Happy 4 year anniversary and 1 million views! Yes the answer really is 4, even the test makers admitted it! But it's a counter-intuitive problem. A lot of people who felt the video was mistaken now see why the video is correct, particularly after reading these links: Sources www.donaldsauter.com/rolling-circles.htm www.nytimes.com/1982/05/25/us/error-found-in-sat-question.html Coin Rotation Paradox en.wikipedia.org/wiki/Coin_rotation_paradox Geogebra demonstrations tube.geogebra.org/m/787131 tube.geogebra.org/m/112822 tube.geogebra.org/m/107691

The center of the small circle travels around a circle of radius R+r. Not R, as the test writers thought. So the number of rotations is 2pi(R+r)/2pi(r) = R/r + 1.

It all depends on what you mean by "completing one revolution". If one revolution is completed when the initial point on A touching B once again touches B, which is what I originally thought, then 3 is the correct answer.

Another way to think of this is that the center of the smaller circle is 4 units away from the center of the larger circle; thus, the center of the smaller circle travels 4 times its original circumference. Since it rolls without slipping, that also means that its edge rolls through 4 times its circumference. Interestingly, it works out in a similar manner if the smaller circle is INSIDE the larger circle. If we think about the center of the smaller circle, it travels through a circumference of twice its radius, which means the edge travels through twice its circumference. OR we can think of it as rolling around itself three times but it travels BACKWARD one rotation because that's how an inner circle rolls. Either way, and in general, the number of rotations of an inside circle will be n - 1.

It rolls 3 times, rotates 4 times, but revolves only once. Hear me out. A full roll is completed when an object progresses around the a surface of another, without slipping, until the contact point between the object and the surface has gone all the way around the circumference of the object and is touching again; a full rotation is completed when an object turns all the way around and is upright again with respect to the frame of reference (yes, this means that rolls = rotations on a straight surface or where the frame of reference rotates with the normal of the curve); a full revolution is completed when an object travels all the way around another object with respect to that second object (most people are familiar with the Earth rotating on it's own axis vs revolving around the Sun). In the video example, you can see that: 1) the "bottom" of circle A, identifiable by the orientation of the text, faces the centre of circle B three times as it rolls; 2) the text is upright with respect to the camera 4 times as it rolls; and 3) circle A travels around circle B only 1 time. Does that make sense to you?

The phrasing is ambiguous. It doesn't give a frame of reference, from the perspective of the bigger circle it is 3 times. From top down it is 4 times.

Visually mathematically: Take Circle B's edge. Cut out all the inside circle (so you only have the rim), cut the rim in any place so you can then stretch the rim out flat on the table. This is the perimeter of Circle B. It has a value that we know, its circumference, which is s = 2*pi*r = pi*d, where pi is obviously pi, 2 is 2, r is radius, and d is diameter. Basically, we're just laying out the length of the circumference of B as a straight line on the table and measuring the total length. Now, take Circle A's edge. Same thing, cut out so you only have a rim, snip the rim, lay the rim out flat. The total length of this circumference of A is also known and can be expressed as s = pi*d. Now, if we put the line we now have from A next to the line for B, the line for A is 1/3rd the line for B. This means it takes THREE circumferences of A to make ONE circumference of B. . Mechanically (e.g. the way you rolled the circles): The problem with your circle is, at the 1/4th point, Circle A has NOT made a full circle. You're lining it up Cartesian like using the table as a frame of reference to say that Circle A has rotated 360 degrees WITH RESPECT TO THE TABLE. But this is not WITH RESPECT TO CIRCLE B's PERIMETER. If you notice, the STARTING orientation of Circle A with respect to Circle B's perimeter is where the "bottom" of the circle is tangent to Circle B. Look at how the A's point is pointing AWAY from the center of Circle B. That is, in order to see when Circle A has made "one full rotation with respect to Circle B's perimeter", you must rotate Circle A until the tip of the A is pointing away from the tangent point/center of Circle B. Each time the tip of A points away from Circle B, you have made one full rotation of Circle A with respect to Circle B. Note that where you stop at the 1/4th point, the tip of the A is NOT pointing AWAY FROM Circle B. It's pointing perpendicular-ish to it. Timestamp ~ 1:44 you can see what I'm looking at. At 1:44, notice how the tip of the A is (more or less) NOW pointing away from Circle B. So at point 1:44, you have completed one full rotation of Circle A WITH RESPECT TO Circle B. Notice also that this is at ABOUT the 1/3rd point... ...meaning that mechanically, we've now arrived at the same solution: Circle A makes it 1/3rd of the way around Circle B when it makes one full rotation WITH RESPECT TO Circle B, ergo, it will take Circle A three full rotations WITH RESPECT TO Circle B in order to trace out the full perimeter of Circle B. . This is kind of some Relativity problem, so you could argue reference frames here, but the fact remains that Circle A needs to "lay out" its circumference 3 full times around Circle B's outside edge in order to cover the complete circle. This is true both mathematically and in terms of Circle A's motion WITH RESPECT TO Circle B. So you can argue that frame of reference can give you different answers (which is true), but 3 is _A_ correct answer, and an entirely valid answer, per Relativity. . That is, you can say 3 AND 4 are BOTH correct answers, but you cannot say that 4 is THE correct answer and 3 is wrong. That would make YOU wrong. Note that I mean this with no animosity, I just dislike people making absolute statements that are not actually dictated by the situation or evidence presented. If you are given a problem that has two answers (say, the square root of 4, which can be 2 or -2), and the list of answers has one of those (say -2) but not the other. Then the correct answer should be the one answer OF THE GIVEN CHOICES that is one of the correct answers. In this case, they presented both 3 and 4, and 3 is ONE correct answer, therefor, of the options listed, 3 would be THE correct answer, since the other four answers were incorrect. Now, if they gave you BOTH answers (e.g. for square root of 4 they gave you -2, -1, 0, 1, and 2), then you could argue that BOTH 2 and -2 are correct, so either you pick should be counted as correct. In which case, going back to the circles, BOTH 3 AND 4 are correct, since it depends on the frame of reference you choose, and the problem did not state a frame of reference. Agree/disagree? I get that this is a 6 year old video, so you'll probably never see this. :)

However, circle A's "tread" only engaged the surface of circle B exactly 3 times. From a topological pov, it only made 3 revolutions.

I'll admit I missed this one as well, but it sorta reminds me of how the Earth rotates and revolves around the Sun. It only takes 23 hours and 56 minutes for the Earth to rotate 360 degrees, but a day is 24 hours because it takes the Earth an additional 4 minutes to turn another degree or so in order for it to be facing the Sun in the exact same way, because it has advanced about one degree in its orbit.

“Rolls around” is actually ambiguous. One rotation could be seen as the point where the same point on the rolling coin retouches the other, in which case the answer is 3.

Both 3 and 4 are correct answers, depending on semantics. When the small circle has advanced 1/4 of the perimeter of the big circle, it has rotated/revolved one time according to our visual reference, but has not finished a full rotation in terms of where it is tangent to the big circle. So if the frame of reference is the small circle by itself, the answer is 4. If the frame of reference is where the two circles are tangent, the answer is 3.

I got to 3, here is how I did it: Every rotation of the smaller circle covers a distance equal to it's circumference, therefore, I treated each circle as a straight line, and then compared them. c= 2pi(r) With r being the circumference of circle A C= 2pi(R) R=3r C=2pi(3r) A/B = 3 So, the A circle needs to travel 3 times it's total circumference to make it around B, which means 3 full rotations. After watching the video and reading the comments: My answer seems to hold true if the circles are stationary and would spin in place, such as in a gearbox, OR the B distance is a true straight line allowing A to move in a straight line. However, they aren't stationary, and B is a circle, so that throws a wrench in it. I'm trying to understand how A rotates 4 times as it travels around B. My only explanation is that the center point of A is travelling a larger circle than the circumference of B as A rotates around. This means that the distance A travels in the question is actually longer than the circumference of B. So the real equation I should use is: c=2pi(r) C=2pi(3r+r)=2pi(4r) A/B=4 Others have pointed out that there is a perspective issue at play. A rotates 4 times, relative to the perspective of the viewer. However, when considering the points of contact between A and B, there are only 3 rotations. ie. the "bottom" of A is only in contact with B at 3 points. I find this argument compelling, because from the perspective of A, it has only rotated 3 times. If I was standing on A, looking directly at B from the start, I would only be looking directly at B 3 times during the trip. So, I suppose it comes down to definition - what does "revolve" mean? Is it relative to B, or is it relative to the viewer? Personally, I would argue that the way the question is written, they meant "revolve" to be relative to B, and therefore 3 is correct. But, it is not explicitly stated which perspective is correct, so it is still vague enough to be open to interpretation.

The circumference that matters here is made by the path traveled by the center of coin A. So you just add the radii of both circles and solve for circumference of the new circle "C" and divide the circumference of C by A. A = 1 B = 3 C = A + B 1*2*3.14 = 6.28 4*2*3.14 = 25.12 25.12 / 6.28 = 4 The above works for any fraction. Any fraction with a numerator of 1 can be solved by simply adding the numerator and denominator together. Example: Instead of A being 1/3 of B, lets make A 1/17 of B. A = 1 B = 17 C = 18 1*2*3.14 = 6.28 18*2*3.14 = 113.04 113.04 / 6.28 = 18

3 rotations relative to its "track', but since its track is oriented as a circle 4 rotations to our eyes. Flatten the track to a straight line, and it will only be 3 rotations relative to both.

The actual correct answer is 1. It rotates 4 times while revolving once around B.😀

If you stretch both circumferences, their lengths are 3 to 1. If you fix the little circle and turn one time the bigger one, the little wiĺl do 3 turns. When the little circle revolves, it do the 3 turns plus one more because the visual change. The initial contact point isn't the same at 1/4 of the turn, but the little circle center trips.

For your experiment with the coins at 2:42, the coin has, in actuality, only rolled half of its circumference by the time it reaches the bottom of the other coin.

I just figured it out. The actual CIRCUMFERENCE of A touches the CIRCUMFERENCE of B 3 times during its trip around. But the angular position goes through (360*4) degrees because it's a circle going around a circle. So despite the circle ending back upright after 1/4 of the trip around or 90 degrees, the original contact point between circle A and B doesn't come back around and touch B until it's 1/3 the trip around, or 120 degrees.

The problem with the cutout example is that where he stopped at "1 revolution" is not a full revolution. The first revolution occurs when the bottom of circle A touches circle B, not when the letter "A" is vertical again. He was counting 3/4 of a rotation as a full rotation, which is why he got 4 instead of 3.

To be honest - I never understood what that additional revolution shall come from. I always thought it must be the same as if coin a would roll on a plain surface with the length of the circumference of coin B. Your explanation helped me a lot! I imagine moving coin A around B in a way its bottom point of its starting position constantly stays touching coin B - that would be equivalent to just SLIDING it over a plain surface instead of rolling. On a plain surface coin A would have 0 revolutions, but around a circle it already does complete one. Thanks, great stuff!

It made 3 rotations. If you are counting from where it is contacting the other circle the original point of contact only touched 3 times. It is an illusion that it rotated 4 times due to the fact we are not standing in the middle of the large circle. If we were we would only see it revolve 3 times.

This Video is incorrect. I wrote this in a reply to somebody, and didn't see anyone else post this. When he rolls the 'circle A' around 'circle b' there's a total of 3 full rotations by Circle A. A full rotation by circle A is when the initial contact point of circle A comes in contact with circle B. An easy way to visualize this is with his coin example at 2:40. You can notice that the initial contact point is directly below the head on the coin that is being rotated. This initial contact point only makes contact once per full trip around the circumference of the stationary coin. The visual representation just makes it look like the coin is revolving additional times. The coin\circle just appears to be revolving more because it's being rotated and moved in a circle. It's not actually revolving more.

It would seem that the writers of the original question had a definition of 'revolve' that was relative to the center of Circle B. It's like the reckoning for an Earth day; it's slightly more than a full rotation each day (2.737% more) because the measure for a day is relative to the sun, rather than to the 'fixed' stars. This is 'sidereal' versus 'solar,' or 'tropical' reckoning. Do the experiment again, but before starting, put a mark on Circle A at the contact point. As you spin Circle A around Circle B, count how many times the mark again contacts Circle B as it completes its trip around. The sidereal measure of it is 4, but the tropical measure is 3.

From an engineering prospective, imagining the circles as gears, the correct answer is 3, as if the big circle is driven, the little circle rotates 3 times.

Three. You have to consider the starting points. Six o’clock for the smaller circle and 12 o’clock for the larger circle. The circles resume there initial positions after three complete rotations. This mistake is counting a full rotation of the smaller coin when it is oriented upward, not when it has traversed 360* of arc.

Maybe I'm misunderstand here... But the question states the RADIUS of circle A is three times the RADIUS of circle b. The radius is the measurement from the centre of a circle to its perimeter. This might make more sense if I give an example. Let Circle A's radius = 1 Therefore circle b's radius = 3 To work out the circumference the equation is 2Pi x r Pi is close enough to 3 So the circumference of circle A = (2x3)x1 = 6 The circumference of circle B = (2x3)x3 = 18 If circle A is to roll around circle B it is the same as opening up both circles to form a line and comparing lengths. So a line of length 6 will fit onto a line of length 18 three times Therefore the answer is 3.

The maths never lies. The answer is 3. It only LOOKS like it has completed 4 revolutions but in actuality, every time it APPEARS to complete one revolution (from our perspective), it has only rotated 270 degrees. Really, it rotates 360 degrees a third of the way around the circle, as expected.

If you assume there is friction between the two circles and both of them rotate at the same time, then I think you’ll find that the small circle will run around the big circle (N+1)/2 when N=1 since the circles are both rotating, and only need to go halfway around to be back to where they started from

Mentally, I was thinking about circumference and that’s 3 circumference matching rolling segments, but after the 1st segment where Circle A is in the bottom-right, it’s not back to upright- what was initially down to Circle A is now up-left (120° CW), so its circumference has been rolled over once, but it’s revolved 1 1/3 times. Multiply by 3 and bam, 4.

Interesting to see where the extra revolution comes from. If circle B was simply a single point with no circumference, Circle A would still take a single revolution to rotate around that point. That is the extra revolution, regardless of how large or small circle B is.

If you draw a spot on Circle A at the point where Circle A touches Circle B at the start, the point will only make 3 full rotations with reference to the circle (in other words, touching the circumference of the circle). However, it will turn a total of 1440 degrees (4 rotations) with respect to a non-moving horizontal line across the page). The question asks how many times "around" Circle B so I'm inclined to say 3 times.

I refuse to believe that no one selected the right answer. There must have been thousands of people who determined it was too difficult and just guessed.

The correct answer IS three because for me "revolve" means returns to the starting point it originally made contact with the larger coin in other words a full revolution.

1 Revolution 3 Rolls 4 Rotations Enough said.

Can you solve it mathematically? Not with coins or cardboards.

another approach to the problem could be "how many circles do you need to have 10.000 people arguing with each other?" and I love it

I counted 3. Draw a line where the two circles meet. Roll the smaller coin around the larger until it's back at its original position. The line on the smaller coin will have touched the larger coin 3 times.

The fact that 4 is not an option removes any ambiguity from the question. It should be clear that the creators of the test expected 3 to be correct, and those who answered 3 got the mark. Anything beyond that is just interesting conversation.

You literally revolved 3 times in your example, you just counted it as 4. Place a mark to where the circle A touches and you will see what I mean. The circumference of the circle * Revolutions= distance traveled. If we let radius of B=1, then the circumB=6.28. Therefore the radius of A=1/3 and the cirumA=2.09. In the original formula, solve for Revolutions and you get Rev.=Distance/Circum. The distance Circle A traveled is equal to the circumB. Therefore 6.28/2.09=3. Not exact because I rounded Pi.

One of the most interesting problems. Very well explained Sir 👍

Looked like 3 rolls to me. The bottom of Circle A touched the outside of Circle B 3 times.

Nice try, but you're picking an arbitrary frame of reference.

What's missing in this question is a specified frame of reference. If the frame of reference is circle B, then the answer is 3. If the frame of reference is the top of the page, then the answer is 4.

The edge travels a circle with a radius of 3, the center travels a circle with a radius of 4. But the question didn't ask about a 3rd circle with a radius of 4, it asked about Circle B with a radius of 3. Circle A rotates 3 times to travel around Circle B, but the guy in the video is looking at a different circle with a radius of 4 which intersects the center of Circle A.

More generally: 1. Take a straight line of length l and a circle with circumference length c. Now bend the straight line to form the required shape while counting how many times you rotate the end of the line (# of rotations is e). It should now be obvious that the number of times the circle needs to rotate to traverse the length of the bent line is l/c+e. For the given example, this would be 3/1+1=4. 2. Take an arbitrary curve, open or closed of length l and some circle with circumference length c. While rolling the circle along the arbitrary curve, measure the distance traversed by the centre of the circle (will be negative for portions traversed of the curve that are concave to a local smaller radius than the circle, since the circle must rotate backwards to progress in such a section) (distance traversed by circle centre is d). The number of rotations that the circle makes is simply d/c. For the given example, d=l+1=3+1 so circle rotations is d/c=4/1=4.

The easiest explanation I can think of is "its always PLUS ONE, because in addition to traversing the linear distance of the circumference of the other coin it is also making ONE rotation around the object. It is the other circle that is responsible for the +1, because it bends that linear line into a circle. The coins of the same size really do translate one to one with each other, its just orientation starts with the top of the coin opposite the edge touching the coin, and when it has gone one half a circuit, it now finally has the top in contact with the other coin, which had not been until HALF the circumference of both was translated through. Both orientations are the head is UP, even though only HALF the circumference has been rotated through. If you were translating it along a STRAIGHT LINE of the same length as the circumference it would rotate exactly ONCE, not twice. Orientation is doubled because the line it is translating along curves around the rotational dimension. The dimensional curve provides the other turn.

"Only using paper and pencil" Me: *starts ripping paper into circles*

That makes complete sense, if you rotated the Center of the smaller circle around the circumference of the coin, you would get 3 revolutions, but because the center is 1/2 a coin difference outside of the larger circle, the circumference is actually that much larger. Brilliant.

The question does not ask for how many times A rotates until it's back facing in the same direction (the writing "circle A" going back to being horizontal in the example). It asks how many times it rolls around (i.e. when the radiuses from the point of contact to the centres of the respective circles are on the same straight line). This only happens three times. The answer "3" is correct. There can be no doubt about it. If the radius of a circle is 1/3 of another, its circumference will also be 1/3 of the other. So you need 3 of one to "cover" the other. There is no escaping that.

Revolve is Revolution, in order to complete one Revolution, all 360 degrees of A has to touch. Therefore the answer is 3. You can't stop short because you are trying to use the background as a reference point. That was where you failed. All 360 degrees of A has to touch B before it even makes one revolution. Or in other words revolve once. As in every single degree on A has to have touch B before it even revolves once, to make a complete rotation, revolution. That is what you are not seeing or thinking about. You are just going off of if the text is upright, of if the coin is upright. Edited, The word you are really looking for is Trips. Circle A makes 4 trips, yet completes 3 full rotations / full revolutions of 360 degree turns as in surface to surface contact.

Real Answer: '1' Reason: Question says 'Revolve', not 'Rotate'! Thanks.

A revolves 3 times around it's own axis. It is also 3 revolutions around the A axis from B's point of view because B also rotates once relative to A (or A rotates once around B). If you were an observer on the surface of B and followed A as it rolls, you would do one revolution and would see A revolve 3 times. In the same time, someone stationary on A would see one revolution around B, and three around itself, it would stop seeing B three times and would eventually see the same starting point once. So it's like 3 days and a year on Earth, if a year was only 3 days. It's 4 from an outside perspective C where you add the 3 rotations of A around it's own axis, plus if you count the rotation around B as 1 (does not matter if it is from A or B's perspective). But 4 cannot work, because you are basically using two different axis and adding them up, which simply cannot be a "revolution", which is always based on one axis, not many. So the answer is either 1 or 3. It cannot be 4 strictly speaking since it requires two different axis.

Important to see that this rule only goes for the whole numbers. You could also reason from the point of view that the middlepoint of the circle has to travel around the middle point of the other circle which makes a circle of r1+r2. That means any point on the circle has to travel the circumference of that joined circle r1,2. The amount of rotations you can fit in there are (r1+2)*2*pi/2*r1*pi. I think that means you can express the rotations needed for a circle to circle a circle as CCC1=(r1+r2)/r1. In the case of a r1/r2 ratio of 1/n you'd get (n+1)/1. Very elegant because it doesnt even matter whether you'd want to calculate the rotations of the small or the big circle.

it'd be hilarious if one of the guys just cut out the circles during the test and started counting

No, you consider a roll based on surface area touching the circles, and in that case it rolled around 3 times. The act of moving the circle across the 2 dimensional plane distorts your perception of what a full revolution is. The "circle A" won't be straight up when it makes one revolution, it will be tilted to the side somewhat and so on... Think of it as if you drew an arrow from the top of Circle A to the bottom of Circle A. One revolution will be when that arrow is pointing directly at Circle B, not when "Circle A" is right-side up and readable. The correct answer is 3.

So the answer is 3+1. The roll distance travelled is 3 times the circumference of the smaller circle PLUS one rotation to remain in contact with the edge of the larger circle.

Circle A revolves 3 times around Circle B. But it revolves 4 times around it's own center. The test makers should have clarified which revolving they were talking about.

You just need to figure out what the testing authority wishes to hear. That is the most important skill for life that the schools teach the kids these days.

3 is the correct answer. There is an error in the coin revolving experiment. One revolution of coin A is not completed yet when the label “coin A” is horizontal again. You need to keep track of the 6 o’clock point of coin A. Another way of thinking about this problem is that there are ropes rolled out on the edge of coin B. Each rope has the length 2 * Pi (radius of coin A is 1).

If you set the circles on an x-y axis and ask how many time the circle will rotate relative to the horizontal or the vertical, the answer is four. If you add a z-axis and make that the point of rotation, or if you ask how many times it rotates relative to the origin (center of the larger circle), the answer is three.

I got it right (4). The centre of mass of the smaller circle (A) is constrained to a radial path at 3+1. The revolutions displace A along this path rather than the path defined by the circle B.

This video is in error, you aren't accounting for the motion of the circle. It will complete one revolution only when the text on the smaller circle is parallel to the tangent line. Otherwise, you are seeing a visual illusion where it looks like the circle is completing a revolution sooner than it really is, because you are judging one rotation based on the text's original orientation.

The options were given keeping in mind the way vehicle gears work, i.e. the circles' centres are fixed. However in the given problem, it is important to note that after each rotation while moving around the big circle, the point of contact on the smaller cirlce is not the same. Initially it was below. After one rotation it is on its left. Actually, the smaller circle does not cover the path equal to its circumference on the bigger circle in one rotation.

To me, the simplest way to solve it is to realize that circle A has a radius of 1, and circle B has a radius of 3. This means that the center point of circle A is 4 units away from the center point of circle B so when circle A is rolled around circle B it is actually circumscribing a circle 4 times larger than itself, i.e. it will take 4 revolutions to go around it,

The centre of the small circle is (r + R) units from the centre of the big (inner) circle, so it must rotate (r + R) times around the big circle to get back to its starting point. The path taken by the small (outer) circle as it rotates around the big (inner) circle is itself a circle of radius (r + R) units.

I saw it roll around three times. I understand what you're saying, but you're using the wrong frame of reference. The coin is rolling around a curved surface, but you're using an invisible flat plane for reference. If you watch the text on the circle relative to the surface of the circle it is rolling around you will see it rolls around three times. I understand the paradox, but it's not a great mystery. You're just looking at it wrong. The answer is, of course, 3.

You’re counting it before it completes a revolution though… when you were counting you were referring to the number of times a becomes upright relative to you, which isn’t correct.

Imagine spinning circle B backwards so that A stays in place. Now A obviously revolves 3 times. Now rotate the whole picture to see that A revolves 4 times in total. Edit: It rolls 3 times, ROTATES 4 times, but revolves only once, because english is hard

I probably got this right away because I recognised it from astronomy, the Earth rotates around its axis roughly 366,25 times per year because it also rotates around the sun once.

I just feel bad for those SAT students, logically speaking, at least one of them got the correct answer (4) on paper, but then couldn't find it on the sheet so they just guessed.

It depends from which system you watch. We see 4 revolutions in the but it made 3.If you draw a contact point it will only touches 3times meaning it effectively only makes 3 rotations. You would see 3 if you turn around B with A.

Anyone else coming back to this video because the Veritasium video gave you Deja vu?

veritasium just did a video on this and i had to go back and check if MYD did one too ^_^

I interpreted the question as "how many times does the circumference of circle a fit into the circumference of circle b" which would be 3. Circle a does rotate by 360° 4 times. I'm not a native english speaker so I might misunderstand the word revolve but it feels as though my interpretation leading to 3 as an answer is more correct. You can in fact prove it with a practical example. Instead of writing "circle a", draw an arrow on the smaller circle. Initiate it with the small circle above the big one with the arrow pointing towards it. Then roll the circle around and count every time the arrow points towards the circle. It's 3 times. What you counted is basically how many times the arrow would have pointed downwards isn't it?

That depends on which way you define the revolve. If you define one revolution as point P on circle A making contact with the ground, the answer is 3.

It depends. At one instant the orientation goes back up to the initial state, but the point touching the larger circle isnt the one that was touching it initially, which means it hasnt actually done a full revolution

In a rotation of the unit circle the contact distance is measure as 2 pi r, r=1. One rotation is 2*pi= about 2*3.14= 6.28 units in distance (it's circumference). The distance determines 1 rotation not the orientation. Circle B radius is 3 units the circumference is 3*2*pi = 18.48 units. 18.48 units/6.28 units = 3. 360° is required for a rotation.

you're only rotating the circle until the word "circle A" is upright, not when the point of contact is the same. So, you're not actually fully rotating the smaller circle until you count it as 1.

Their are 3 Rolls around the circle B 1:34, 1:45, 1:57, 2:08 However, their are 4 revolutions of the coin.

I found a beautiful way to solve this question with physics! We can consider circular and rotational motion with pure rolling. I will refer to big circle as A and small as B. Now we consider A to be fixed and B to be performing pure rolling along the circumference of A. The center of mass of B is performing circular motion at radius (r+r/3). So it's linear/tangential velocity would be -> (r+r/3)w1 where w1 is angular speed of COM of B. Now applying the condition for pure rolling, we get (r+r/3)w1 = (r/3)w2, where w2 is the angular rotational speed of B. From here we get w2=4w1, then applying basic kinematics, time taken for w1 to cover 2pi, is the same time taken for w2 to cover n(2pi). we equate the equations (just use time*speed=distance) and we get the number of revolutions (n) to be 4.

If you think of it in terms of gear ratios, your assumptions are wrong. If you try to get leverage by using gears, a gear with 3x the radius of a smaller gear will rotate the smaller gear 3 times for every full rotation of the large gear.

Wohoo! I got 4 right away! Then... I remembered I've watched this a few months ago. :(

The ambiguity in this problem parallels the definition of a day in astronomy. One way gives you the solar day, which is 24 hours. Counting it the other way gives you the siderial day, which is about 23 hours and 56 minutes.

In 1982 I was in freshman Algebra class and our teacher talked about this question. A friend of mine talked to one of the students who were protesting and he got the explanation from them, and he showed it to me.

While circle A is rotating around circle B, circle B is rotating around circle A, which makes it harder to visualize. The circumference circle B is still only 3 times that of circle A. Likewise if you rotate one quarter around another quarter it would rotate two times, although the two quarters are the same size.

Well I got three like the test preparers intended, so I count it as a win.

Next time make a mark on circle A and do it again. You'll quickly see that A revolves (not rotates) 3 times. I used algebra to show the maths behind the answer, it's hard to follow in the KZfaq format but bear with me. If you write it out yourself it should become clear. I'm afraid I also don't have the symbol for Pi on my IPad thus why I use 'Pi'. rA means radius of A, rB, radius of B. DA diameter of A, DB, diameter of B. In order to go from rA to DA one must do the following: 2*rA*Pi=DA rA*3*2*Pi=rA*6*Pi=DB The Pi's can be cancelled to give 2*rA=DA 6*rA=DB DA*3=DB Thus the answer is 3 times. The numbers don't lie.

Ah I see. I was visualizing it as if the big circle was flatbed out which would mean 1 rotation would only count once the top of the small circle was facing away from the center of the big circle. Kinda like if the smaller one was effected by gravity. But when it asked for a rotation it means like a literal full rotation

We used to have to design a new "test" to locate our best field technician each year. Writing totally unambiguous questions is pretty hard to do. Sometimes, we'd even have to give up when writing a question meant to address a real issue from the field. The result was that many of our questions ended up being pretty simple. One of the issues with test writing is, in my opinion, a lack of enough proofing of the test. I had one field tech who wasn't all that bright, but he was a hard worker. EVERYTHING that went to the field was tested first by him. If he "got it" we were good to go.

It is like a train crossing a platform. If the length of both is the same, still, the train will have to travel twice its length, to completely clear the platform. Basically, it will have to clear the platform, and then cover its own length.

I know it's 7 years old and an irrelevant fact, but I enjoyed that the thumbnail of the video showed enough of the problem to start thinking of it before "spoiling" the answer by watching the video. Seems like a great format for educational videos (when the problem is simple enough for a thumbnail to show it, anyway).

so, if the areas of A and B are 1/3 and 1 respectively, we can divide by pi and multiply both by 2 in order to get the diameters, so we get 2/3/pi and 2/pi. We divide B / A: pi's cancel out, so we're left with 2/(2/3), which is 3.

Image not drawn to scale?????? 😂😂

This question is wrong on even more levels. From a physics standpoint, the answer is 1. The reason being is that the word "revolve" means to orbit something, and Circle A orbits around circle B once.

The general solution would be: R - radius of the rolling circle S - Radius of the static circle N - number of rolls N = (S+R)/R For a general explanation of why this is the case, note that the path made by the center of the rolling circle is itself a circle of diameter 2π(S+R). For a circle rolling inside, N=(S-R)/R

It is like the smaller circle turns 3/4 of its own circumference for each revolution.

If you spin both circles without changing their position, the answer is 3.

Just imagine rolling it on a straight line. You get three revolutions. Then take the end of that line with the circle at it and curve it around into a circle. It rotates an extra time as you do this. Or imagine the smaller circle sliding around, without rolling, just keeping the same point (bottom of the text) touching the larger circle as it slides. It will rotate once despite not having rolled over any of its circumference.

I saw a neat graphical solution to this: Starting with the first diagram, unroll circle B so the circumference becomes a straight line. Looks a bit like this: O______________ where the O is circle A. Roll circle A along the line. It makes 3 rotations to get to the end: _______________O Then roll the circumference plus circle A back around to form circle B. Circle A makes another rotation, so 4 in all.

unless you account the friction and thus the counter rotation of circle b...what would happen in that experiment, I would like to see that outcome. I think others have argued this point of view, I would just like to see if it comes out to 3 or an even different number than is given as an option.