controversy

on a

mathematical and geometrical

puzzle with 2 circles

on a

mathematical and geometrical

puzzle with 2 circles

the puzzle with 2 circles

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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)

my answer

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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

the answer of Mind Your Decisions (Presh TALWALKAR)

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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?

Answer is n+1

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 ...

the good answer was 3 ...

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 ...

the reason I agree

the good answer is 4 ...

Presh T. was right

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I exchanged emails with Presh T. and got explanations and links (added at the end of this page).

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

Conclusion:

How many times will circle A revolve in total?

Answer is

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).

links (some of them have been given by Presh T.)

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Link (URL) to the YouTube video: (_)

Link (URL) to the website of Mind Your Decisions: (_)

Link (URL) to the Wikipedia topic about the Epicycloid: (_)

Link (URL) to the Wikipedia topic about the Coin Rotation Paradox: (_)

Link (URL) to an interactive graph about the Coin Rotation Paradox: (_)

Link (URL) to an interactive 'spirograph': (_)

Link (URL) to an article in the New York Times: (_)