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?

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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.. – Josh Silverman Staff · 3 years ago

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– Mathh Mathh · 3 years 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.Log in to reply

– Josh Silverman Staff · 3 years ago

Typo, you are rightLog in to reply

– Mathh Mathh · 3 years 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.Log in to reply

– Josh Silverman Staff · 3 years ago

indeed, thanks for pointing that outLog in to reply

– Victor Spirou · 3 years ago

That's BrilliantLog 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. – Ivan Koswara · 3 years ago

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@josh silverman provided more than enough justification – Victor Spirou · 3 years ago

Thanks for the clear explanationLog in to reply

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

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– Victor Spirou · 3 years ago

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

– Pankaj Joshi · 3 years ago

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

– Victor Spirou · 3 years ago

How many papers are written in India?Log in to reply

– Victor Spirou · 3 years ago

Please send link :)Log in to reply

here – Pankaj Joshi · 3 years ago

u can get themLog in to reply

– Victor Spirou · 3 years ago

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

24 – Dinesh C · 3 years ago

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– Victor Spirou · 3 years ago

I also thought that,but remembered the second hand.Log in to reply