Thanks to Riddles.com for this this quick clock brain teaser:
At noon and midnight the hour and minute hands are exactly coincident with each other. How many other times between noon and midnight do the hour and minute hands cross?
Will unmask answers in a day or 2
10. The first time it happens is a little bit after one, then after two, and so on until a little before eleven. After that the hands meet up again at midnight.
58
11
10
11 times. since it starts at 12:00 it will never cross again at any 12:XX but it will at all up till 11:XX. then they cross again at 12:00.
I count 10, corresponding to the crossings that occur after 1 through 10 PM. The post 11 PM crossing corresponds to the midnight crossing.
At every hour, so not including noon and midnight: 11 times
12
10 times, it does not cross at 1:00
yes, 10
1:05.454545…
2:10.909090
3:16.363636
4:21.818181
5:27.272727
6:32.727272
7:38.181818
8:43.636363
9:49.090909
10:54.54545
The answer is 10
how did you manage to work out those #’s, Ken?
The angle that the hour hand makes is t/2, where t is in minutes. The angle that the minute hand makes is 6t.
Now you can use tan(6t) = tan(t/2). Then you solve that (you can’t just set 6t=t/2 cuz it’s a trigonometric equality with an infinite number of answers). I forgot all my high-school trig, so I had to dig around the web for the method.
Anyway, you find that the hands overlap roughly every 1 hour, 5 minutes, and 27.3 seconds.
Thanks, Ken.