At what time, exactly, do the two hands of a clock overlap for the first time after 12:00 noon?

Let

*d*= the distance between numbers (on the arc). It’s clear that the hands will cross at a few minutes after 1:00, let’s say at*d*+*x*.Now, the minute hand (the big one) has gone the same fraction of the whole circumference, namely 12

*d*, that the hour hand has gone the distance between 1 and 2.That is

d + x |
= | x |

12d |
d |

*d*(*d* + *x*) = 12*dx*

*d*^{2} + *dx* = 12*dx*

*d*^{2} – 11*dx* = 0

*d*(*d* – 11*x*) = 0

*x* = *d*/11.

So the hands cross at exactly 1/11 of an hour after 1:00.

But 1/11 of 3600 seconds is 327 3/11 seconds, or 5 minutes, 27 3/11 seconds.

So, the hands cross at 1:05:27 3/11.

(Now, what about the next time after that? and the next? And, uh-oh? Does this suggest another way to go at the problem?)