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## Question

In the figure below, you can see a green circle and a blue circle. Assume that they have the same radii. Now, what you do is roll the blue circle round the green circle (without sliping) and arrive at the same spot. After you have completed this work, Tell me, how many rounds have the blue cirlce rotated about its own axis (its center).

Not be too quick, its quite tricky. And one thing ---> I am more concerned about how you describe than the numerical answer.

Edited by the-genius

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Two times. By the time the blue one has reached the bottom of the green one, its top point is touching the green one (that was the first complete revolution). By the time it gets back to where it started, the top is on top again (that's the second complete revolution).

By the way, I like CIRCLES but I don't like ROUNDS (see in Games)

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2...will get to describing why when I get back from my meeting (if I don't forget)

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one time around of 360 degrees the axis may turned on each quarter by one full turn to give an answer of 4???

In the figure below, you can see a green circle and a blue circle. Assume that they have the same radii. Now, what you do is roll the blue circle round the green circle (without sliping) and arrive at the same spot. After you have completed this work, Tell me, how many rounds have the blue cirlce rotated about its own axis (its center).

Not be too quick, its quite tricky. And one thing ---> I am more concerned about how you describe than the numerical answer.

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my guess is one

Edited by hungryjeff
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I still think the answer is one. If they have the same radii, they have the same circumference. Suppose you could peel of the outer most layer of each circle, and flatten this layer out. They would each be the same length. If one revolution is one cycle around it's own axis, then it will take one blue round to go around the green circle. If a round is 180 degrees, then the answer would be two. But I think a round is 360degrees.

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And it is not really needed, imho. If the circles have equal circumference, then the point on the blue circle that touches the green now will be in exactly the same orientation after one complete revolution around the green one. 360 degrees and not a fraction more. As for a math proof, I will leave that for others.

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Of course, you could try it with two coins...

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Of course, you could try it with two coins...

If we were to roll the coin alone a straight line equal in lenght to the circumfrence then there would be only one revolution, but the curve of stationary coin also affects the turning like gears turning together would. See, this is why math and I don't get along.

P.S. love the Getafix avatar.

Edited by Grayven
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I'd say 2 times. it seems that people aren't reading the question fully. it asks "how many rounds has the blue circle rotated about its own axis", not simply "when do the 2 starting points meet again". think of the blue circle as a clock face and you may see that it does spin fully twice during it's one trip around the green circle.

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I'd say 2 times. it seems that people aren't reading the question fully. it asks "how many rounds has the blue circle rotated about its own axis", not simply "when do the 2 starting points meet again". think of the blue circle as a clock face and you may see that it does spin fully twice during it's one trip around the green circle.

If the green circumferance is laid flat, and the blue circle is rolled across it, it will go through one full revolution when it reaches the end of the flattened circumferance. Unless the green circle is supposed to be turning as well, but I didn't get that from the question, It just reads "no slipping".

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take two coins out of your pocket (like i did) and

2 times!

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If the green circumferance is laid flat, and the blue circle is rolled across it, it will go through one full revolution when it reaches the end of the flattened circumferance. Unless the green circle is supposed to be turning as well, but I didn't get that from the question, It just reads "no slipping".

just as others have suggested, like size coins is a rather easy way to understand this. I took a sharpie marker and marked a line on one coin and placed the mark at the 12 o'clock position then rolled it around the other coin. the mark hits the 12 o'clock position twice during the full revolution around the static coin.

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just as others have suggested, like size coins is a rather easy way to understand this. I took a sharpie marker and marked a line on one coin and placed the mark at the 12 o'clock position then rolled it around the other coin. the mark hits the 12 o'clock position twice during the full revolution around the static coin.

You're right, I see it now. I did not have any change in my pocket, so I was trying to conceptualize it. I suppose I should stop leaving change in the take-a-penny,leave-a-penny dish.

Edited by hungryjeff
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Well, it seems that several have hit on the answer, but:

If you roll a circle along a straight line that is equal in length to the circle's circumference, it will turn 360°. If you roll it along a circular path instead, it also turns through that circle (another 360°). Thus, two full rotations.

An interesting twist might be to ask how many times the blue circle would rotate if you rolled it ¾ (or some arbitrary amount) of the way around the green one.

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There is two coin puzzle here, which seems very similar.

To find a puzzle, which has not been posted here yet is a puzzle in itself.

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Ok, I did't find that the puzzle was already there as it was differently called ("two coin puzzle"). But What I am still confused is that, the blue circle makes 2 revolutions only from our prespective. Suppose the blue circle is a 1 wheeled veichle which records the rotation its wheel has made in its meter. In if the veichle goes round the cirlce it I think won't record 2 revolution of its wheel, but 1. (because for the veichle, it makes no difference on what type of path it moves for counting the revolution of the wheel, it only depends on distance travelled by the wheel. If it travelled a distance equal to the circumference of its wheel, then one revolution is recorded)

What people will say when the veichle goes round the green cirlcle is "The wheel made 1 revolution and the veichle made 1 revolution". Peoples here are adding these revolution. What we are concerned with is the rotation of the blue circle about its own axis, the same thing the meter on the veichle records.

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... Tell me, how many rounds have the blue cirlce rotated about its own axis (its center). ...

Technically, if we adhere strictly to the statement of the problem -- the answer is 1.

But that's the first answer that comes to mind. Typically, a puzzle of the type takes into account the additional rotation of one circle around another.

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Ok, I did't find that the puzzle was already there as it was differently called ("two coin puzzle"). But What I am still confused is that, the blue circle makes 2 revolutions only from our prespective. Suppose the blue circle is a 1 wheeled veichle which records the rotation its wheel has made in its meter. In if the veichle goes round the cirlce it I think won't record 2 revolution of its wheel, but 1. (because for the veichle, it makes no difference on what type of path it moves for counting the revolution of the wheel, it only depends on distance travelled by the wheel. If it travelled a distance equal to the circumference of its wheel, then one revolution is recorded)

What people will say when the veichle goes round the green cirlcle is "The wheel made 1 revolution and the veichle made 1 revolution". Peoples here are adding these revolution. What we are concerned with is the rotation of the blue circle about its own axis, the same thing the meter on the veichle records.

Actually, the-genius, I disagree. If the statement of the problem had said that a unicycle was ridden by a rider (with enough gravity in the green circle to attract the unicycle), then it would make sense that the wheel made one revolution in the frame of reference of the unicycle.

However, without that very special frame of reference, we have no reason to believe that the blue coin has any other "up" than the one we viewers all share. In that frame of reference, as we have seen, the "top" of the blue circle comes up to face "up" twice in its travel.

As you point out, the unicycle rider does not see the additional rotation because he/she is undergoing it.

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Ok captainEd, I din't mention about the special frame of reference. But however I mentioned, the axis of rotation, the center of the blue circle. I think mentioning this has the same effect as mentioning that the rotation should be counted from the frame of reference of the blue circle(the rotating circle).

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Ok captainEd, I din't mention about the special frame of reference. But however I mentioned, the axis of rotation, the center of the blue circle. I think mentioning this has the same effect as mentioning that the rotation should be counted from the frame of reference of the blue circle(the rotating circle).

Strictly speaking, in the frame of reference of the blue circle itself, it goes nowhere, the green circle and the rest of the entire universe performs two complete revolutions around it.

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O.K. CAPTAINL i AGREE. bUT WITH THE FRAME OF REFERENCE OF THE CENTER OF THE BLUE COIN, ( AS THE QUESTION SAID) ITS JUCT ONE REVOLUTION BY THE BLUE CIRCLE.

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I think there is a bit of confusion regarding frames of reference and this question. I drew up a graphic that depicts what some of you have been trying to explain. The first colum on the graphic shows the revolutions of the blue coin with respect to a rotating axis (polar system, origin being the center of the green circle). The second column shows the revolutions with respect to a shifting axis (cartesian system, origin being the center of the blue circle).

Since the OP never specified the definition of the blue circle's "own axis" by means of polar or cartesian coordinate systems, both answers (1 and 2) must be accepted as correct.

Note that at phase 2, the circle on left column has only made a half rotation with respect to the axis, but the on right column, the circle has made one full rotation with respect to it's axis. The difference is the coordinate system utilized, or "frame of reference."

Edited by DanCDow

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