# Around the clock.

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?

Note by Victor Spirou
3 years, 11 months ago

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The 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..

Staff - 3 years, 11 months ago

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.

- 3 years, 11 months ago

Typo, you are right

Staff - 3 years, 11 months ago

That means you'll have to change the expression of $$t$$ and then we have $$t\approx 1.09 \text{hr}$$ ($$\frac{13}{12}$$ becomes $$\frac{12}{11}$$) instead, but that doesn't change the answer.

- 3 years, 11 months ago

indeed, thanks for pointing that out

Staff - 3 years, 11 months ago

That's Brilliant

- 3 years, 11 months ago

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.

- 3 years, 11 months ago

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

- 3 years, 11 months ago

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

- 3 years, 11 months ago

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

- 3 years, 11 months ago

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

- 3 years, 11 months ago

How many papers are written in India?

- 3 years, 11 months ago

- 3 years, 11 months ago

u can get them here

- 3 years, 11 months ago

I feel you may not know there are 20 questions in the SAMO. I saw most of yours have 6

- 3 years, 11 months ago

24

- 3 years, 11 months ago

I also thought that,but remembered the second hand.

- 3 years, 11 months ago