When the clock strikes 12, the hour and minute hands exactly line up with each other. Are there other times when this handshake happens? We could say six-thirty, but nope, the hour hand is a little further than the half-hour mark. Not a quarter past three, the hour hand is ahead. Not a quarter to nine, the hour hand is behind. So is it just twelve o’ clock? Are there no other times like this when the hour and minute hands line up perfectly?
If there weren’t any, how could the minute hand cross the hour hand once every hour? The answer, as you might be knowing, is that there are other such times, only they are not so neatly expressible. What are those times?
I have developed a very simple algorithm which I have used to find those times. They are given below, the seconds correct up to two places of decimals (not approximated). Actually, they are recurring decimals. Obviously, since they are in terms of a 12-hour clock, they hold for both AM and PM:
So there are 11 such ‘handshake hours’. Actually, this means 22 such times over a period of 24 hours.
An important note: don’t take the digits after the decimal place in the seconds very seriously — not because they are wrong, but because this whole concept of the hands meeting is associated with an analogue clock, not a digital one. And I think the smallest movement possible for the second hand in an analogue clock is shifting by an entire second, not any fraction of it. So that analogue clocks can never show such fractional times. A clock can never exist in such a state Those decimal places have come about because the algorithm assumed the motion of all hands to be continuous, not discrete. That is, it assumed that the clock’s state can be all intermediate times. Those decimal places do show in digital watches, but in digital watches there are no hands to line up. You can, of course, round up those times to the nearest integer in seconds. You can test these times now with an analogue clock you may have lying around. You don’t have to go all the way to fixing the seconds, though. (I don’t think the second hand can be moved anyway.)
I cannot give you the algorithm right now because it involves mathematical type and it’s going to be a hell of a work for me to get that laid out and uploaded as images and all. However, if you’re interested, why don’t you try finding it out yourself? It’s not very hard.
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