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

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

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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/CoinPa...

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

Sidereal time via Wikipedia - ve42.co/Sidere...

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

Images & Video:

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

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

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

Revolution Definition via NASA - ve42.co/Revolu...

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

Earth motion animation via NASA - ve42.co/OrbitNASA

Satellite animation via NASA - ve42.co/SatNASA

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Directed by Emily Zhang

Written by Emily Zhang and Gregor Čavlović

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Y’all are overcomplicating a simple problem as an excuse for flunking out of community college

@@LJ3783I think there’s a greater theme here - there’s a certain hubris to the belief that questions such as this represent “intelligence.” There are…certain large tech companies that exclusively leverage SAT type philosophies in hiring to the exclusion of allowing nuance, and it doesn’t actually work that well in my opinion. Problems in the real world often don’t look like an SAT question and more often there literally isn’t a “correct” answer. If we condition people on these sorts of problems they don’t end up adapting well to an engineering trade off, nor are people who view the world from an SAT lens necessarily good at solving trade-offs in the context of a team. I think this type of criticism is that the SAT quite obviously fails to support its own philosophy of the existence of “correct answers” when the wording is wrong. I don’t say that to explain away my life failures, rather I say that because I have learned the importance of hiring people in a more nuanced way that allows for these different dimensions. Not sure if you’ve ever tried to work with an arrogant math PhD before 😂

@@LJ3783not really. The wording here is objectively bad, and dare I say, wrong.

@@justarandomguy8694I'd say that's the real issue, it comes down to semantics.

@@LJ3783if the wording is bad enough that most everyone got it wrong, then perhaps there needs to be an evaluation instead of brushing it off as semantics. Usually with tests like these it is expected for some people to get it wrong. But not a vast majority. If you say things poorly, then it makes sense that you get misunderstandings. Also, you cannot flunk out of community college if you’re not even in college. These exams are meant to loosely determine how ready you are for college. I’m not sure what your first comment was meant to say

This was a mentally challenging video to watch first thing in the morning. I'm awake now

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

To all the 1st posters: KZbin 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. . .

What makes it intuitively easiest for me to understand is to think about it this way: if the circumference of Circle B were 0 (i.e., a dot) then rotating Circle A around it would result in one revolution. So any addition to the circumference of Circle B would simply add on to the starting number of 1 rotation needed.

I came up with a similar scenario: what if the outer wheel didn't rotate at all, or was a sqaure, but was rather DRAGGED around the inner circle. It would still complete one rotation on its own, right? That's the rotation of the reference frame itself. Incidentally, from the POV of any point on the "tread" of the outer circle, they do indeed make contact 3 times during the rotation around the inner circle.

That's the kind of simplified thinking that gets to a good solution with minimal fuss. Well done.

So, is it safe to say that B could be as small as one atom and this is still true? And that it stops being true only if B gets larger than it is? I'm from the visual learner, math challenged peanut gallery.

@@velvetbees hold a quarter in your hand and keep it facing you. Turn yourself in a circle and note how (from an aerial view) the quarter would have rotated 1 time. But relative to you, it wouldn't have rotated at all. This experiment simulates how a circle with a radius of 1 to ∞ would orbit (without slipping) a circle with a radius of 0. I may not understand this well enough to explain it clearly, but that's my best attempt.

You're making the "A" (or whatever marking is on the little circle) either face upward (our perspective) or outward (circle's perspective) as the determining factor for a full rotation. So, when you start the rotation, from our perspective, you want to face the "A" marked on the circle upward, and then count a full rotation every time it comes around to face upward again. But if you want to count it from the perspective of the circles themselves, you only count a full rotation every time the "A" points outward from the circle, instead of upward.

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

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 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.

I’ve waited 45 yrs for this explanation. When learning how to make gear trains in tech school, we were given the n-1, n+1, and n solutions, regarding what application we were designing for. Thank you 😊

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!

Having the small circle rotating 3 times with the camera rotating is the best intuitive explanation of what's going on I've ever seen for something like this

It was the perfect explanation

I solved the question at the start of the video by pausing the video. MF I got 4 and then wondering the whole video why did people mark 3💀

@@anonymousguy5694 because they just wanted to answer something and there was no checkbox for the answer 4? so they assumed they are missing something and marked 3.

I didn't watch the video. But it is 3 right? Because the small circle would spin 3.141592653589etc x(radius x2) for about 6.28 before going a full rotation, while the bigger circle would spin closer to a distance equal to 18.84.

@@EmmaSquire-ks9nuWatch it and you’ll see it’s 4. But 3 from the perspective of inside the circle (i think that’s how you word it)

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

Good job, very clear. My professor in computer science used to say that if you are a good programmer and you have an error in your code chances are you are usually off by 1 somewhere.

I have found that in confusing situations in math the key error, or answer, is -1, 0, or 1.

I was a professor of computer science. My school, in the midst of my career, did a study of graduation rates, and used that to set a minimum grade on the math SAT for a Computer Science major. I could now teach more Computer Science because I was teaching less remedial math, and that in turn increased our school’s reputation.

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.

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

As an engineer, I made the same answer mistake just like anyone else till realized yeah it is the center of the circle ⭕️ which + 1 because it is running outside then yeah it makes sense.

I knew this was the case because I visualized it immediately, but I still didn't know the answer until he said it increases the distance traveled by exactly one circumference of the circle, then I was ashamed of myself for forgetting curvature introduces an extra rotation. I had learned this during mechanical engineering school and missed my opportunity to say "I know the answer!"

The part here is that it's rotating around not with it like gears then they both become flat lines and 3 to 1 ratio. How is that to blow ones mind.

Yeah

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 don't like math, but any visual explanation like this makes me engross in it for hours, replaying multiple sections to fully understand it and appreciate the fact, that how amazing it is.

Start learning Houdini. You'll be jumping down an endless rabbit-hole.

Same bro

3blue1brown gang

Idk why but my brain sucks at handling visual models

Truly, that’s the case with me too!

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 🤣

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

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??

Dear Seekers, I have an interesting story to tell. I'm nearly 70 now. As a boy, I was a total math nerd. After school I dropped math altogether (except for practical daily needs). Surprisingly, I got the answer 4 (before watching the presentation). I don't regret working with my hands for most of my working life. I've kept math as an interest but not an obsession. I have learned to often do "thought experiments" rather than using formulae. It works for most needs. Have a good life. --Doug.

I learned about this problem when calculating gear ratios of planetary gearboxes, using exactly same 1:3 ratios.

Don't read my nameDon't read my name

The same thing happened to me

That was my exact thought was gear ratios lol.

Learned about this when we talked about the moon slowing down its rotation in high school and I realised it still made 1 rotation around its own axis for every lunar month, so it could always show the same face towards Earth.

I was just wondering this. It is only for planetary gears or all gears?

It will never fail to amaze me how seemingly simple questions can turn out to go against common sense when studied further, and then can be used to add to knowledge and laws that are used to greatly change or enhance our world.

This is why common sense is not a thing

@@GameTimeWhy That's not at all what common sense is. Common sense is an ability to intuitively solve simple everyday problems such as "It is cold outside, I will wear warm clothes" or "it is raining, it is better to dry clothes inside". It is certainly not something you can use to solve complex math.

@@anteshellTrue. The major problem with "common sense" is that too many people equate "I think that...." with "It is common sense that....".

This channel information starts where common sense end. And there are many people who dont have common sense to start with

@@anteshellThis is a a hand-wavy explanation. Common sense is usually used to describe something that should be simple and intuitive and known by many people within a given area. This video shows why common sense doesn't map easily to reality and we should study things further. This also isn't complex math its basic geometry, the fundemental of math.

I am currently 6 weeks from earning a Purdue Aerospace Engineering BS, I have completed the requirements for a physics minor, ive taken 2 graduate level astronomy courses and a graduate level Space Traffic Management course that dealt with sidereal time on every assignment, but this is easily the best conceptual explanation of sidereal time I have ever seen. Genuinely incredible educational content, I'm blown away.

Hear, hear!

Keep It Simple Stupid KISS

Damn I wish to do aerospace/astrophysics too

Out of curiousity, how often do people pronounce it side real and how often do you hear cider eel? I'd seen the word before and assumed it was a compound word - and Astrophysicists seem like exactly the kind of people to read a word and understand its meaning before hearing it out loud.

@@rosly_yt You're hilarious.

I took the SAT in 1982 and scored a 780 in the math section, indicating that I likely missed just one math question. When faced with this question, I initially thought that it needed to roll around three times and then once more, leading me to select 4 as my answer. However, upon realizing that 4 was not provided as an option, I doubted myself and ultimately chose 9/2. As I doubt that I would have chosen 3, this is probably the question I missed.

But SAT authority discounted the problem. Was your score corrected?

That makes you a very sharp cookie.

Did this question stand out, or do you remember all the questions from 40+ years ago

@@JayJay-ye7ic No. I don't remember anything about the test in 1982. I'm just guessing.

Congratulations Mark. I was very smart like that too (but not exceptional). I value it, but not too highly.

I have a 1st class degree in Physics and clicked on this thinking it would be simple algebra, I had a huge grin on my face whilst being explained to how I was wrong. I love these kind of videos, I love learning something new. Never stop learning!

the phenomenon he describes is true, but it does not apply to astronomical observation the way he makes it out to be. According to their own theory, the tilted axis of supposed ball earth always faces into the same direction (towards the star polaris) in this 360 degree orbit which supposedly gives us the seasons. That means the earth is independently rotating ACCORDING TO THEIR OWN THEORY which contradicts this presentation completely because in this presentation earth is dependently revolving around the sun as if there was a mechanical connection between sun and earth, like a carousel, which we know from actual reality that it is not like this.

@@theswordofthespiritspeakstoyou Apart from getting everything wrong, it does apply to astronomical objects. I'm not sure if you're being serious though. A lot of people, people who never had a chance at education (surprise surprise), repeat stuff from other people who pretend that they believe "earth is flat" to make money of such people. I personally find it hard to believe that anyone who older than 5 can believe "earth flat".

the typical response of denial or paid actors: personal attack without arguments. You can't even stick with the topic. There is no point in having a conversation with you. Good luck.@@josephh891 btw I am seeing this channel has a few million followers making money off of spreading lies. None of the people I talk to make these amounts of cash! You might want to reconsider your insults, they don't stand the test of time... but then again so does the heliocentric model not

Yeah, I paused the vidoes calculated and divided the circumference(even did it on a calcluator and made myself realise after getting the answer how unecessary that was) and thougth the answer was obviosu and ez. Then after already calling myself dumb I got even more corrected :) But as U said "Never stop learning"

Flat earth websites are largely a creation of the intelligence community. There are legitimate conspiracy inquiries that point the finger at national and international BIG LIES. So one of the ways of getting people to ignore said theories is to "muddy the water" (a CIA term), by confusing the population. Let me give an example. Suppose the JFK assassination was really a plot...a plot by "deep-state" people who wanted JFK dead because his policies were threatening military or financial goals of the deep state. So you create a very slick "Flat earth" website, in which you also show evidence that JFK was murdered by a conspiracy, and you also mention evidence that 9-11 was an inside job, also designed by the deep state. In this way, people who don't like conspiracy theories will conflate "flat earthers" with JFK conspiracy theorists or 9-11 theorists, and just come up with the conclusion, "Hey, those conspiracy theorists are all nuts." thus ignoring two conspiracy theories that have some merit. Believe it or not, there are propagandists who work full time at this sort of thing. That's why it's called the Information Wars.

I really liked the graphic when Jungreis was explaining his proof at 9:49. The additional +1 radius from the smaller circle added to the larger circle is super clever. Awesome video

Geometry is the best mathematics, and I will never be convinced otherwise.

@@M4TCH3SM4L0N3Instead of adding +1, you can allow the vertex to follow sine or cosine and the circumference to follow sine or cosine. Circumference measurement is one rotation for 2 Pi and vertex measurement is two rotations for 2 Pi. You're just changing the path and starting point of the measurement. He used trigonometry, and could have just kept using it for his proof.

@@ADUAquascaping I understand that you CAN use trigonometry for the proof, and I'm not saying that isn't valuable; I'm simply saying that I prefer the branch of mathematics that only requires a straight-edge and compass and its corresponding axioms and proofs.

@RepentandbelieveinJesusChrist5 Sad how religion turns you into a mindless drone

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

as soon as Dr. Doug drew it on the white board with dots, it instantly clicked that the point touching the circles is moving 3r but the center of smaller circle moves 4r wow such simple yet mindblowing question.

I paused the video with the question before the multiple choice answers came up. I debated with myself but decided the answer was 1 (because of the term "revolution"). I was disheartened when seeing the choices, deciding it must be 3, and then excited again when you said the answer was not an option. Then disappointed again when you said it was 4, and then excited again when you said 1 was a possible answer . . . a real rollercoaster of a video.

Literally same❤

Exactly. Rotation and Revolution are pretty different imo. Pretty ambiguous

Revolutionary comment

That coin rotated once in the first demo, I don't understand how it was 2? With its head up, it went around once before its head was up again.

Was mostly with ya till 10mins….then i felt like a toddler afterwards 🤦🏼♂️🤷🏼♂️😜

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.

I'll always remember when in my freshman astronomy lab, we directly measured the sideral period of the earth. The rooftop-dome telescope was aimed at a patch of sky with it's tracking motor turned off. Over the course about 20 minutes, each of us would peer through the eyepiece (no computer screens back then) and pick out a star that came into view, quickly making a sketch of it amongst its neighbors. When our chosen star passed behind the crosshair (we made sure no one rotated the eyepiece) we each started our stopwatch. Once everyone had their turn, we labelled each of our watches and put them in a cabinet. Then next night we all returned, and one-by-one, observed our star slide across the view, and stopped our stopwatch when it again went behind the crosshair. Mine read 23 hrs, 56 min, 3.92 sec. Across the class, we were all within a quarter second of the actual value. Yes, really simple (and dependent on there being two clear nights in a row), but how many people can say they've done that?

Yes! Sidereal time! Thanks

Beautiful

me, I've done that with timelapses over 24 hours. really cool stuff.

More schools should do this, and similar experiments that require minimal outlay but reconfirm "known" results. For example, I would expect most schools to be able to find someone due north/south who could set up a vertical pole and measure the length of the shadow at solar noon on a specific day. Which, with some trig, is all you need to confirm that the Earth is curved (at least along a north/ south path), and the circumference (if you assume a sphere).

wow ur ancient, did u shake hands with trexes back in the day?

so glad i kept watching. what helped it click for me was imagining 2 coins, one rotating around a circle and one rotating across a line, and comparing the positions of the two as they rolled in sync. the first “rotation” of the quarter is only seen because the curve of the path rotates the image itself. when the quarter on the curved line is at its first “rotation”, the quarter on the line is facing to the right. but if you bend that line into a circle, the quarter will be facing down. this broke my mind at first so im thankful for the great explanations here.

It’s so impressive how you made this seemingly basic math question into a really interesting and well thought out video. I hadn’t even considered the idea of a Siderial day, it’s so cool!

Thou ne maketh a full point, anything of mathematics must be really interesting.

Agreed

@@aniketmeshram6598 reconstruct your sentence. Please.

@@bill5197 i mean to say that he/she/pronouns wants to defy this Cosmic phenomena which was discovered by that great mathematician and astronomer who gave us "Zero"

The best thing about Veritasium videos are that they keep giving. The video could have been ended at multiple occasions, but they make an amazing, extensive learning out of it.

I'm really glad Veritasium included the astronomical part. The moment I realized my mistake (which happened when I gave it some more thought after he confirmed that 3 was wrong), I noticed the connection to sidereal days - as a kid, I spent ages wondering why my astronomy books claimed a day was only 23 h 56 minutes long, so that's pretty firmly imprinted on my mind.

Leave it to Veritasium to make a 45-minute fascinating video on a seemingly trivial topic!

I think the explanation here is confusing, its actually pretty simple if we use SUPERPOSITION: take the number of rotation ("revolution" along the circumference flatted out as a line) we call it "linear". and the number of the revolution of center point of circle A along the circumference from start to end (the given is 1). to be less confusing, lets just say the single revolution of the circle A, along B. we call it "given". linear = 3 given = 1 total = 4 this is true for all radii. ex. 2: for 2 coins of the same radius for about 1 revolution. linear = 1 given = 1 total = 2

If you learn real math go to mathologer. Veritasium is rookie compared to him

That actually blew my mind. It was so great to see how a simple math question with two circles can be related to space observation. Thank you for such a great content!!

Wait till they figure out how it ties in to space travel too =)

Thank you for including the practical application of this math problem. One of the most frustrating part of my education was asking the teacher why I needed to know something and getting the answer "because I said so".

That's why you don't study in American schools lol

As an aerospace engineer, once I realized this is sort of a trick question, I visualized it as I do with sidereal and solar days. I'm happy you talked about those in the video.

Same thought. How is it possible that not one of the test writers/editors etc. had even the most rudimentary understanding of astronomy? I solved it from the thumbnail, before watching the video and wondered how I could be wrong, since my answer wasn't listed.

I wish Derek had rolled his coins in the other direction to match solar system's rotation. My head is stuck on the astronomical visual (and I have a hard time dropping that out of my head).

ABSOLUTELY NOT A TRICK QUESTION. Saw the answer just by looking at the problem, only to watch the video and see that I was correct. The problem with average minds is that when they become highly educated, the tend to Believe that they are way more intelligent than they really are, when in all actually they are just smarter than than rest of us.......... in one specific area.

@@basildraws it was a trick question they told u it made 1 revolution then they asked u howmany revolutions it made if ppl misread question and answered how many rotations it made well thats like being asked if 2 trains are traveling at x speed and start from station x & y at time x when will they meet and deciding to submit a answer on wind speeds over tracks instead

@@hamasmillitant1 No, it wasn't a trick question. If it had been, then "1" would have been on the list of choices. So even if they HAD intended it that way, they still made a mistake. It's pretty clear they meant for the student to calculate rotations based on the choices given, and it's clear they still failed to calculate the answer correctly themselves. The use of the word 'revolution' instead of 'rotation' is just an ADDED mistake on their part.

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.

Undergraduate astronomy student here. The idea of solar vs sidereal time was something I had heard about before, but never properly understood until now. Thank you for all that you do!

I still don’t understand exactly how the movement of the earth affects the rotation time.

@@temple69 Watch a 3D demonstration of it

But why should we add 1 day for Sidereal year, if Earth may not "slipping"? But it was correct only for slipping case

@@igarazhaQuite the opposite. It works only if there is no slipping. Which is exactly the case with the Earth's movement around the Sun.

Thanks for not misusing any comma.

I think the easiest way to see it is as two fixed gears. Circle A achieves +3 rotations while circle B achieves -1. If the camera tracks circle B, circle B appears stationary (0) while circle A has now achieved +4. Beautiful question.

You can also arrive at the N+1 solution by considering the case where the radius of circle B is zero. Circle A would not roll at all but still hinge around the point and make one full rotation.

Great idea!

🤯

Or leave circle A and B attached at the same point and rotate circle B clockwise. This is effectively the same as having circle A orbit circle B without any rotation.

Makes me want to research gears now

We know...

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.

Veritasium is ridiculously talented at making videos.

How many sidereal minutes does KZbin take?

@@tombiby5892I have no idea but for a production like this it's not uncommon to have multiple hours of side reel just in case

I can't believe how well the explanation is made.

Good

Fggg

Very good

Excellent video. Thank you !!! This is possibly the best STEM video I have seen. The link to sidereal time etc is brilliant! To see practical application of "tricky maths" is how people learn. Well done! And thank you again

As a machinist, we deal with this quite a lot. When milling around a circular boss, you have to do a calculation how much you need to increase the feedrate to keep the same speed at the outside of the end mill. The same goes for milling inside a hole, except you calculate the smaller diameter caused by the size of the tool instead, since everything is based on the center of a circular tool.

Super interesting!

Dude how fast are your feeds for this to matter?

@@appa609 On a production machine this matters. For one offs who cares.

As a CNC programmer, that's not really true. I just asked a couple other programmers/machinists at my shop this question and nobody got it right. The thing you have to deal with is varying chip load, which isn't the same at all.

@@fresheFresse Yeah doesn't matter at all for one offs and low volume stuff. When you need a machine running 24/7 for years to make 12 million of something, a fraction of a second quicker could save days

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.

Mathematically it‘s actually really easy to calculate. 1) Calculate the distance d that point A has to travel. If you think about it, that‘s not the circumference of circle B, but the circumference of the radius B plus the radius of A, creating a third circle C around point B with point A now on it‘s outline. The circumference of this new circle C is now the distance d that point A has to travel to get back to its starting position. Formula for circle circumference: 2 * r * PI Now we know that r = rB + rA and rB = 3 * rA -> d = 2 * (4 * rA) * PI = 8 * rA * PI 2) Calculate the distance dRev that circle A covers with one revolution, which is equals the circumference of circle A. Again we use the formula for circumference, this time on circle A. -> dRev = 2 * rA * PI 3) Lastly, calculating the number of revolutions needed, we divide the total distance point A has to travel on d ( = circumference of circle C) by dRev (the circumference of A). -> R = d / dRev = (8 * rA * PI) / ( 2 * rA * PI) rA and PI cancel each other out. -> R = 8 / 2 = 4

"easy"

@@blinkers88 I mean it's pretty easy if you understood the video, it's more or less exactly this part 9:45

Yeah "easy" SATs dont have enough time to calculate that you know

Ikr Also... U didn't really need to write the whole second half The answer is simple The R is 4r then it's 4 times

@@tanez778the video is a waste of time for such a simple answer

I’m glad you chose 3 at first. I didn’t feel so stupid because of it. 😂 The triangle shape was what helped it click with me. When the circle is going around one of the corners, the point it touches the triangle doesn’t move, but the circle rotates by a third before carrying on. Third multiplied by 3 corners equals 1 extra rotation.

Did you even watch the video? Did you miss, that it is always just +1? So 365,24 days of rotation about the sun becomes 366,24 from a different view point? +1 exactly even there.

Yeah, that makes it a lot more intuitive for me as well. Especially since you can easily in your head generalise it to rectangles, pentagons, hexagons, … So the circle intuitively follows.

@@gardenjoy5223 I mean, he saw the whole triangle part, didn't he? The concept is not the easiest to fully grasp, and I also agree that the triangle part helped to make it make sense to me, a simpleton.

I thought 3 immediately, backtracked because it had to be a tricky question if it were on Veritasium, recalculated 4, didn't see it on the list and decided to just watch the rest of the video.

No, not 1/3 at each corner. Less than that.

I've been amazed over the years how vaguely, or just poorly worded, tests questions or assignment questions are in K-12 education. It's also a problem in higher education. When I was in school I was sometimes frustrated at how the teacher who wrote a poorly-worded question seemed incredulous that anyone would misunderstand. Sometimes the problem was that the teacher was unable to account for more creative thinking than their own.

I find it's especially problematic with multiple choice tests. I grew up in a country where they are barely used at all (only for tests that are meant to give an idea of how students as a whole are progressing. They are more meant to test the school and education system as a whole and the grade doesn't account for much) and when I prepared to take the Cambridge Certificate (basically like TOEFL) most of that time was spent learning how to answer multiple choice questions bc well, all important exams we had ever taken up to that point allowed you to explain your answer and what was graded was the whole answer and as long as what you did made sense and was well explained.

Not sure about others, but this was really bad for me, as I had major issues taking the problems (as i am autistic) extremely literally with very little wiggle room. To others, it may have been very easy to "tell what they meant", not for me though.

But this time it's not about wording it's about a wild paradox!

@@fragophilefiles9976 And wording. As he stated the wording of the question allowed for 3 different answers two of which and arguably the most relevant answer wasn't an option.

The most ironic thing is that the testwriters can make questions as ambiguously worded as possible but as soon as you missed a unit or misused one word you lose a point

In 1976 my maths teacher gave us the 2 (identical) coin problem. She insisted the answer was 1. I got 2 coins out and demonstrated that it was 2, but she could not be persuaded. It seems like this was a common mistake amongst teachers of that era.

Literally seeing it in front of her and _still_ insisting it's not true is wild

She sounds like a useless teacher.

@@orangenostril "Your coins must be faulty. The answer *is* 1. Now go and sit down!"

Still true today for many teachers, especially in Asia. Teachers are often drilled to "teach what's correct" but never consider what happens when they are wrong. I've been teaching for the past 10 years and the way I look at teaching is, I don't teach. I share and learn at the same time. I share what I know with my students, and encourage them to seek their own versions of the knowledge, and I feel great when they come back with alternative perspectives to the same subject, or other versions that they've found. Then we explore the differences together. This fosters an atmosphere of collaborative learning and students are much more willing to engage the subject, because they own the learning process. For me, I grow with them.

@@bunface 💖

The motive of the question is that the point A should return to its original position So if we consider Angular velocity of the circle A is omega(ω) rad/sec Time taken by circle to reach the initial position is t seconds Angle rotated by circle when it reaches its initial position is θ radians Radius of Circle B = r Radius of Circle A = r/3 Now centre of circle A will trace a circle of radius 4r/3 So if we apply a basic equation of a point travelling a whole circle of radius 4r/3 is Speed = Distance/Time Since, Speed of circle A = ω(r/3), Distance=2π(4r/3) units, time=t seconds ω(r/3) = 2π(4r/3) / t Now we know, ω= θ/t: (θ/t)(r/3) = 2π(4r/3) / t θ = 8π This accounts for 4 revolutions, because 1 revolution is of 2π. Correct me if I am wrong somewhere, but I solved this before watching the answer, and I wondered that my answer was correct

That's the great thing about math. You can get the correct answer more than one way.

There is an anecdote of a professor in the math department of the university I went, who wrote in a final exam of calculus something like "do you dare to calculate the sum of the series?" to which a student answered "No". The professor said he had to give the student full marks since the answer wasn't wrong, and he started being veeery careful in the wording of the exams

That happened to my junior year English teach in high school (but a year before I took her class). The exam question was "describe the book 'The Scarlet Letter'". As I'm sure you've already guessed, one student wrote a 5 paragraph essay about the size and shape of the book, the various artistic properties of the cover art, the texture of the paper and the font used, etc. According to her, she took it to a faculty meeting for help, and the other teachers concluded that she had to grade it as a correct answer.

Both of your stories are amazing!

once I wrote a paper for a friend who said "I didn't know anything about the breakup of the soviet union, so I asked a friend, and HE said: " then she put my entire paper in quotes, ending with "I couldn't have said it better myself." She got an A.

can't get hung up on small quibbles, quickly scrawl the "F" and move on

I took a 3rd year math course called numerical analysis. We had to "Prove a theorem" on an exam that involved a set of given variables in relation to the error when solving differential equations numerically. The intent of the question was to basically memorize a theorem about the minimum error produced we proved in class and reproduce it on the exam. Except the question said nothing about proving a minimum - it just said prove A theorem. I thought I had understood the process of the theorem so I didn't have to memorize it, but I just couldn't get it to work out to show a minimum. I ended up proving a maximum to the error which was correct (we did not do this in class), and he had to give me full marks as he didn't specify which theorem to prove. I ended up with 100% on the exam, and he learned to more carefully word his questions!

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

Watched this with a friend and they really struggled with the extra rotation per revolution until I showed them a coin rolling along the edge of a rectangle. It's getting around the corners that causes the additional rotation - angular movement is required without any linear movement. The circle is just the limit with an infinite number of infinitely small corners. On the inside of the circle (or any concave corner) that corner rotation is in the opposite direction, so in one loop of any size and shape it will result in -1 rotation.

Thank you!!! I finally understand 😊

Actually get it now!

that made it easier for me thanks. Pity the guy doing the vid couldnt explain as easily.

@@mk1cortinatony395 He showed how the rounded path around the corners of the triangle could be pasted together to get a complete circle.

What dontou mean on inside lf circle the rotation is in the opposite dorection..the circle.rptsripnal.direction doesn't change so notnsure what you meant..and how does a circle have infinite number of corners..you mean because it has an infinite number pf tangent lines?

Measuring angular rotation is another approach. If one circle has circumference 11 cm and another has 17.5 cm, their angular rotations on a flat surface are 360/11 or 32.7 deg per cm and 360/17.5 or 20.6 deg per cm respectively. Adding these rotational rates together describes the rotational rate of the smaller circle rotating around the larger circle of 53.3 deg per cm. Multiplying the angular rotation rate around the larger circle by the circumference of the larger circle gives 53.3 deg/cm x 17.5 cm = 932.75 deg of rotation. The number of rotations is 932.75 deg/360 deg = 2.59 rotations. This matches the circumference ratio + 1 formula as shown in the podcast 17.5/11+1= 1.59+1 for 2.59 rotations.

Ironically, the problem identified by the three students, was essentially the same problem that ETS faced when it had to account for converting scores based on 79 questions, to the 80 question scale. "Where did the extra question go?" is a lot like "where did the extra day go?"

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

They rotated the scores around themselves

they converted the score from solar to sidereal but all the good schools were only accepting solar scores. hang it up, son.

Pretending the question was never there is improper and creates 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 some other questions correctly because they wasted time on this question. 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.

😂 That is so funny. Where did the extra question go? Oh, no! Where did my application to Yale go?

Quite enlightening! To me, a more intuitive understanding of why the +1 rotation for Circle A rolling around Circle B is to imagine that Circle B has a radius of 0 (just a point). When this happens, Circle A will make a full rotation once to return to its original position. From there, you just expand Circle B and when its radius is r, matching that of Circle A, then you need 2 rotations and so on. Then the equation of (P + C) / C as in 11:04 makes more intuitive sense.

I really like your idea. It is a common trick to get to extreme values (0 being the case here) in order to clarify things. The movement of the moon is another great example, as the rotation of the moon around the earth matches the rotation around itself. Thus, we always see one side of the moon, but still the moon rotates around itself.

He turned it into a rotation orbit thing but the practical application would be cogs in a gear like inside a clock or in a production line, the real answer would have to lie there, which suppose is 3. The quarter had made 1 rotation but the outside of the quarter had only managed half a rotation. I’m fascinated this had 3 possible answers.

This could made the video a 1 minute short! Great explanation!

Oh , that's a good explanation!

Your explanation at first sounds like a great idea for explanation of this coin paradox, but then it got me thinking, since a point has no circumference, you cannot rotate (or "roll" is better word) a circle around it. Another paradox? Not quite. You can rotate a circle around a point, but it has to be in a fashion, as the point and the center of the circle are connected by an invisible shaft. In other words, for one full rotation each point on the circle will draw a circle around that point, by which you rotate. Whereas in that SAT question each point on the circle A will draw three "petals" around circle B. In two equal coins it would be a circle twice the diameter of the coin and tangeant at one point to the stationary coin. This was a cool problem to think about.

I’m a retired Judge and when starting my career as a Prosecutor, first I had to take the Bar exam. I prepared very hard as I knew three smart fellows who each had failed the Bar three times and finally gave up on practicing law. I remember an ethics question that had no fully correct answer. It had the two parts of the answer listed separately with no choice answer that combined them. So one had to figure which part of the answer was more important to whom ever was grading the test. Apparently I guessed correctly more often than not.

Same. I was raised by a father who was an attorney. He taught us to think very critically and to try to consider all possibilities before drawing conclusions. Word problems on tests were very difficult for me because of my inability to begin from a point of a blank slate. Before I had even finished reading most questions I would recognize the answer will depend on one or more unstated assumptions. Most of the time the question would not include all the necessary information to eliminate enough assumptions to be sure of the answer, unless only one answer choice was even reasonable. The worst were problems that seemed to be designed to lead you to two possibilities. Not knowing the assumptions the test author was making, you just had to guess between them. I don't fault my father though; critical thought is the foundation of healthy skepticism. Today I am astounded at the inability of people to parse information logically, failing to recognize when relevant facts are being omitted

The LSAT had an entire section that was subjective back when I took it. Not long after, they quietly eliminated that section.

I'm sorry but nobody has pointed out the joke in the comment? Ethic question on an exam for lawyers? Before you string me up and then nail me to cross, I know not all lawyers are bad, just the bad ones give everyone a bad reputation.

@@daviddrake5991 "Ethics" has a different meaning in a corporate context. Ethics means playing by the rules some governing body has set, at the cost of losing one's license or facing severe penalties. Not so much about doing the morally correct thing. One does their business ethically, or they lose their livelihood.

@@daviddrake5991 I get it. My dad used to say (tongue in cheek) most of the lawyer jokes must have been made up by dead-beat dads who were paying alimony and child support. He, having grown up on a farm and the son of a sharecropper, had a special place in his heart for poor people. He used to take homegrown veggies to various poor folks in his home town. I know because I was the bond servant in his gardens, lol On the other hand when he served in the state Senate he spoke of a senator who had a postmaster in his district intercept social security checks in the mail so he or his staff could personally deliver them to well-placed locals. Big ethics problem there IMO

What a cool proof. I am so glad that Dr. Jungreis went on to become a mathematician and is doing well. Excellent video!!!

I'm really glad you added in the part about the sidereal year - that's always bugged me! I always thought it was about how the solar system moves within space but couldn't find any satisfying answers about it when I first searched. The coin paradox actually unlocked the mystery of the SAT question for me early on in this one, since that example is so simple and yet counterintuitive. Seeing the quarter right side up on the bottom and wondering why made me think of things from George's perspective, and then I realized he was actually upside down!

The simplest way to look at it it is, if you look at the center of circle A revolving around the center of circle B, then in the circular path, the center of circle A has to travel (3R + R) distance, while in a straight line, the center of circle A only has to travel 3R distance. Interesting problem that I have never come across before. This was an amazing explanation on the paradox!

That’s literally what he explained in the video.

@@Bot28111Instead of adding +1, you can allow the vertex to follow sine or cosine and the circumference to follow sine or cosine. Circumference measurement is one rotation for 2 Pi and vertex measurement is two rotations for 2 Pi. You're just changing the path and starting point of the measurement. He used trigonometry, and could have just kept using it for his proof.

You keep saying paradox. I do not think it means what you think it means

@@Bot28111He saved us 15 minutes though

I think the simplest visually is: we know the point on A that starts tangent to B will touch it again 1/3 of the way around. at that point A will be on the top right of B, and the point will be aiming down and to the left 120°, an extra 1/3 rotation. So, every time A travels its perimeter, it does 1 and 1/3 rotations, which it does 3 times.

I needed to watch the video twice for the intuition to settle, but I think I finally got it. It is one of those paradoxes that is hard to understand even when you know the answer. Then it does click and it makes total sense. It is so obvious that you are confused as to why you were wrong the first time. The world is fascinating.

This blew my mind. Even incorporating sidereal time into this just topped it over the awesome scale. Glad I clicked it. Thank you for the explanation.

Here's the thing...I understand sidereal time, and I still got the problem wrong the first time I thought about it! (Whoops. Of course, TBF, it doesn't help that the correct answer was not one of the options.)

it‘s still wrong, earth is not a rotating ball revolving around the sun, the constant speed theory of the rotation of earth contradicts this whole video. Since constant speed theory of earth is essential to justify „we don‘t feel any consequences of such a rotation“ this house of cards folds in. There is so much evidence out there outside of the realms of astronomy that prove the earth is flat and not a ball. As Jesus says blessed are the eyes of those who see. What He means is that it‘s God‘s creation (flat), not a people’s creation (ball) and truth is with God in Heaven! Friendship with the world means enmity with God. It‘s the blind leading the blind in this world.

This was a lot more interesting than I initially expected. Great explanation and visuals that made it easy to understand all of the facets of the paradox. Kudos!

Use cosine and sine. Set the edge as cosine (0,1) and the center as sine (0,0). 2 Pi is one cosine rotation. 2 Pi is two sine rotations. Cosine as the circumference has four 90-degree rotations and sine as the vertex has eight 90-degree rotations within 2 Pi.

Learnt this when studying gears and cams in engineering, so i'd say the full solution is best since it also helps to figure out distance/ time travelled when revolving / sliding over an uneven shape e.g. eggshell. That makes it easier to start understanding variable valve timings (VVT) and such. In class, realizing that we needed to account for the radius of the moving circle as well was a shock-LOL-eureka moment, the kind that leaves you feeling a little more enlightened. Hence thank you Veritasium for this wonderful reminder of the joys of learning!

1. A complete rotation is a 360 degree turn about the centre of an object. 2. A full rotation of circle A can be identified when the initial point of contact on circle A touches B again as it roles without slippage over B. 3. It will take three such turn for A to get to it's original point around B. 4. The misconception of the casual observer is to expect circle A to return to a 0 degree parallel orientation with respect to its initial position. For the identical coins, this corresponds to rotation through 180 degrees (180 × 2 ÷ 360 = 1 complete rotation). For circle A rolling over circle B it corresponds to 270 degrees (270 × 4 ÷ 360 = 3 complete rotations) That is the supposed circle paradox. It doesn't exist. There. Nice, elegant, straightforward, simple, and aligned with mathematical commonsense.

@@markverani5088 Could you explain further e.g. how you chose 180 degrees vs 270 degrees and the multiplier too, please.

@@midaz7 It is not choosing. It's observing keenly. I started with the coin. Logic says half a rotation will cover just half the distance, right? If the coin is fixed at the center and you mark the turning point, at exact half a turn, the coin will be upside down (this is the source of the false expectation that the upright coin has done a full rotation). However, the coin is rolling. So I now expected only the back of the head to ever touch the fixed coin as it rolled to the lower position which is what I saw. Now the coin is exactly upside down relative to coin B. Like you standing on your head instead of on your feet. I think we can agree that is a 180 degree flip. That's the instant I realised I had been false expecting the coin to be upright as it rolled to have done a complete turn like everyone else watching the video. Realising what happened when it was 180 degree turn for the coin, I immediately spotted Circle A was upright at 90 degree angles against Circle B every time it seemed it was a full rotation. Since a 90 degree turn is clearly too short a distance, I automatically knew it was a 270 degree turn instead. That's a cheap conclusion for anyone whose done a little trigonometry. I proceeded to verify mathetically.

@@midaz7 As for the calculation, that's easy. We know the number of times it seems a rotation is complete (when the coin or Circle A is upright). So the total rotation to return should be that angle multiplied by the number of turns, right? We also know that a full rotation is 360 degrees. So dividing the total rotation to return by one full rotation (360) should give the number of rotations to the start point again, right? That's why the calculation is consistent when you use the real angles that represent the upward positions.

Learning about gear formulae was wild. I really enjoyed drafting gear diagrams and drawings even though it was time consuming.

*Here’s how I got 4.* If A circumference is 1/3 of B, and both are circles (i.e. 4 * 90 degree angles or effectively 4 circumference points). Then A’s circumference would rotate 3 * 4 = 12 times around the outside of circle B. But, the question asked for A’s rotation not A’s circumference’s rotation. Hence, you divide 12 total rotations by 3 = 4.

5:57 that was my thinking from the beginning and i was confused when you started to count them like 360° on the first try. That just proves how important it is to make sure the example is well-writen and understood by all students.

For real... i didn't even though like this before

I had an error on my SAT too (in 2016). Half of the exams had a misprint that switched the time allowed for each section with another section. They ended up throwing away both entire sections of the exam, I was pretty mad since it was parts in my strongest subject getting tossed. Timing is a big part of the SAT and I feel bad for folks who may have spent longer on this problem since the real answer wasn’t listed which may have cost them more than just the one free point in the end.

This is why skipping questions you can’t immediately solve is such an important standardized test strategy

If someone was dumb enough to continue wasting time on one question that was stumping them instead of moving on and finishing everything else and returning, I doubt it made much of a difference to their end score.

@@PoopiDScoop That's not necessarily true. Some questions require more time, so a person might just assume this is a harder question. Thus, instead of spending their normal 1 min, they'll spend 3 minutes. Generally, the skipping questions when you're stumped is good if you don't know where to start or if you think the problem will take too long, but otherwise, spending an extra couple of minutes is usually worth it, as otherwise you'll just lose your train of thought if you skip the question. Since the question had a misprint, it's entirely possible that some people had the right idea, and were sure they knew how to solve the question, so they spent that extra bit of time to hopefully solve the question, instead of completely discarding their train of thought for that problem and moving on (since moving on effectively resets their progress on that question to 0).

@@Boltclick Skipping then returning tends to be the better option as there may be later questions with similar reasoning that will simplify the harder question. It also allows you to divvy up your remaining time more equitably between any other questions skipped.

It’s also a good strategy because you are penalized for wrong answers but not for blank answers.

One of my SAT questions (on the verbal test) still bothers me. It was the analogy questions "A is to B as X is to . . . " and they were asking for the meaning of "sanction" and both "to approve" and "to punish" were options. I wonder who sanctioned that and if they were sanctioned. 😆

Don't read my name

My goodness.

or sectioned

This could be valid. I assume 'sanction' is the 'X' in the above. We would have to know what the A is to B part is.

Autoantonym

When I saw the coin example, my thought was basically “but the coin only did half a rotation…clearly they want the coin to do a full rotation”. Then you essentially covered that difference, I’m glad

"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.

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

life in general is exactly that. on repeat. this is why most nerds who make it are autistic or agreeable or naiive and basic. others get depressed. and society deteriorates.

except 3 is a correct answer - the problem that the 3 students who "corrected" the SAT - was that they overthought the question - in fact "revolution" as more than one meaning - and i would have interpreted it the way the questioners intended it - on a flat surface - the rolling coin that starts with the head upright - will have made 1 full rotation/revolution/roll (when the circumference has fully played out) when it's upright again - but put it on a curved surface - and that no longer applies when rolling a quarter around a fixed quarter - the coin has NOT made a full rotation when George's head is upright again at the bottom of the fixed coin - if you mentally straighten out the edge of the fixed coin - you'll realize that the head of the rolling coin is UPSIDE DOWN when it's on a straight line if at the start you placed a dot on the edge of the rolling coin where it was touching the other - that dot would not be touching until it has gone all the way around to the top of the fixed coin

What’s crazy to me is when I tried to solve it, I intuitively did one rotation of the little one on the big one in my imagination and saw it only go a 1/4 of the way. I then thought to myself, “wait that must be wrong”. Mind blown

I did the same thing and guessed 9/2 since it was the closest answer haha

I snipped the small circle into a string and draped it over the larger circle in my mind, giving me the answer of 3

Yea but it's just a visual representation of the problem, you're supposed to use the data given in the problem. The actual size of the "coins" in the image is meaningless

@@oneilljames1 The image is to scale

Use cosine and sine. Set the edge as cosine (0,1) and the center as sine (0,0). 2 Pi is one cosine rotation. 2 Pi is two sine rotations. Cosine as the circumference has four 90-degree rotations and sine as the vertex has eight 90-degree rotations within 2 Pi.

I love how Derek goes the extra mile and tracks down one of the people that called the problem out, who just so happens to be a mathematician now 😂

Right?! As soon as I saw his title, I was like, ok that checks out lol

At this point, I just assume Derek has a 'Sherlock Holmes'-esque filing cabinet of every mathematician, professor, and scientist he can call on for collabs XD

matched so perfectly, like a well written script from a movie😂

is it really a coincidence that the person who called out the test creators on a math problem is a mathematicion

Always has been

I solved as Φ=circumdistance/radius of circle circumdistance=2π(1) Radius of bigger circle=3 Φ=2π/3 The bigger circle now divided into 3 equal parts with each subtending an angle 120° at the centre.This means small circle has 3 revolutions on big circle

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.

The Eureka moment for me to get the problem was when you showed the rotation from the viewpoint of the larger circle. That was brilliant, well done!

@@seekerofthemutablebalance5228 It makes sense because the number of rotations depends on your frame od refrence. From the viewpoint of the larger circle the small circle rotates 3 times, but from the viewpoint of an external observer the small circle rotates 4 times. The difference comes from the "camera" from the viewpoint of the larger circle rotating too. The "camera" has to rotate so it always sees the small circle.

Poe content creator in the wild.

@@seekerofthemutablebalance5228 The center of the circle rotates 4 times because, if you draw a line from the center of the larger circle to the center of the smaller circle, it forms a radius of 4 (1+3) but the outer portion of the circle rotates three times, that's why his demonstration at 6:15 has the blue arrow.

This is one of the most interesting "simple" math problem videos on KZbin. Amazing job!

There is a caveat sometimes the d_center < P_shape + C_circle, if circle is bigger than features of the shape: At about 10:50-11:00, it is stated that the distance traveled by the circle's center around a shape is the shape's perimeter P + the circle's circumference, C or a total distance dcenter=P_shape +C_circle. But there is a caveat that should be noted (probably someone else has already (if so, sorry)): That equation is true only if the circle is small enough to "fit into" all nooks and crannies in the shape's perimeter like all tight inward cracks or concavities, or corners say at very abrupt outward points (abrupt convexities). Else, if the circle is too big to fit into all of those, it doesn't touch the whole perimeter, but "bridges" or rolls right over the concavities, or misses the heart of abrupt steps like a gear shape. promontories thus the center of the circle doesn't travel as far as the actual perimeter plus its circumference. dcenter < P_shape +C_circle in that case, if circle is larger than the smallest nook and crannies.. It probably seems like I'm quibbling but there practical examples like treaded or gear-toothed shapes, or the situation of railroad wheels or say bike or truck wheels going over smaller bumps or cracks.. in general large circles roll less distance than all the smaller texture, if any, of the full perimeter of the shape. I hope this is useful. BR

It’s cool how this problem has so many practical implications that most people wouldn’t even think about.

what amazed me is it's as simple as putting the smaller circle on the inside of the larger one and seeing it makes less rotations

Yes; the entire industrial revolution relied on a precise understanding of gears.

The fact that the main issue was a poorly worded question is the exact issue I've had in school with so many tests being poorly written. So often the test writer(s) understand the questions they wrote but they don't have them vetted properly so they can be understood by the test takers.

This so very, very much. The countless pains of trying to figure out whether to answer what's literally being asked instead of answering what would seem to be what the maker of the question wanted to ask. It's ridiculous how such a thing exists so pluralously in tests, questionnaires, forms and medical examination papers etc.

I don't think the question writer knows what a revolution is.

Well I guess if anything it better prepares you for life

That’s not a fact. The main issue, is that the correct answer wasn’t even there. The wording of the question was poor also.

Math word problems are more often English problems which is why they are often criticized as being racist. You shouldn't need to be an English major to sold word problems. They should be written like people naturally speak. And the answer should reflect that as well.

I love how science channels, this one especially, can take you from what you think is a pretty clickbaity title, into a deeper appreciation for the sciences. These videos are just the right amount of learning to interest ratio for me. I feel like I'm just enjoying any old video, but at the same time learning *how* to think, and not just *what* to think.

Well said!

The thumbnail is clickbait because it shows the wrong question -- it pared down the original SAT question to be very incomplete. Note how different the actual question is with more details.

@@oahuhawaii2141lol he had a whole video about this. It’s called “clickbait is highly effective.” Idc if he wants to do clickbait as long as the content is actually good 😂

@@kirbya9545 I agree. Flashy thumbnails is the game you have to play. It's like a really flashy bag of chips...If you open it and it's full of delicious chips, then who cares?!

@@Stevelemontrudy and unlike a bag of chips 40% of this video isn’t air 😂

One has to consider that the small circle makes one revolution initially. Then, one has to take the ratio of circumferences which is 2(pi) * r{lg circle}/2(pi)*r{sm circle} and one adds these and that is number of revolutions of the small circle around the large circle.

I was so confused because of the word "revolution," "1" is what I thought the answer was because of that

Yes, there's a distinction between "rotation" and .revolution" The outer coin makes one revolution but several rotations.

Same, I felt dumb when 1 wasn’t listed at first then thought about it again and thought 3 was the next “logical”…ist. Because the original wording is legit asking after how many revolutions…will a revolution be completed lol

same

4 toppers meet

@@JustNuggie4 toppers meet

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

I think the most intuitive way to understand this is to imagine the inner circle is just a point. Even though there is no circumference, the outer circle would still have to rotate once to go around it.

👌

bro. this is amazing,thank you

That's one way to see it. Another is to try it with two equal rolls of ducttape. As you unwind the outer one around the inner, you will notice that the point of contact only travels at half the speed of the center of the outer roll. If you start from 12 o'clock and roll it clockwise down to 6 o'clock, the outer roll has done a full revolution around its own center. But it has only unwinded half of its circumference, because the point of contact has rotated _in the opposite direction_ at half of the speed - it went from 6 o'clock over 9 o'clock to 12 o'clock of the moving roll of tape. So the actual length of unwinded tape at this point is 1x the circumference (for the whole rotation of the outer tape) minus 1/2x the circumference, for the counter-rotation of the point of contact. So 1/2 the circumference for 1/2 revolution, even though the outer tape has spun a whole 1x around its own axis. But if you imagined the outer roll of tape to be made out of super-thin tape and be infinitenessimally small, then this counter-rotation would make up practically 0 distance. Like if it only has 1/100 the circumference of the inner one, then it only needs 101 revolutions, so only 1% more than if the counter-rotation was no factor at all.

Yeah, I was surprised they didn't include that example.

Yeah good way to visualize it

You can also use the straight line to understand it. When the coin reaches the end of the line (after 3 complete turns) and now you rotate under the line to go inverted on the opposite direction, you´ve just added half circle to the 3 complete turns. (and another half circle if you rotate to the initial position)

HI thanks for this. But would that mean 7 complete rotations? So you mean the two sides of the line gives 3*2pi+3*2pi or 6pi perimeter and then two end half rotations is pi+pi so total is 7*2pi or seven total circle rotations 7*2pi... i think I see it even works for a line cool. thx

@@foxlies0106 Yes that´s what I think is correct. In the end 7 complete rotations.

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.

In a lot of other schools, I may have been the best student in math. But with Doug in my class, I was far from that. I remember that SAT and I really didn't think about that question because I had thought of the rotation relative to the circle itself when I answered (The fact that 3 was a choice and 4 wasn't may have contributed to my tunnel vision). When Doug came out with his findings, I realized that there is a big difference in the way he views math than how I did.

I had a good teacher at school - just the one - he showed us a whole bunch of bad exam questions, the coin paradox example here included. Years later as an engineer I would come across the same thing when it came to some machinery having 3 sets of nested orbital gears and trying to work out the required input rotation from the expected output rotation, even though I worked that out eventually due to remembering my teacher talking about it, it wasn't easy. Now, after watching this, I feel I actually understand it and could explain it to someone else.

👍⚙️

It can be done using concept of rolling in physics Let velocity of centre of smaller circle be v, radius of bigger circle be r, angular velocity of smaller circle be w, no. of revolutions of smaller circle be n Now, the centre of smakler circle is revolving in a circle of radius r+r/3=4r/3 about the centre of larger circle So, time for 1 revolution will be 2π×(4r/3) / v = 8πr/3v =t (let) Since smaller circle is in pure rolling motion, velocity of point of contact with larger circle is 0 So, v=w×r/3 , so w=3v/r So, 2π×n/w = t So, n=wt/2π = 4

My mate Paul used to do all his rolling in Physics. It was the period before cafeteria lunch. He could sell about 15 doobies in our 50 minute lunch without catching the attention of the monitor.

I've always had a tough time visualizing the sidereal vs solar time. I love what youve put together her. Your guest is an amazing guy.

he's got great presence on camera too

The question should have said "rotations", not "revolutions"...or maybe I don't understand what the world "revolution" is supposed to mean, I'm basing my idea on what we say for the earth around the sun, etc...

Nobody says "the earth revolves 365.25 times around the sun in a year...we say it only does so once, lol...because we mean the whole thing going around it...and circle A only goes once around circle B, regardless of rotations, which are supposed to be different things than revolutions...

7:13 that's exactly what they are bringing up...I need to watch these videos entirely before commenting, etc...

Holy cow - I remember that question! I also remember that I skipped it, intending to go back, and never did. I'm one of those people who skips around answering the easiest questions first and going back to the less obvious questions later. There was something about that question that made me decide to come back. Watching the video, it seems likely it was the awkward wording that did it. Kudos to the students who corrected the test. I would never have done that. Especially on the SAT. Heck no.

Just showed this video to my grandpa. He went to get an object in his mancave right after, a Breguet Siderometre Type 102. Turns out pilots, before any radio navigation was invented, could have used siderometres to keep track of their navigation at night. It seems quite complex but from what I understood the time is divided in 360º and not 24hours, and navigators would pick a star visible for the whole trip and mark down it's position every 15mins. The tool also has a clock named "Rattrapante" that allows to compensate for the plane's movement during those 15mins. Pretty neat stuff!

That was really cool, thaxxxx 4 sharing it with us! Cheers!!!!

Ocean vessels also navigate using 360* even underwater ones.

So, is a sun dial using the same 360 degree concept as navigators?

absolutely ingenious, and even if I heard about the "sidereal time" 45 years ago through one of the Jules Verne books, I never got into this insight.

Another way to visualize the extra rotation is by making the inner circle infinitely small. The outer circle will still take one revolution to go around. Just use a pin as the inner circle and it will take roughly one revolution for a coin to go around.

Brilliant!

Pls, tell me if I am wrong so I know I understand this. I think the pin would still give it an N+ number. You wouldn't use an infinitely small point either. Just an imaginary zero point where there is no there only then can you create a perfect N number.

I half understand on an intuition level what's going on. The amount of rotations are relative to the plane so three is technically correct from the pov of the line circumference. Sorry Mr Doug, don't understand your previous point about velocities. Also the triangle aroused something in me and I thought of another way to explain it, this only works in shapes with no depressions, but more or less the angle of a triangle's outside is always 360, but you could very well also talk of a circle having a total angle(outer that is) of 360, as weird as that may sound. Although if we're thinking of curves as angles, that may be another way to solve it.

Thx Marcello it totally makes sense this way. If the edge of the circle turns around an axis, il would make 1 x 360° circular distance and 0 x linear distance (the axis is consider a point), hence 0 + 1 = 1 rotation.

There is no extra rotation of A, it is an illusion because you are viewing it from your frame of reference. Though I like your example, the reality is that if you make the inner circle infinitely small then the thing you are actually moving around that point is the frame of reference of A(ie the axis of A is doing a 360 or full rotation around that point). That point will not traverse one iota along the circumference of A and therefore A will not rotate at all. The thing rotating will be A's frame of reference relative to us and more aptly it's axis. Essentially you would be rotating circle A about a point on it's circumference. A itself will not be rotating at all about it's own axis. A better way to look at it wrt the original question in the video would be to understand that by the time A makes a full revolution/rotation about it's own axis the axis of A itself will have rotated 120 degrees. By the time it makes 3 revolutions as it moves around B the axis of A will rotate 360 degrees which is where the extra PERCIEVED rotation comes from. In actual fact A has ONLY made 3 rotations about it's own axis plus the axis making 1 full rotation makes it look to an independent observer that circle A has made 4 revolutions. Strictly speaking the revolutions are 3. The three students are wrong, the answer is indeed 3. For the answer to be 4 the question would have to have been framed differently: How many revolutions does circle A appear to have made to an observer in an independent frame of reference. For the answer to be 1 the question would have to be framed: How many revolutions of A around B.... As it is framed "How many revolutions of A..." implies revolutions about it's own axis and the only answer to that is 3(ie divide circumference of B by circumference of A). In fact ANY shape would produce that singular extra rotation of the axis of A around the shape(360 degrees rotation) giving the perception of the +1. Note: In astronomy a revolution about one's own axis is called a rotation. This is in order to distinguish it from a revolution around another object or point. However this is an SAT Math question not an astronomy question. There is no expectation that students know or incorporate that into their interpretation of the question while if a revolution of object A about object B in implied then object B is named. If object B not named then the implication is that object A is revolving about itself or rotating about it's own axis to be precise.

11:37 The "Not to scale" broke me 🤣

I just retired with over 2 decades flying around the world. It took flight school instructors 2 days (solar) to explain sidereal days which it took 2 minutes for this video to clearly explain. AWESOME!!!🎉

Sidereal days kicked me in the butt at the end of my physics PhD work. We had to multiply our results by 366.24/365.24, not to the end of the video yet, but I think sidereal and the coin paradox are related but not the same. We'll see.

@@stevenknudsen7902you were right

The outer center (A) has to go 2pi ( 4r) to complete one round. A's circumference is 2pi (r) When divide these values, you get 4.

When I saw the question, my first thought was that the answer is 3, because of the logic you described. Then I noticed the word "revolution" in the question, and realized that, if inferred literally, the answer is 1, because "duh". Then you explained the "plus-1" rule for the rotation count, and I was absolutely delighted. Also, thank you for explaining sidereal time in a way that I can finally understand. I tried reading the definition in a dictionary years ago, and was more confused than ever.

My instinct was 1 too. Our bwois confused between revolution and revolution

I thought it was 4 as 3r + r = 4r saw the multiple choice thought I was wrong and said A

Bro wrote a whole book

I remember trying to explain the solar/sidereal day discrepancy to some friends in high school over 50 years ago, only to be met with bafflement and indifference. As a maths tutor for high school and uni students I have probably seen dozens of examples where the exam setter has either been ambiguous or just downright wrong.

I read about sidereal time and year long ago in school geography textbook, didn’t quite understand it or its use. Until now. Wish learning was this fun in schools.

Because people are stupid. Even when they think they're smart.

People have different interests.

To understand, study... To live in peace, watch this 👉The Connections (2021) [short documentary]💖

A "year" defined by the gregorian calendar is NOT the time for the earth to make one orbit around the sun ! It is to fit the seasons and was created to fix the equinox drift. The sidereal year is longer because of the precession of earth's axis. 365.24 days is the gregorian calendar year, 365.256 days is the sidereal year. I think perhaps Veritassium could have used 365.256/366.256 days for a distant fixed observer.

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