Anne: "The minute hand takes one hour to make a complete circle. Therefore, it will pass the hour hand once every hour!"

Mike: "But by the time the minute hand has made a full circle, the hour hand will have moved 1/12th of the circle ahead. And by the time the minute hand gets to that spot, the hour hand will have moved forward yet again. This will continue indefinitely, so it is obvious that contrary to what it may seem like, the minute hand will never pass the hour hand."

In reality, how often does the minute hand pass the hour hand?

Let us assume that the current time is 12:00, so that the hour hand and the minute hand are coincident.

Let the respective number of minute marks traversed by the minute hand and the hour hand by the time of their next meeing be M and H.

Then, it follows that:

(i) M - H = 60, and:
(ii) M = 12*H, since the minute hand travels 12 times faster than the hour hand.

Solving for (i) and (ii), we have: (M, H) = (720/11, 60/11).

Thus, each successive coincidence of the minute hand and the hour hand occur every 720/11 minutes or 12/11 hours.

Consequently, the minute hand will pass the hour hand precisely 12/(12/11) = 11 times in every 12 hours.