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# Rolling a circle within a circle

## Question

We have two circles. One circle has a diameter of 12 cm's while another is D cm's. The circle with the diameter of 12 cm's fits inside the other circle. If rolled within the larger circle, the smaller circle rotates three times before coming back to rest at the same starting point. That is, Point p in the smaller circle starts and stop at Point P' on the larger circle after three revolutions.

What is the diameter of the bigger circle?

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To rotate three times, the point p must touch the larger circle 4 times.

At 12:00, 3:00, 6:00 and 9:00.
D therefore is 48.

It's the old roll a coin around another coin puzzle.

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36 cm

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Just visualising it, won't there be multiple valid answers?

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Just visualising it, won't there be multiple valid answers?

elaborate please.

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the answer is unique.

P and p could align after multiples of three rolls, if D were 36n cm,

but if after exactly three rolls p aligns with P, then D must be 36 cm.

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36

Though I got such a simple answer that I probably forgot to account for something lol
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be careful with your analysis. In this image we have a large circle and two smaller congruent circles that have 1/3 the diameter of the larger circle. When both circles travel around the larger circle, do they in fact travel the same distance?

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To rotate three times, the point p must touch the larger circle 4 times.

At 12:00, 3:00, 6:00 and 9:00.

D therefore is 48.

It's the old roll a coin around another coin puzzle.

Wait, what?

p touches the circle four times: at P, one third the way around, two thirds the way around, and then P again.

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To rotate three times, the point p must touch the larger circle 4 times.

At 12:00, 3:00, 6:00 and 9:00.

D therefore is 48.

It's the old roll a coin around another coin puzzle.

Wait, what?

p touches the circle four times: at P, one third the way around, two thirds the way around, and then P again.

How big must the bigger circle be to do that?

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To rotate three times, the point p must touch the larger circle 4 times.

At 12:00, 3:00, 6:00 and 9:00.

D therefore is 48.

It's the old roll a coin around another coin puzzle.

Modifed though, as the circle is within your circle. Also if it rotates from 12 to 3, 3 to 6, 6 to 9 and comes back to start, that is four rotations, right?

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The small circle makes a full revolution when the point p is in the same orientation as before, not when point p touches the large circle again.

Black circles show the locations of full revolutions of the small circle. Grey circles show the locations of point p touching the large circle. At 36 cm, the small circle makes 2 full revolutions and point p touches the large circle 3 times before coming back to P. At 48 cm, the small circle makes 3 revolutions and point p touches the large circle 4 times (as bonanova stated)

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To rotate three times, the point p must touch the larger circle 4 times.

At 12:00, 3:00, 6:00 and 9:00.

D therefore is 48.

It's the old roll a coin around another coin puzzle.

Modifed though, as the circle is within your circle. Also if it rotates from 12 to 3, 3 to 6, 6 to 9 and comes back to start, that is four rotations, right?

Yes, modified; yet similar.

Both require the consideration that traversing a circular path (inside or out)

affects the coin/circle in the same way that a rotation does, as it traverses that path.

No, not four rotations - see k-man's picture.

A rotation occurs each time the point p returns to its initial azimuth.

Not each time the point p touches the outer circle.

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To rotate three times, the point p must touch the larger circle 4 times.

At 12:00, 3:00, 6:00 and 9:00.

D therefore is 48.

It's the old roll a coin around another coin puzzle.

Modifed though, as the circle is within your circle. Also if it rotates from 12 to 3, 3 to 6, 6 to 9 and comes back to start, that is four rotations, right?

Yes, modified; yet similar.

Both require the consideration that traversing a circular path (inside or out)

affects the coin/circle in the same way that a rotation does, as it traverses that path.

No, not four rotations - see k-man's picture.

A rotation occurs each time the point p returns to its initial azimuth.

Not each time the point p touches the outer circle.

nicely done! I forgot to consider, in my own calculation, that the point p touches the circle more often than it does coming back to its starting position. which makes sense since it goes to the line before it comes back to its starting position on every spin.

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