October 7, 2010

Clock Angles

How many times are the hands of a clock at right angle
in a day?

André Meneses said...

40 times

Igor Santos said...

Twice a minute.

Secret Squïrrel said...

If you mean just the hour and minute hands, 44.

This is easily determined once you realise that the hands coincide 11 times every 12 hours, therefore 22 times a day. The hands are at right angles a little more than 15 mins before and after they coincide, so 44 in total.

ron mccuen said...

2232 times a day in a 24 hour day.
The second hand will right angle both the hour and minute hand 3 timee every minute x 60 minutes x 24 hours
Add the right angle occurence of the minute hand to hour hand = 3 times per hour, add to second hand angles =
4392 right angle occurences every 24 hours.

Secret Squïrrel said...

@Ron - a right angle looks like an 'L'. The hands are not at right angles at 6 o'clock.

All of the hands move in a repeating cycle so the number of "matches" is always one less than your intuition might tell you (this is a common error). As an example, the hour and minute hands coincide 11 (not 12) times in any 12 hour period. You know it can't be 12 times because then they would coincide exactly once every hour, and this can't be the case as the hour hand has moved 30° in that time.

To work out the number of right-angle occurrences, count the number of coincidences and double.

The second and minute hands coincide 59 times an hour which is 1,416 times in 24 hours.

The second and hour hands coincide 719 times in 12 hours, so 1,438 in a day.

Add in the minute and hours coincidences from before and you get 22 + 1416 + 1438 = 2,876 times a day. Double that for the number of right angles and you get 5,752.

Anonymous said...

I'm assuming the clock has only 2 hands, with no second hand.

Visualizing the hands moving from midnight, the first right angles of the day occur around 12:16AM and again around 12:48AM. Two more occur each hour of the morning through approximately 11:11AM and 11:48AM for a total of 24 right angles during the AM hours. The same angles repeat in the PM hours for a total of 48 in a day.

Secret Squïrrel said...

@Anonymous - no, you have also over-counted. There are certainly not two right-angles each and every hour. They creep around the clock-face; eg 12:48 -> 1:54 -> 3:00 -> 4:06, etc. See? - the 2:xx one was "missed".

If there really were 48 occurrences every 24 hours, then they would have to be exactly 30 minutes apart which is nonsense.