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?

#Time #Clock

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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..

Josh Silverman Staff - 6 years, 10 months ago

That's Brilliant

Victor Spirou - 6 years, 10 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.

mathh mathh - 6 years, 10 months ago

Typo, you are right

Josh Silverman Staff - 6 years, 10 months ago

@Josh Silverman – 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.

mathh mathh - 6 years, 10 months ago

@Mathh Mathh – indeed, thanks for pointing that out

Josh Silverman Staff - 6 years, 10 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.

Ivan Koswara - 6 years, 10 months ago

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

Victor Spirou - 6 years, 10 months ago

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

Pankaj Joshi - 6 years, 10 months ago

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

Victor Spirou - 6 years, 10 months ago

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

Pankaj Joshi - 6 years, 10 months ago

@Pankaj Joshi – How many papers are written in India?

Victor Spirou - 6 years, 10 months ago

@Victor Spirou – Please send link :)

Victor Spirou - 6 years, 10 months ago

@Victor Spirou – u can get them here

Pankaj Joshi - 6 years, 10 months ago

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

Victor Spirou - 6 years, 10 months ago

dinesh c - 6 years, 10 months ago

I also thought that,but remembered the second hand.

Victor Spirou - 6 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$