But it is becoming a tad monotonous to continue writing about it, and I still have not uploaded the photos. Hence I am taking a 473 radian turn here, and writing about something entirely unrelated. In any case, as the journey has already been completed, let the travelogue remain an open chapter, with the option of being closed sometime in the future.
Maybe you have already seen this one before, or maybe you have not - either way there it makes no difference to the simplicity and charm of this puzzle, given to me,like many other interesting ones, by a very tall and very thin (genius) friend. This was in the context of those cycloids and trochoids we learnt in first year graphics.
The puzzle itself is very simple.
There are two wheels of same diameter. One is fixed and the second one is rolling around this fixed wheel, touching it. The question is, how many rotations will this second wheel undergo to complete one revolution around the fixed wheel?
If you are seeing this puzzle for the first time, please, please, think of your answer.
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The solution
The answer is One.
The approach to find this answer is normally like this - We find the total distance the rotating wheel has to travel. To do this (the engineering approach :D) we assume the circumference of the fixed wheel is cut open, and laid flat. Then we can see that the distance the rotating wheel travels to complete one complete revolution is 2pi*r.
As its radius is also same, to travel this distance of 2pi*r, the rotating wheel has to complete just One rotation, right?
Wrong.
Try it out with two coins.
It takes not one, but 2 rotations.
The Proof as suggested by Newton - not Isaac Newton, but an (equally brilliant) alumnus of model school - is as follows.
Just find out the distance the wheel travels - only, without cutting the fixed wheel open.
We can see that, when revolving around the fixed one, the centre of the rotating wheel actually travels a distance of 2pi*(r+r), compared to simply 2pi*r that it travels on the flat ground, when the fixed wheel is assumed to be cut opened and flattened.
Thus,the curvature of the path has introduced an additional 2pi*r distance, to travel which, the rolling wheel an extra rotation.
Ya, it really does take 2 rotations. You can keep the two coins back now (If u tried with them in the first place :P).
Another explanation to this is based on relativity, and different frames of references. This is quite difficult to imagine and validate, especially if you have not put your grey matter to much rigorous use for some months. Hence I am not explaining it here. If you are really interested, there must be some sites (I couldn't find it in any puzzle forum i normally visit) that give you the explanation.
Regarding the implications of this puzzle - again too complicated to write about here.
If you are wondering why I had the sudden impetus to explain a puzzle in my blog , it is because I watched the rather stupid and extremely self indulgent Aronofsky movie called Pi , last night. Some dialogue in the movie reminded me of this puzzle.
