The following is taken from the 2nd round of the 2014 South African Maths Olympiad:

How many times in a 24-hour day do the hands on a 12-hour clock point in exactly the same direction?

I am unsure about the answer which they give as 22.

What answer do you get?

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

TopNewestThe angular velocity of the minute hand and hour hand are \(\omega_m = 2\pi\text{ hr}^{-1}\) and \(\omega_h = \frac{2\pi}{12}\text{hr}^{-1}\), respectively.

The first intersection will occur when \(\omega_m t - 2\pi = \omega_h t\), i.e. when \(t = \frac{2\pi}{\omega_m- \omega_h} \approx 1.09\text{ hr}\).

As soon as an intersection happens, we have the same problem again, and the next intersection will take the same amount of time as the first. \(24/1.09\approx 22.02\) which means the crossing will occur 22 times..

Log in to reply

Why isn't it \(\omega_m t - 2\pi = \omega_h t\)? I mean, the LHS shows the distance traveled by the minute hand minus one circumference and the RHS shows the path traveled by the hour hand, which is speed times time.

Log in to reply

Typo, you are right

Log in to reply

Log in to reply

Log in to reply

That's Brilliant

Log in to reply

An intuitive answer that I haven't found justification for:

In \(12\) hours, the minute hand travels \(12\) times the circumference and the hour hand travels \(1\) time, so the minute hand will cross the hour hand \(11\) times, and those crossings are exactly when the two hands point in the same direction. Just multiply this by \(2\) since there are \(24\) hours.

Log in to reply

Thanks for the clear explanation @josh silverman provided more than enough justification

Log in to reply

Do they seriously give such easy question in the S.A M.O? I could solve this even 3 years back!

Log in to reply

2nd round 2nd question but there's also a 3rd round and a camp in the junior section

Log in to reply

U must take a look at India's M.O papers at AoPS.. They are far more tougher than this!

Log in to reply

Log in to reply

Log in to reply

here

u can get themLog in to reply

Log in to reply

24

Log in to reply

I also thought that,but remembered the second hand.

Log in to reply