mathschallenge.net

Problem

The angle between the hour hand and the minute hand is 90^o at 9 o"clock and 3 o"clock:

How many times between 9 a.m. and 3 p.m. is the angle between the hour and minute hand 90^o?

Solution

After 9 o"clock, the next time it happens is a little after 9:30, then around 10:05, 10:35, and so on.

It can be seen that it will happen twice during each hour, at roughly the following times:

9:00, 9:33, 10:06, 10:37, 11:11, 11:43, 12:17, 12:49, 13:22, 13:54, 14:27, and 15:00.

That is, 12 times between 9 a.m. and 3 p.m.

At what time exactly does it first happen after 9 a.m.?
What about the other times?