
But here is one more small confirmation of this argument: Let's say, that the minute hand will catch up with the hour hand in x minutes after the hour 1. Way, which the hour hand will pass during this time, is obviously equal to x / 12. Corner, who will circle the minute hand”, is about 5 minutes greater than the angle, which will pass the "hour". Hence