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How an SAT question became a mathematical paradox. Head to brilliant.org/veritasium to start your free 30-day trial, and the first 200 people get 20% off an annual premium subscription.

I invented Snatoms, a molecule modeling kit where the atoms snap together magnetically. Try it at ve42.co/SnatomsV

Huge thanks to Dr. Doug Jungreis for taking the time to speak with us about this SAT question.

Thanks to Stellarium, a wonderful free astronomy simulator - ve42.co/Stellarium

Thanks to Newspapers.com, a database of historical newspapers - ve42.co/Newspapers

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References:

Summary of this problem by MindYourDecisions - • Why did everyone miss ...

More cool math about this problem by Kyle Hill - • The SAT Question NO ON...

Discussion of a solar day by MinutePhysics - • Why December Has The L...

Murtagh, J. (2023). The SAT Problem That Everybody Got Wrong. Scientific American - ve42.co/SATSciAm

United Press International (1982). Error Found in S.A.T. Question. New York Times - ve42.co/SAT-NYT

Yang (2020). What's the hardest SAT math problem that you've seen? Quora - ve42.co/SATQuora

Coin rotation paradox via Wikipedia - ve42.co/CoinParadox

Simmons, B. (2015). Circle revolutions rolling around another circle. MathStackExchange. - ve42.co/CircleRoll

Sidereal time via Wikipedia - ve42.co/SiderealWiki

Solar Time vs. Sidereal Time via Las Cumbres Observatory - ve42.co/SiderealLCO

Images & Video:

Zotti, G., et al. (2021). The Simulated Sky: Stellarium for Cultural Astronomy Research - ve42.co/Stellarium

Newspapers from 1980s - 1990s via Newspapers.com - ve42.co/Newspapers

SAT Practice Test via the College Board - ve42.co/PracticeSAT

Revolution Definition via NASA - ve42.co/RevolutionNASA

Revolution Definition via Merriam-Webster - ve42.co/RevolutionWebster

Earth motion animation via NASA - ve42.co/OrbitNASA

Satellite animation via NASA - ve42.co/SatNASA

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Special thanks to our Patreon supporters:

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Directed by Emily Zhang

Written by Emily Zhang and Gregor Čavlović

Edited by Peter Nelson

Animated by Ivy Tello and Fabio Albertelli

Filmed by Derek Muller

Produced by Emily Zhang, Han Evans, Gregor Čavlović, and Derek Muller

Thumbnail by Ren Hurley

Additional video/photos supplied by Getty Images and Pond5

Music from Epidemic Sound

Wait, it is night

It is 10 PM where I live and now I can't sleep😂

Bruh it’s 16:46 where I am Got back from school and just did some homework now I’m eating snacks then I will play games

You're mentally challenged

@@zayansaifullah2008 same

"I just put 3 down. I figured that's what they wanted". So depressing if you stop and think about it.

That's what the school system teaches you

@@Magst3r1 Which is a good lesson for the real world: Learn to pick your battles. When it's a trivial issue, don't waste your time raising a big stink about any concerns you have. Just do what you're expected to do and move on.

I thought the sentence "I found a mistake, no one cares." Was great too 😢

You (as I) initially analyzed from a gear-ratio perspective. The problem is more subtle. (see below addnl comments)

My brain didn't fully accept this until I pictured a circle going "around" a straight line segment in the same manner. Picture a horizontal line segment, circle positioned above it at the left end, bottom (not right or left side) of circle touching the end of the line segment. The circle travels to the right along the length of the line. Then to flip itself around the right tip of the line to the bottom side it has to undergo a 180 degree turn, but while doing so it travels no additional distance along the line. (Its centre travels a distance along a semicircle, but the part touching the tip of the line does not.) Then back along the bottom of the line to the left, then another 180 degree rotation back around the left tip, to the top again. Total distance traveled is just twice the length of the line. Number of rotations is some amount to accomplish that traveling, PLUS one additional complete rotation. Same thing for any convex shape that it travels completely around.

I hadn't watched this far when I wrote that, but he almost describes this at @11:15, though for some reason he stops after only one side of the line.

This is a good explanation.

Thank you, that really helped put the broken pieces of my brain back together. 😂 Much appreciated. ❤

Thank you so much. Was going mad

Great explanation thanks

Man... The diagram at 10:50 in this video is so great! It is so awesome to see the reason visually like that. Simple and perfect.

Reminded me of Minkowski sums (addition of convex shapes). He only talked about convex shapes in his example, but if the shape can be concave, then the formulas do not apply anymore interestingly enough! Because rolling on the outside where the shape is concave is the same as rolling inside a convex shape! So you'd have to sort of do a mix of both formulas in those situations !

I found myself understanding the problem less and less until that exact diagram

Yeah, that one really helped me understand this. I was super confused otherwise

Yes, I agree! So good.

What is so interesting about your videos is that almost 100% of the I couldn't care less about the topic. Yet, I'm still enthralled through the whole thing. That is most definitely a compliment just to be clear. I love that you love to teach. That's all that matters.

This is great! I want to point out that the more general solution doesn't seem to apply when the shape is not convex. I need to explore this more deeply and check the literature for details and edge cases. To see why convexity matters, take the example of a small circle (circumference c) rolling around a large circle (circumference C), and cut out a segment of the larger circle that the smaller one can fit through, leaving a circular arc of length of C-c so that it rolls around the outside and then the inside before returning to its starting place. The smaller center then travels (C-c+c)=C for going around the outside + (C-c-c)=C-2c for going around the inside and c/2+c/2=c for moving between inside and outside 2 times, so a total of 2C+2c-c= 2C + c. The perimeter of the shape is 2(C-c)= 2C-2c, so the difference isn't c, but 3c, meaning the smaller circle revolved N+3 times if C=N•c. N+3 isn't a general rule for a circle rolling along nonconvex shapes, just for this example. This will characterize nonconvexity in an interesting way that seems like it must appear in the literature but I don't remember a name for it. Reminds me of the Euler characteristic, but this is not exactly a topological property.

ouu glad u told it

This was a great video! Blew my mind when I realized how I was wrong!! Good to know question wordings can be so important, eh?! 😁😉

I was confident that I was right, but because of that, I was then confident I was wrong

I'm just glad I got the correct wrong answer

same

Are you familiar with Symmetrical Sequence Component theory created by Charles Fortescue in 1928? In this work he proves why 3n+1 harmonics are positive sequence (rotate in the same direction as the fundamental) and why 3n-1 harmonics are negative sequence. This comes down to this very coin paradox problem

what was you trying to imply here bro 🤣

Love this channel!! Considering the appalling quality of education in general, and it's quite obvious when reading comments in various social media platforms, I think that "Brilliant" should be completely free and not just for a thirty day trial. This world needs all the help it can get to raise the level of intelligence and not just knowledge.

But if it was completely free how would they pay their instructors Or pay for their servers? A better idea would be for schools and school boards to put their budgets towards things like that, use all these resources on the internet like brilliant or veritasium or three blue one brown as teaching tools in the classroom

5:58 mind blowing moment right there. I love it.

I've played it so many times so many times so many times

7:44 whoever came up with that headline deserved a raise.

Not sure of how it works out, but after initially thinking it was 3, I started thinking about the specification that it was the *center* of the circle A that has to return to its starting point. So I redid the math using that center, and the path it would "draw" as circle A moves around, and that path would create a circle with a circumference of 8pi. So 8pi divided by 2pi (the circumference of Circle A) equals 4.

I did the same lol

Ahh, yes. Standardized Testing: learning to try to guess what the questioner means and hoping they are not incorrect/have bad wording, and then guestimating what they are looking for, all in an allotted amount of time that *requires* that you don't actually think through the problem in multiple ways or do some testing yourself - quantity over quality. Surely people haven't been denied learning opportunities based on this. That said, awesome video, as always. Challenging us to think in ways *literally* outside our own perspective. The best type of education.

Back in the late 90s, I skipped the SAT entirely, went straight to 2-year college, and then transferred to Columbia. I asked admissions if I would have to take the SAT, and they wavered it because I was transferring from 2-year college and they would base their admissions decision on my combined HS and JC transcripts. So... yay me! Didn't have to stress over the SAT or take placement exams. Was just a straight shot without any extra stress into a BA in the prestigious field of General Studies. 🤣 Although, a lot of universities are no longer requiring SAT or ACT scores, so that's good. I know at least one person who is an otherwise incredibly intelligent person, but just doesn't test well. Anxiety messes with their recall ability, making it very difficult to regurgitate facts during tests. And then the answers to everything floods back to them when the test is over, which is sort of like salt in the wound.

Great video. I have been working for months on my own updated video with the topic of sidereal days, so those interested may find a few surprises not covered here. It's a wonderful topic and makes me realize how little astronomy we learn.

I really felt the absence of "Hey, this is Presh Talwalkar here!" at the start of this video

Ic thanke thee.

Doing some math with sidereal days and orbits of moons are what made me figure all of this out as well. Was figuring out phases of the moon with sine waves and stuff and noticed that indeed, if you tell that the moon does 12 orbits in 1 year, you actually end up with 11 full moons per year because it needs to orbit a bit more than a full orbit to end up at full moon again. This is what made me realize that the number of full rotations in an orbit (which the SAT question effectively is asking about) is the amount of "days" + 1. So 'lo an behold how confused I was when the SAT question didn't have a 4 in the answers lol

I have not been taught this in school. I had no idea this existed at all

Way to slide in your own video. Kind of spammy if you ask me.

Your way to develop content for your video from a question on a quiz is impressive. Thank u so much XD

My brain just melted in a shower of understanding. If anyone besides veritasium explained this I’d have been lost lol. Veritasium is a legend

This is mindblowing, so simple yet so important

I think this is the most interesting video to date on this channel... well done!

Fantastic! Only just caught this channel and love the fascinating subjects it brings up and the beautifully clear explanation.

To all the 1st posters: KZfaq takes up to 15 minutes to gather data on a video before showing stats. Everyone in the first 15 minutes all think they're first.

😂

Nuh uh

I’m 9 minutes in and I says 12k views and 150 comments

haha

Yeah but I was first before you even wrote this. . .

With the coins its easier to see (6:23) If you count the rotating as a rotation when its facing the same rotation as it started then its 2 But if you look at it from where the outer coin was touching the inner coin (the bottom of the coin) that only touches the coin again after it full goes around Depends on how you look at it Its very confusing

Interesting yet simple, as a person who uses logic rather than actually doing the question to find the answer. This video was fun to watch .

I think that we should consider the point of contact of both coins at the beginning of the circular motion to define how many times the small coin has used its entire length to complete that circular motion around the big one. The return of the coin in the upside position does not mean that the small coin has fully used its length. 2 type of movement should be considered in this scenario: The first one where the small coin moves around the big one while keeping a fixed upward position and the second one where the small coin keep goin upward and downward continuously and alternatively. What do you think?

OMG thnks for clearing the mists on this.... I never was able to work it out and had a great aha moment when you showed the path tracked by the center of the rotating circle. I thought of those measuring wheels that real estate agents used to measure lot sizes back in the 80;s.. running the wheel across the distance of a flat back yard, not a problem, the center of the wheel has to travel the distance of the yard length, and that equates to the number of revolutions.. no. x wheel circumference = answer. But when the center of the wheel has to travel around a circle, its a longer distance, because its centre is following a longer circumference path versus the circumference of the original circle.

huh ?@@fenrirgg

this was amazing! please make more stuff like this

The 1872 novel “Around the World in Eighty Days” had a plot that depended on this kind of situation. Phileas Fogg traveled around the world eastward, against the earth’s rotation. Though initially he thought he’d missed the 80 day deadline by some hours, in fact only 79 days had passed in London. One extra rotation had passed beneath his feet. He won the prize, married the girl and lived happily ever after.

Fun!

That is what first came to mind when I first saw this problem. I didn't immediately jump to 4 as the answer, but I knew 3 wasn't correct.

There's a recent TV version starring David Tennant that I remember that from.

@@Mark73 Really? I might have to check that out

Glad about him.

Awesome video. I remember learning about the paradox in high school in the 90's with the ribbon and wheels. didn't think much about it then, but it's cool to know why it was on the agenda!

Thank you for this video and explanation. It gives me some validation to my school days experiences with educators and my diverse thinking methods and answers.

Final additional (hopefully helpful) comment: Consider the smooth rolling disc on the INTERIOR of a circle. Start again with a square (see below) and roll the disc on the inside edges. Moving from L to R along the top (inner) edge the disc rotates counter clockwise until it finds the first corner (upper R). To continue, the disc must pivot 90 deg in the clockwise direction. And so on for the other vertices. The result of the vertex pivots in a net pivot-related 360 deg clockwise rotation which negates the length consuming 360 deg counter clockwise rotation for a net result of ZERO rotation (if the square perimeter equals the disc circumference). Again, go the N-gon version and increase N to create a circle. The motion always introduces a reverse full rotation no matter what the perimeter length of the inner disc, coin, shape.

Dude! That threw me off for sure, and I feel a lot smarter for knowing how it works. However, I can see myself making this same type of error again in the future. Still, it's brilliant! Solid work and great explanations.

an easier way to wrap your head around this is to notice that the the center point of the rotating circle actually travels along a larger circle. In order for the smaller rotating circle to travel the exact distance equal to the larger circle you have to first center its center point onto the actual path its traveling, which is the line representing the larger circle. The lower half of the rotating circle should be inside the larger circle and the upper half should be outside of the larger circle leaving the center point to travel directly along the actual larger circle directly.

In college, I took a poetry class and once had an answer marked wrong on a test. Confident in my response, I reached out to the poet themselves, who affirmed I was right and even communicated this to my professor. Despite not being a fan of poetry, that moment made me quite proud!

Did the professor change your grade?

@@QYXPI had a question marked wrong on a chemistry test that the professor refused to accept was actually right. The head of the chemistry department came to our class and embarrassed him in front of everyone showing why I was right and he was wrong.

literature tests: q.e.d.

@@Sciguy95 very cool, but also unprofessional

@@pongmaster123 We don't have the full backstory and never will, it might have been well deserved. Don't feel offended for some random obtuse chemistry teacher that may or may not even exist.

The centre point of the turning outer circle travels the perimeter of an unseen circle that has the circumfrence of r1 + r2. Thus in the first qurstion the circle should revolve 4 times around itself.

If a reference point is chosen on the smaller circle which is then rolled on the bigger one. What is arc length on the bigger circle when the reference point comes in contact with bigger circle? It is circumference of the smaller circle. That's 2.pi. That's one rotation. Hence it would take 3 rotations for reference point to get back to starting point on bigger circle which is as per intuition.

Rotations aren't based off reference points and surface length. That's why everyone thinks it's three. A rotation is based off the smaller circle doing one physical rotation, which is not the same as it's entire surface area touching the bigger circle one time. Rather the right most point returning to the right most position after spinning.

In other words, a rotation is an object rotating 360 degrees. When object A has a radius of 1 and object B had a radius of 3, the point of contact will repeat every 480 degrees and will repeat 3 times. 480*3 is 1440. 1440/360 = 4 rotations.

The correct wording for answer to be 3 would be so complex 😂, a point to travel around the bigger circle is terms/units of circumference of smaller circle is ? XD@@shawnmeehan8962

So one question: on turning machine if we have workpiece diameter of 40mm, and for smoother surface we are pressing over workpiece with rolling tool in holder with it’s own drive, translation of revolutions is not 1:2 but actualy 1:3? If main spindle (40mm) is spinning with 500 revolutions/min, a driven tool should have 1500 because it would make 3 turns for 1 turn on main diameter? Or is it right 1:2, because as seen in video, from circle’s perspective there is seen theoretical difference between circumferences? Edit: if both diameters are spinning than we can take the normal ratio between both r’s whitout adding r.

Love his videos, informative as always

I thoroughly enjoyed that! Thank you for helping make physics and maths enjoyable for my kids and I.

I really like the space applications. I imagine though that this would be applicable to anything involving (more complicated) gears too? (The first thing I thought of were watches.)

That part about the circle rotating around the triangle was mind-blowing. You instantly understand why it's not the same if the circle rolls on a flat line or rolls on a curved line

That was the "aha" moment for me too.

This

There were 3 aha moments for me

if you divide the straight line in half and start to roll along it at the "top" to the end you then can make a 180, roll around to the "bottom" and then go in the other direction, make another 180 and keep going until you reach your starting point. These two 180 needed for the direction change add the 4th rotation 🤯

The earth around the sun was a fantastic example for why the frame of reference matters, especially with the graphic

Always look forward to your videos👍

At time 3:50 he does a rotation but if you follow the rotation he only does a half rotation for the coin to be head side up and a full rotation when the coin reaches the top with the head side up... watch the corner of the quarter as it rotates. So it maybe 3 but I am not sure even though I did the problem out on paper, in my head and by calculator and it came out to 3 but wow the explanation through the whole video is quite amazing

Sir its rotating once but only half of its circumference cover distance (in contact with other coin)

The answer is 3 and 4-- it simply depends on the chosen frame of reference or vantage point from which you observe or "measure" the relative motion (rotation in this instance). If for instance you were a tiny denizen sitting stationary on the smaller circle as it orbits the (larger) stationary circle, then you'll observe a total of 3 rotations by the time your tiny planet returns to its original starting position. Make this same observation from space, and you'll observe 4 rotations of the smaller circle. From this distal vantage point, a complete rotation, from your perspective, will not be the same as that of an observer sitting on the rotating smaller circle. It will in fact be 1/3 less the number of degrees/radians of radial motion. A "complete rotation" therefore is a function of frame of reference and vantage point from which observations are made/measured. There's really no "paradox" here, but rather a difference of vantage points, and therefore what constitutes a complete "rotation" from the perspective of the observer or "measurer" of those rotations. The exact same "paradox" could be said to exist in examples involving linear motion. Stand by the side of the road and observe a car as it passes-- it will appear to move linearly very quickly, from your vantage point. Stand in the middle of the street, and observe that same vehicle as it approaches, and it will not appear to be moving linearly at all as it rapidly closes the distance on you. The difference in apparent motion experienced by the observing is due to the difference in vantage point, not because of some "paradox" which does not exist.

"The answer is 3 and 4-- it simply depends on the chosen frame of reference or vantage point" If this is true why did they say the question was wrong. If 3 is a valid answer.

@@linsqopiring6816 Not sure how a question can be wrong. An answer perhaps, but not a question. To understand why 3 is a valid answer, please read my original response, above.

@@BetterByThePound1/videosThen you didn't watch even one minute of this video as he clearly explains how none of the answers are correct and that's not even talking about the bad wording. Since you seem to be unable to find this information in the video I'll spell it out for you and give you the time stamp. You're welcome. 0:52 "I'll tell you right now that option B or 3 is NOT correct".

@@linsqopiring6816 Wow, you are really lost sport, and don't even seem to be able to recognize it. I'm familiar with the problem, have watched (and understood) the entire video, and have written out the explanation which clearly explains the alternate answer. Not sure if you're just bored or triple boosted, but go find something better to do that suits your limited cognitive capacity. You're welcome.

Yo, i haven't watched the video yet, i already calculated the answer which is obviously 3. Why don't we consider perimeters and just compare them?

It’s interesting to apply this to objects that revolve around their axis in the the opposite direction they revolve around the center shape. Also interesting to apply rotations to the center object; and to do this in 3-space around different axis.

I was confused for a second until I realized that if you set the radius of the big circle to 0, or in other words rotate the smaller circle around a point on its circumference, it takes 1 full rotation for the circle to end up back at the start.

this comment helped me solidify ny understanding thank you

Thanks. This is a great way to think about it! ❤❤

Genius comment, thank you!

finally! i got it

That idea helped me as well

How many times does the point on the circumference for circle A, touch the circumference on circle B when it returns to the starting point? There's how the question should have been asked at the end, after giving the necessary information in size difference of the 2 circles 😊.

i swear when i say this, i learned maths way easier from you and other maths videos on youtube than i ever did anywhere else, especially school. Thank you guys!

Who would have thought that some of the best educators in the world are better at explaining stuff than your average teacher

Mind boggling. I guess its similar to how our vehicles track distance! Super interesting. Love your videos, Derek!

This was fantastic man! 👏

I thought of it as the distance the center point of A travels is 8pi (assuming the radius of A is 1 and the radius of B is 3). Then divide that by the circumference of A: 8pi/2pi= 4. Is this actually how this works or was this just a coincidence?

Another fun way to conceptualize the N+1 is to ask what happens if the circumference of B is 0. A still has to rotate around that point, one time. Great video.

Brilliant. Wish I'd thought of that!

I thought of it as a circle rolling three times along a straight line, and then one more time as the straight line is curled into a circle itself

That's actually a great example.

Yes because by measuring from the center of the circle, you are offsetting by the value of the radius. So you essentially just add up each circle's radius to get the number of rotations of circle A. So if Circle B's radius was zero, the centre of circle A still has to travel around it's own radius of 1.

this helps a lot!! thanks!

And then I asked myself, how are you going to make a relevant video about THAT? And you nailed it.... Bravo sir!

I guess, if you think about it. . . it's not just traveling across a straight line, it's also traversing past a shape's vertices. It's very cool. This is the kind of KZfaq I live for.

I didn't catch any of the issues brought up, but purely from a wording standpoint, there were two primary issues I saw immediately: 1. it doesn't specify which circle's centre needs to reach its starting point, it only implies it 2. it specifies when the centre "first" reaches its starting point, which is after zero revolutions - that's how a starting point works... thought that was amusing

4:00 I think that's one revolution but it may seem as 2 because you are not focusing on edge of coin you are focusing on arrangement of the coin. If you focus on points on the edge of coin and mark them(imagine) you would see it makes only one revolution.

Simply the centre of the smaller circle is rotating about the circle with radius (3+1)R . 3 for a bigger circle and 1 for its own 🙂...

Thinking about this yesterday and I realized the extra rotation becomes intuitive if you shrink the large circle down to a point, and rotate around that. Even though the diameter of the circle it's rotating around is zero, the "small" circle still has to make a full rotation to return to its starting point.

Imo this is a more immediately intuitive explanation than what was in the video!

This is a dumb fake question to convince you that the Earth is turning. These two clowns couldn't solve the time of day.

I also thought of this same explanation

Excellent!

Great visualisation. This should be pinned

vertasium explanation is effed up to say the least.. Doug’s explanation is spot on

10:44 The circle traveling on the outside of the triangle helped me visualize the solution best.

6:12 I watch this multiple time still in amaze

Would this apply to gears and cogs in mechanical systems?

Solved it on a tissue-paper in under a minute. After seeing how many people fall for this I will start using it as an interview question, thanks!

Three of them got it right by saying that the question was wrong.

likebot more

1k likes within 5 minutes? Wow!

Also 3 is still a correct answer to the problem it’s just badly worded. So everyone who answered 3 still got it right.

The question is incomplete. It should ask how many rotations does the small circle make, on its centre point, to rotate exactly once around the large circle.

@@stevejones1318they forget an A. If there was one more A in the question, it would be correct.

Love these kinds of videos!

≈ 10:22 into the video there is an animation running incorrect at the "or at the inside". With V=0 it the inner zircle has to rotate around itself against it "driving" direction. If it goes clickwise (as it does) along the inside, then it at itself would have to rotate counter clockwise. On the diameter 1 to 3 ratio it would be 3pi-1pi=2pi.

thank you; really enjoyed this vid

I was wondering why this video was recommended to me until you got to the Sidereal section. As an astrology fan, this is super useful to the ongoing arguments between Sidereal and Tropical astrology. 😂

“(B)3” is the correct answer to a question that the writers of the question believed they had asked, but hadn’t actually asked. At least your post included an accurate illustration of the radius variation that matches the “(B)3” answer. Someone else posted a less than thorough version of this SAT question but he had only a fraction of the detail you’ve provided.

I loved the "I hope so" answer from Doug at the end. It highlights the most important lesson I learned during my education: "I might be wrong."

I feel like I already had that lesson before education. I feel like the most important lesson for me - that helped me grapple with how to be effectively wrong - is how to think in terms of probability than binaries.

@@hieronymusbutts7349❤

A harder lesson still is, "I might be wrong and I'll never know it." This is why people who fear the Scientific Method really shouldn't. It's also a primer in the Scientific Method, perfectly demonstrating why the goal isn't to prove a hypothesis is correct. Rather, the goal is to prove a hypothesis is NOT correct. Similarly, it demonstrates why the strongest theories are those derived from inductive reasoning (multiple specific cases lead to a generalized conclusion), rather than deductive reasoning (a generalized case leads to multiple specific conclusions).

Agreed! The most important thing I learned when learning math or physics or any objective knowledge is that by admitting the probability your are wrong is the best you can do to advance in those fields. I love to think that the physics, as we human know and define it, is always more correct than before but never (at least in the foreseeable future) completely right.

I always thought this way, but I learned in the working world that if you acknowledge that you could be wrong other people will assume you're wrong.

This brings to heart the real life experience of two wheels of a vehicle on an axle with a differential gearbox and one without a differential box.

Very thought-provoking. I have a question. Every other examples and the mathematical proof were about a circle rolling around something, as Dr Jungreis mentioned the "no-slip" condition. Does it mean that the earth's rotation speed about its axis is the same as if it were rolling around a huge wheel the same way it orbits around the sun? Does it just happen that way?

Here is another way to intuit this: Run this experiment yourself using coins, but rotate both coins against each other in opposite directions. Notice if you rotate both coins the entire time, they each rotate exactly once before they are perfectly aligned with tops of heads at the top again, as you'd expect. Note the total rotations between both coins was two rotation. Also note the heads were actually aligned twice (once upside down) Now play around with this some more by changing how much you rotate the left coin. If you rotate the left coin 0.25 rotations and continue rotating the right coin, you will rotate the right coin 1.75 before their heads align the second time. A total of two rotations between the coins Similarly, if you rotate the left coin 0.75 rotations and continue rotating the right coin, you will rotate the right coin 1.25 before their heads align the second time. Again, a total of 2 rotations Taking this back to the original problem at hand, if don't rotate the left coin at all, you must of course rotate the right coin two times before their heads will align the second time Now introduce different ratio circles and observe, but be careful not to break your brain 🤯

the thing that bothers me about this problem, and the idea that you expanded it to, is that it relies on some visual indication of "upright". But in reality, restoring the "uprightness" of a circle only results in a full rotation, on a flat line. Once you place a circle on any kind of curved line, using the "uprightness" of the circle to judge a rotation gives you a false answer. In the example above, with one circle having 3 times the diameter of the other circle, the "A" is restored to "uprightness" 1/4th of the way around the larger circle, but if you look at the original contact point, it has not yet come back in contact with the larger circle. Therefor the smaller circle has NOT actually completed one full rotation, even though the letter "A" is upright again. The original contact point is to the right of the letter "A", and that contact point won't arrive back on the larger circle until the smaller circle gets 1/3rd the way around the larger one. Another way to think of it is, remove ANY markings from either circle, and ONLY place a dot where the contact point is. Once this is your only reference, the confusion caused by "uprightness" disappears, and the fact that the circumference of the smaller circle is 1/3rd that of the larger circle, becomes clear. If it has 1/3rd the circumference, it would obviously take 3 full rotations to return exactly to its starting point. Had the SAT question simply placed the "label letters" OUTSIDE the circles, the false sense that the question couldn't be answered as written, would not have existed. The correct answer truly WAS "3".

Could be helpful if you showed what extra points on the moving circle touch the other circle vs the flat line

what if the circle rolls inside a square? the circle does not touch the entire surface, only a part. Is there formula N+1 or it somehow changes?

See below. Rolling on a the perimeter of a square leads to some insight into what is happening here. For a square perimeter, the disc must pivot at the corners which effects a rotation of 90 deg WITHOUT PROGRESS along the perimeter. Rolling on the outside introduces an addnl clockwise rotation of 360 deg. Rolling on the inside introduces the same but in the opposite sense of the rotation related to length consumption along an edge --- which results in a reduction in the net rotation. Extend this picture to an N-gon and consider N being large (circle) and you can see what is happening. The thing to recognize is that the map from the interval defined by the rotating coin surface to the fixed coin surface --- although one-to-one is not length preserving (so to speak, as would be the case for gears which is why we get this one wrong --- folks are thinking about gear ratios). This is a fantastic physical realization of an interval to interval map that is one to one but results in some stretching hence the 1 to 2 resultant ratio for case of equal sized coins.

There's been a couple of videos on this particular SAT problem before. I'm an engineer and a bit of a math nerd myself, so I understood the point the other video was trying to make. However, Derek uses both computer graphics and real-world cut-outs to explain things, and that sets this video apart from the others. Very elegant, as always, Derek. Love your vids!

I haven't watched this video yet, but based on the thumbnail, it is one that super annoys me because the answer depends on perspective, how you view the english language. I should go find my comment from the past, but first I should watch the video. I just know I will get annoyed when I do, lol

Thank you, for a great YT comment!

haha, good point@@Redmenace96

@@gruangerhave you watched it yet?

Watched it :) The video didn't annoy me but it is the problem I remember@@Alpha_Online

I thought I would need an Excedrin, but you explained it great, thank you!

This may be the coolest video I've seen on this channel.

This is amazing. Thank you.

It makes sense now! At first I also thought 3, and was so confused about the answer being 4. But when the guy said "it's always the distance from the center to the point of rotation", it all clicked. To travel 3 radii, circle A would have to travel along the perimeter of circle B. But it would mean circle A has no radius at all. A dot can do it, but not a circle. Or the other way to look at it - if we remove the circle B and place circle A on a flat surface, how many rotations does it make until 1 full "revolution"? One. Not 0, like the answer 3 would suggest. In short - we have to account for both the circumference of the larger circle *and* the smaller circle! And that's why we get 1 added to the ratio - it's the radius of the smaller circle divided by itself

My own interpretation is that we need to trace a line from the outer end of the smaller circle as it gors around the larger circle. Now image the much larger circle formed as a result of tracing the line from the above sentance. The diameter of the new large circle is the sum of the diameter of the two circles inside it (2r + 6r, so total of 8r). The circumference of this circle is 8πr. The circumference of the first smaller circle is 2πr. Divide the two and we get 4.

I came up with the answer, 3, in a second or two, and then wondered "how could that possibly be incorrect". I spent the next 18 minutes learning how. Great video!

An actual honest response, lol at those who said they instantly concluded it was 4 rotations

It is the kind of problems which when you see the solution you feel dumb because the solution is so obvious

You weren't incorrect

i was surprised cause my intuitive answer was 4 by looking at the circles but it was not an option so i thought 3 XD

The answer is 3 only the video is useless

I found that if you take the ratio of (big radius + small radius)/(small radius) will get you the correct amount of turns every time. (1+1)/1 = 2 for the two quarters, (1+3)/1 = 4 for the SAT question.

You've just increased my self-confidence in maths, thank you. I too looked at the other possible answers and thought - as it's American PI may as well be 3 :D The thing is, I totally understand why the answer is 4, having been shown it. I am becoming more convinced that I was taught maths badly.

12:35 blew my mind - awesome vid

It's amazing to me that a maths question at such a senior level would tlstill be multiple choice 😮 Most of my high school and university maths I got the answer wrong but passed from the marks I got for the various steps of working it out. If it was just marked on the answer I'd have been screwed 😅

If the center of A were directly on the circumference of B, would it be exactly 3 rotations? If a 4 inch separation is 4 rotations, a 3 inch separation (between centers) should be 3 rotations.

It makes the story even better to know that one of the students who found the SAT error became a mathematician.

They should have offered him a job making the tests.

The fact that he corrected a mistake from the very test that they use to determine if you were good at math probably is a good point to bring up to get hired or accepted for a job or university Its also nice to see that they aknowledged their mistake, admitted it to everyone in news, and dismissed the question from everyone’s test. They have admitted to everyone their mistake, knowing well that it would impact their reputation for having made the mistake Only 3 people in the whole country sent a letter to correct them, likely not many noticed or cared about the mistake. They could just “ignored it and pretend it didnt happen” like so many goverments and corporations do regularly. Even more so considering people were not sharing everything instantly using internet on a global scale

dude if he became a social worker i'd be more fascinated

@@FlorenceSlugcat Removing the question was improper and created more inaccuracy in the scores. The question was part of the test and consumed time that could have been used on other problems. At least some students failed to answer other questions correctly because they wasted time on this question. For example, a great math student could have spent 5 minutes on this question totally stumped that no correct answer was there. Now, that great math student gets this question thrown out and also gets some other questions wrong because of time. So, any student who answered 3 should have been given full credit. The test makers who allowed this faulty question also administered a faulty correction.

@@jakemccoy I agree the question should have been thrown out. When every student in one of my classes misses a question, I eliminate the item. This rarely happens, however.

This was awesome. Thank you!

this is just one of my favorite videos of all time.

claiming the plus one and min one sure it happens but have nothing to do with the center that is only a calculation point, what is creating what you call a center 2 lines from N to Z and E to W using only one you have only a line but that line is doing a lot more than a center point,and were you going to put that center in the air on a needle on a iron bar or on a planet? gearing and orbit is a lot more it is the answer to a lot of questions you have but can you ask the question from students if you do not know the answer?

Why isn't it the circumstance of the larger divided by that of the smaller circle. What does the center of it have to do with it. If you made the perimeter of the smaller circle a string would it take 3 pieces to wrap around the larger circle.

Wasn't quite ready for this on my break lol! I haven't done the math but by watching the video I have to wonder if after going 360 degree's and adding +1 if that mean's If it hits a 90 degree angle I should add (.25)(2rpi) or (.125)(2rpi) if it followed a line that had a 45 degree angle? Then again it may not work because it would still have to travel the same distance as if it followed a closed triangle. It makes me wonder if you had a closed circular object with waves in it creating many more angles if the negative angles would cancel out the positive ones. Because you closed the loop in the end you'd still wind up with +1? I'll have to either do some math or make a model lol!

One way to see the extra rotation -- shrink the inner circle to radius approximately 0, so it's like a thin wire. The circle still has to do a rotation to roll around the wire, even though the wire's circumference is negligible. (The rotation disappears from the "circle's perspective" because the "camera" does that one rotation along with it.)

You’re clever 👌

That’s some pro level thinking🔥

but why is it one? why cant it be anything else?

@@munkhjinbuyandelger10:10

Where is the paradox, when started rotating around same sized coin, point under neck of face picture was touching, after halfrotation at 180 deg where narrator started speaking again, point above head of face picture was touching the stationary coin, that means half rotation, full rotation will be when same point that was touching the stationary coin will again touch it, and in same sized coins, that comes when coin reaches starting point again. So where is paradox?? Cant they see that point that was touching at start, touches the circle again at whole 360 rotation, in same size coins. What is confusion??

My mind was actually blown watching this! Like my brain couldn’t believe what my eyes were seeing. I had to rewatch parts of this video many times…

For complete rotation the a point shouldn't always in upside, Start point as tangential direction and 1 rotation will get complete when another tangential line will achieve

The circumference of the circle is a curved 1-D space. So the question should also state with respect to which reference frame the rotations should be counted.

This is a great video. Extra points for jumping to sidereal time.

When I first read the question, my answer was 1. Then, when you came back and started explaining it, I thought ok now I know what they are asking for. But then felt reassured when you showed it could be 1.

4:20 Fun fact, the SAT actually tells you to assume all diagrams are drawn to scale unless otherwise indicated. Definetally made my life easier when I took it.

Thats convenient. In Jee they purposefully distort it

It didn't help you in the Writing and Language section...LOL, JK😂

@@scramjet7466According to my experience most of them are close, if not to scale. Anyways scale doesn't really matter for the questions in JEE

techgeek2625 was right - whether it was drawn to scale (or not) - it didn't matter in this case. The outcome is always the same. total # of rotations = ratio between inner circle to outer circle + 2πr

@@attsealevel Idk much about the questions of SAT, but judging by the level of SAT Maths, maybe some questions will be easier to solve with diagrams which are to scale.

This was way more math than I bargained for, but it was fascinating! Thank you.

Very thought provoking video, thank you

Can this explain planetary gearsets like the ones used in automatic transmissions?

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