Puzzles
2 December
In a week (Monday 12:01 to Monday 12:01 a week later), how many times will the minute hand of an analogue clock point in the same direction as the hour hand?
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In a 12 hour period, the hands will meet at 12:00, between 1 and 2, between 2 and 3, ..., between 9 and 10 and between 10 and 11. This is 11 times.
Therefore in a week, the hands will meet 154 times.
Ticking clock
Is there a time of day when the hands of an analogue clock (one with a second hand that moves every second instead of moving continuously) will all be 120° apart?
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The second hand will always be pointing at one of the 60 graduations. If the minute and hour hand are 120° away from the second hand they must also be pointing at one of the graduations. The minute hand will only be pointing at a graduation at zero seconds past the minute, so the second hand must be pointing at 0. Therefore the hand are either pointing at: hour: 4, minute: 8, second: 0; or hour: 8, minute: 4, second: 0. Neither of these are real times, so it is not possible
Extension
If the second hand moves continuously instead of moving every second, will there be a time when the hands of the clock are all 120° apart?
8! minutes
How many weeks are there in 8! (\(8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\)) minutes?
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$$8\times 7\times 6\times 5\times 4\times 3\times 2\times 1 \mbox{ minutes}$$
$$= 8\times 7\times 4\times 3\times 1 \mbox{ hours}$$
$$= 7\times 4\times 1 \mbox{ days}$$
$$= 4\times 1 \mbox{ weeks}$$
$$= 4 \mbox{ weeks}$$
Extension
8 is the smallest number \(n\) such that \(n!\) minutes is a whole number of weeks.
What is the smallest number \(m\) such that \(m!\) seconds is a whole number of weeks?
Burning ropes
You have two ropes and some matches. Each rope, if lit at its end, will burn for 60 minutes. But the rate of burning is not regular, so cutting a rope in half doesn't result in a burn time of 30 minutes.
How can you use the ropes to time exactly 45 minutes?
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Light one rope at both ends and the other at one end. When the first rope finishes burning, light the other end of the second rope. The second rope will finish burning 45 minutes after the start.
Extension
What are all the possible times you can measure with two ropes? How about three ropes? Four ropes? n ropes?
