controversy

on a

mathematical and geometrical

puzzle with 2 circles

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the puzzle with 2 circles
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The puzzle is proposed (on YouTube) by Presh TALWALKAR who owns the website called Mind Your Decisions.

PUZZLE:

The radius of circle A is 1/3 the radius of circle B.

Circle A rolls around circle B one trip back to its starting point.

How many times will circle A revolve in total?

(see picture here below)

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Here is my mathematical (and geometrical) reasoning:

 perimeter of any circle (recall): 2 x Radius x 3.1416 (Pi)

 perimeter of circle A (radius = 1): 2 x 1 x 3.1416 = 6.2832

 perimeter of circle B (radius = 3): 2 x 3 x 3.1416 = 18.8496

 number of times circle A revolves in total = 18.8496 / 6.2832 = 3

 my answer is: 3 times

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The Mind Your Decisions website answer is a YouTube video which shows revolutions of circle A around circle B and doesn't demonstrate the answer mathematically.

See here below a screenshot of the The Mind Your Decisions video ...

Here are the explanations you can read on the video (a copy-paste):

 The radius of circle A is 1/n the radius of circle B. (n=1,2,3,...)

 Circle A rolls around circle B one trip back to its starting point.

 How many times will circle A revolve in total?

 Circle B has n times the circumference so circle A rolls around n times.

 Circle A also makes 1 trip around, so this adds 1 more revolution.

 Then, correct answer is 4.

the reason I thought

the good answer was 3 ...

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demonstration (answer is 3 and not 4)
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here below are pictures extracted from a topic of Wikipedia dedicated to the epicycloid (geometry).

To illustrate the epicycloid, there is a GIF (animated pictures) on which a small circle turns around a 3 times bigger circle.

This illustration can be used for the puzzle with 2 circles to get evidences the number of revolutions is 3.

It's obvious to see the number of revolutions is 3 (and not 4).

(from Wikipedia)

and

the reason I agree

the good answer is 4 ...

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Presh T. was right
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The two here below pictures show an example with 2 circles with the same radius (note: whatever are radius sizes, reasoning is the same).

The circle A goes around the circle B and will be in the same position at halfway.

Conclusion:

 How many times will circle A revolve in total?

 Answer is 2 (and not 1).

Why I was wrong?

 Bad use of the epicycloid model to solve the puzzle.

 There are 4 rotations (and not 3) of the small circle around the big circle in the graph dedicated to the epicycloid (see here below the last picture of this page).

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links (some of them have been given by Presh T.)
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click on the eyes (_) to activate the corresponding link