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?

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

What answer do you get?

Note by Victor Spirou
5 years, 1 month ago

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1 vote

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

Pankaj Joshi - 5 years, 1 month ago

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

Victor Spirou - 5 years, 1 month ago

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

Pankaj Joshi - 5 years, 1 month ago

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@Pankaj Joshi How many papers are written in India?

Victor Spirou - 5 years, 1 month ago

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@Victor Spirou Please send link :)

Victor Spirou - 5 years, 1 month ago

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@Victor Spirou u can get them here

Pankaj Joshi - 5 years, 1 month ago

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

Victor Spirou - 5 years, 1 month ago

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24

dinesh c - 5 years, 1 month ago

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I also thought that,but remembered the second hand.

Victor Spirou - 5 years, 1 month ago

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

In 1212 hours, the minute hand travels 1212 times the circumference and the hour hand travels 11 time, so the minute hand will cross the hour hand 1111 times, and those crossings are exactly when the two hands point in the same direction. Just multiply this by 22 since there are 2424 hours.

Ivan Koswara - 5 years, 1 month ago

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

Victor Spirou - 5 years, 1 month ago

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The angular velocity of the minute hand and hour hand are ωm=2π hr1\omega_m = 2\pi\text{ hr}^{-1} and ωh=2π12hr1\omega_h = \frac{2\pi}{12}\text{hr}^{-1}, respectively.

The first intersection will occur when ωmt2π=ωht\omega_m t - 2\pi = \omega_h t, i.e. when t=2πωmωh1.09 hrt = \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. 24/1.0922.0224/1.09\approx 22.02 which means the crossing will occur 22 times..

Josh Silverman Staff - 5 years, 1 month ago

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Why isn't it ωmt2π=ωht\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.

mathh mathh - 5 years, 1 month ago

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Typo, you are right

Josh Silverman Staff - 5 years, 1 month ago

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@Josh Silverman That means you'll have to change the expression of tt and then we have t1.09hrt\approx 1.09 \text{hr} (1312\frac{13}{12} becomes 1211\frac{12}{11}) instead, but that doesn't change the answer.

mathh mathh - 5 years, 1 month ago

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@Mathh Mathh indeed, thanks for pointing that out

Josh Silverman Staff - 5 years, 1 month ago

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That's Brilliant

Victor Spirou - 5 years, 1 month ago

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