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Accepted Solutions
@KevinFinity wrote:
Here is a simple math challenge for you. Solve with MathCAD!
On a regular 12 hour clock, how often does the minute hand pass the hour hand?
(Hint: The answer is not 60 minutes.)
Since you are asking for a number, the solution cannot have the dimension "time" anyway 🙂
Also you sure mean "during a time period of 12 hours".
Anyway - challenge accepted. Lets use Mathcad 15 to simply watch and count (and stop before we reach 12:00):
Maybe I wasn't clear enough describing the problem. It was not meant that the clock with the exchanged hands shows THE correct time. This would be the case when the hands pass and so would yield the same answer as the original question.
What I had in mind was that this clock with a large hour hand and a small minute hand will have most of the time its hands in a position which would never occur on a "normal" clock and someone watching would immediately(?) realize that its not a standard clock. But sometimes in the period of 12 hours the hands would be in a position which could be a correct time and someone watching may not notice that the hands are exchanged (he may notice that the time is not correct, though).
Of course when the two hands pass we have a correct hand position and we already know that this happens 11 times in 12 hours. But there are also other hand positions where the clock with the exchanged hands shows a correct time. So the solution sure has to be quite greater than 11.
Here is an example of a hand position which ressambles a correct time no matter how the hands are exchanged:
BTW, an example for a clock where the hour hand is larger than the minute hand is the clock on the clock tower in Graz. Originally it had only one (large) hour hand, the smaller minute hand was only added later.
https://cdn.pixabay.com/photo/2020/02/23/21/30/graz-4874700_1280.jpg
https://www.blounge.at/wp-content/uploads/2013/03/uhrturm-graz.png
@Fred_Kohlhepp wrote:
"What I had in mind was that this clock with a large hour hand and a small minute hand will have most of the time its hands in a position which would never occur on a "normal" clock "
I'm having trouble envisioning a set of hand positions on a clock that wouldn't happen normally at some point in a twelve hour period. Please give me an example.
You can't demand an arbitrary hand position because the position of both hands depend on each other, Whenever you move one, the other has to move accordingly. The minute hand moving 12 times faster than the hour hand. So every position you set the hour hand to forces one and only one position for the minute hand.
Lets say its 9:00 'o clock and now you exchange the two hands.
This position can never been seen on a regular watch (unless its broken and needs a repair).
The very same applies to 4:30 and an infinite number (most) of other times
But there is a finite number of times where exchanging the hands results in a valid clockface.
OK, the contest seems to be over so its time to solve my additional riddle, too.
The interesting part is that its very similar to the original question.
This puzzle was probably printed for the first time around 1880 and exists in different versions. In one of them a watchmaker has mistakenly installed two identical hands and the question is how often within 12 hours one does not know what time it is (because the position of the hands allows two valid interpretations). Here you would have to subtract the 11 positions where the hands are on top of each other (the original problem in this thread) from the solution to the problem as I formulated it here.
Of course, the task can be solved in many different ways (periodicity, systems of equations, ...). A very elegant approach goes back to the mathematician David Gale (https://de.wikipedia.org/wiki/David_Gale_(%C3%96konom)). He introduced a fictional third hand that moves twelve times as fast as the hour hand (which in turn moves 12 times as fast as the minute hand). And he went on to explain that a valid position of the reversed hour/minute hands is always found when this third hand and the hour hand are on top of each other. This means that the task is essentially traced back to exactly the one that was initially set here in the thread and one can analogously consider that such a hand position is always obtained at intervals of 12/143 hours. So within 12 hours there are 143 valid hand positions - here they are (I couldn't resist):
@Fred_Kohlhepp wrote:
Simple problem; simple answer:
Maybe not that simple 😉
Your reply is missing the actual answer to "how often".
Furthermore only every second point of intersection is a valid solution!
12 of those 23 solutions are not valid!
E.g. the "solution" t = 12/13 hr = 55.385 min corresponds to 00:55:23.077.
At this time the handles do not pass but are mirrored by the vertical
The times we are looking for simply are all multiples of 12/11 hours.
Your reply is missing the actual answer to "how often".
The times we are looking for simply are all multiples of 12/11 hours.
really?
But you get credit for a really nice animation!
I thought it'd be fun to think of it as distance travelled instead of angle. Similar to Fred's otherwise.
Hello, everyone. Here is my solution. I modeled it as an infinite series and came up with the same results. It was very interesting to see the variety of different ways to get the same answer. Everybody thinks a bit differently, and that's what I wanted to see. Thank you all for participating.
