“I was amazed how badly it’s worded,” literally half of the SAT problems.

To all the 1st posters: UKposts 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.

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

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.

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

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

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!!!🎉

It's all a matter of how you look at it, (perspective) as explained so well with the graphics at <a href="#" class="seekto" data-time="351">5:51</a>. Thank you for this video!

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.

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!

This explanation is the best i have found. The idea of the distance the centre of the planet circle travels and then deriving ratios makes the most sense to me and your graphics helped me to grasp this. It's fascinating. Not too mention the perspective element influencing the answer!! What i didn't realise is that the extra rotation is accounted for as well if observing from the perspective from the centre of the "sun" circle by the fact that the observer has to rotate once to continuously observe the "planet" circle

This concept is quite important while solving Rotation problems in Physics. Instantaneous Centre of Rotation, given that it is pure rolling i.e. there is no slipping at the point of contact. It at this centre of instanteous rolling the entire circle or rigid body is pure rotating. Thanks for sharing.

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.

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.

I appreciate every one of your videos, they always make me think, and sometimes make my head hurt. Thank you.

I've just found your channel and it is really brilliant. Keep up with the good work :)

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."

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.)

I wasn’t the best at math in high school not because I didn’t get the right answers but because I could visualize the problem. My problem is I couldn’t show how I got the answer. This one is easy. In my head I just rolled it around and got the right answer. This is geometry for me. Visually simple.

no wonder everyone got them wrong, there was no correct answer

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

I learned about this problem when calculating gear ratios of planetary gearboxes, using exactly same <a href="#" class="seekto" data-time="63">1:3</a> ratios.

Honestly one of the best random fact and knowledge shows I have seen in quite a while.

aaaauuuuggghh!!! i was going crazy because i thought the answer was 4 and it wasnt one of the options. took me waaayyyy too long to realize that the reason everyone got it wrong was that the right answer wasnt one of the options. thanks for making drawing the connection to solar vs sidereal time and the practical applications in GPS time. i really appreciate how you relate the abstract puzzles and theoretical questions to real world situations. but this also opens up a whole can of worms about the equation of time vs the mean solar day and how the shortest day of the year, the day of latest sunrise, the day of earliest sunset are three different days.

<a href="#" class="seekto" data-time="260">4:20</a> 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.

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.

Love this video. Good explanation of this math magic. Great job.

This is really interesting! I initially interpreted it as the revolutions answer (1) and was immediately very thrown off by the answer options. Very cool to learn about this type of math problem!

<a href="#" class="seekto" data-time="644">10:44</a> The circle traveling on the outside of the triangle helped me visualize the solution best.

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

Very impressive video! I really enjoyed how the information was presented.

What an awesome video,full of knowledge and images, lots of explanations,very good job😊

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!

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.

Fascinating. , thank you. Especially the circumference versus the linear pathway is something that keeps me wondering about our perception of reality 😮 What if I drive my car around the planet thinking I am on a flat road all the time ?

I love these sorts of things where you can make it as complicated, or as simple as you like.

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?

The explanation you gave just showed your point. Pov determines internal and external proportion. +1 and -1

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!

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.

This got my mind really spinning! 😉 So I did my own little experiment, but using rectangles. I found the number of rotations for same-sized rectangles is the same as for the same-sized sized circles (i.e., the quarters) shown in the video. I used a couple Chipotle napkins that were sitting next to me on the couch lol. The outer napkin rotated 2 full times to get back to the original location. And sure enough, when I figured out how to alter my perspective to that of the inner napkin, there was only rotation from that perspective. This was a fun simple way to reinforce a key principle in this video.

By far the coolest UKposts video I’ve see in a while

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.

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!

Fantastic explanation! You are one of the clearest and most creative instructors on UKposts. Thank you for your content.

i've learned more stuff from textbooks or reading on the internet than i have in school. i've seen 'sidereal' for more than a decade and never had any good understanding of what it was nor even how it was pronounced. now i learned both. thanks!!

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?! 😁😉

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!

Amazing paradox😮 I mean the fact you can turn a straight line into a circle & observing the circles rotation from different perspectives changes the amount of rotations the outer circle makes in both cases. The reality of physics is awesome

This really hurt my brain until the whiteboard explanation. Now it's so clear!

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

I really liked the graphic when Jungreis was explaining his proof at <a href="#" class="seekto" data-time="589">9:49</a>. The additional +1 radius from the smaller circle added to the larger circle is super clever. Awesome video

I got this right at first glance by doing with visualization what you did with the cut-outs at <a href="#" class="seekto" data-time="285">4:45</a>. No math brain here, but a lot of practical application in life.

Fantastic analysis...great video!

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.

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 got the answer using the center-distance proof right off the bat!! Thanks for the self esteem boost, really needed that today

This is such a delightful error! Gave me a good chuckle!

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.

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.

This was an excellent video. Not only did you explain the coin paradox and sidereal time, but you also showed some of the pitfalls in experimental, survey, and test design.

The main idea is the "center" of the small circle, not any point on its circumference has to return to the starting point. It has to travel a longer path.

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 😂

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

The way I'd think about it (yes, I figured it was 4) is: if they were cogs, both circles rotating around their axle, the small one would do 3 rotations and the big one would do 1. In order then to fix the large circle, we can imagine rotating the sheet of paper once in the opposite direction, so that the large circle would look still. So that would make it 4 rotations for the small one: 3 in the paper and 1 because the paper itself (the reference system) is rotating.

Simply while the small circle is moving, keep the camera rotating counter-clockwise for a total of 360 degrees, so it will look like 2 hooked gears with the big one taking one full turn and the small one three. After that, make one 360 degree rotation clockwise of the camera, during which the small circle will make the 4rd rotation.

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.

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.

If the big circle had paint on its perimeter and the rotating circle was getting painted as its edge touched the painted portion, there would be a blank space (with no paint) when the smaller circle is right side up (<a href="#" class="seekto" data-time="270">4:30</a>). To paint the whole circumference of the small circle, you would need to go a little further which would reach 1/3 of the big circle. So even though the small circle has "rotated" 4 times, it has not matched up with the circumference of the big circle.

Another way of solving this could be making a line X, on centre A, parallel to point B and then rotating the circle A with respect to line X, such that line X will be always parallel to point B… it is basically same as solving with respect to point B as you said but I just noticed that while I was observing you demonstrate the question…

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!!

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.

Finally now i can relax. Huh this was satisfiying 😌. Thanks🎉🎉

I knew it was too simple. this was such a good video thanks for broadening my mind.

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

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 figured 4 right away: 3 turns for the gear ratio, and one more because the small gear is going around the circumference. Then spent half an hour trying to figure out why I was 'wrong'. Even dug out old Meccano gears to confirm I was not mistaken (which let me confirm 2 for a <a href="#" class="seekto" data-time="61">1:1</a> ratio, 3 for a <a href="#" class="seekto" data-time="121">2:1</a> ratio, and 4 for <a href="#" class="seekto" data-time="181">3:1</a>. And then I watched the rest of the video and learned a few other things that made it all worth while. Thanks.

Man I live these videos so much! Brain food, love it!

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.

I love you and what you do Sir. Keep them coming. 🥳 You always blow my mind

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.

This was fascinating. I would not have caught the error but I imagine there were plenty of others who either thought the answer should have been 4 but didn’t contact the SAT; or thought it was 4 but talked themselves out of it. These three just happened to be both confident enough and motivated to contact the SAT.

To count the revolution of smaller coin we should put a mark on circumference of it. At touching point of 2 coins. It gives us simplest way of understanding the revolution.

I can say from experience, pointing out flaws in a test is such a double edged sword. I pointed out 3 bad questions on a science test in 7th grade, and the entire class hated me because "I" messed up their scores.

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 one of my favorite videos on yt, ever

Loved this man! 🤯

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.

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.

<a href="#" class="seekto" data-time="207">3:27</a> i decided to watch the quarter being rotated around the fixed one again - this time i noticed something that i didn't notice before - instead of watching George's profile - i watched the point on the circumference of the rolling quarter that had touched at the start - that would be the "r" in "quarter" - the quarter had clearly only rolled halfway along the circumference when George's head was poised upright again at the bottom of the fixed coin try this - with the rolling quarter glued to the bottom of the fixed quarter - use you imagination and unroll the fixed quarter's outer edge into a straight line - see how the rolling quarter is now SITTING UPSIDE DOWN - it's correct to use the head's orientation when rolling on a flat surface - but roll on curved surfaces - and the upright orientation of the head becomes less accurate as an indication of a full rotation/revolution/roll thinking in terms of POV - is the question asking if the coin rotates from head upright-to-upright from the reader's POV - or has it rotated when it goes from touching "r" to touching "r" - ie from the fixed coin's POV too bad images or lettering is being used in the circles - instead remove them and leave the face blank - how are you going to tell when it has rotated/revolved/rolled 1 time - how about by placing a mark at the edge of the rolling coin where it first touches the fixed quarter - we would clearly see that the rolling quarter's circumference doesn't spool out until it is at the top again - that's 2 rotations of George's head - but 1 span of the circumference for the small and large circles - if you focused on a dot placed at the edge of the circle - that dot would touch the fixed circle 3 times - but the letter "A" would be pointing up 4 times - since i would have understood "revolution" to mean the playing out of the entire circumference of the rolling circle - therefore to me 1 revolution would mean when the dot has returned to touching the fixed circle - the answer "3 times: is natural the College Board had a dilemma - by invalidating the question due to the question's ambiguity - they probably invalidated students who got the correct answer because they had interpreted the question the way i described above - - a survey should have been done to find out if students were usually confused about the questions meaning - i suspect the 3 students who complained were overthinking the astronomical definition of "revolution" is not the only definition of the word - it is an absurd one in this instance since the astronomical revolution has nothing to do with the outer body's rotation - whereas the circles here are touching - and remain touching while the smaller coin is rotated - or revolved - or rolled using the wordage "how many rolls" would have been clearer - however i asked ChatGPT that - and it said one and a half rolls (?!) - - despite this - i asked ChatGPT "does a carousel rotate or revolve" - and it replied "A carousel typically revolves, meaning it rotates around a central axis or pole" (?!) the moral of my story - "everyone got wrong" is the wrong title - the other people probably interpreted the question in the way the author of the question intended - that "revolution" is defined in relation to the edge of the fixed circle - the students who interpreted the question as the rotation of the rolling circle relative to the page's fixed axis - came up with a different answer - not the one & only correct one --- there's a youtube channel called "MindYourDecisions" that tackled this topic too - since it's a math oriented channel - it has many math people viewing it - and they have deluged the comments section with assertions that the correct answer is 3 - some have added conditions - eg "in topology it is 3" - "depends on the POV" "tech-science" channel has animated the issue - and uses something called the "Willis Equation" - to explain the "Rotation Paradox" (the title of the video) - again - he distinguishes between the POV of the fixed circle - and what he calls the "outsider's POV" - which i consider the "page's or screen's axis or grid" - which is the one used when the edge of the circle is straightened out someone pointed out - using the astronomical definition of "revolution" - the moving circle revolves once around the fixed circle - so another possible answer is "1"

Woah that is a weird one, the center of circle A is moving along the radius of B plus the radius of A, which is the translation. On the Flat path the translation is only as far as it rolls in that direction. That's wild. Great video!! I liked the proof he gave with the slipping, really breaks it down!

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

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!

<a href="#" class="seekto" data-time="152">2:32</a> This statement makes me feel better, because I TOO answered B, but I honestly couldn't tell you what the question was actually asking me to do. Atleast not at the pace the question would demand on the SAT/ACT

Here’s my guess, if the wheel A is revolving like a wheel then you divide both the circles circumference. Circle A has a radius of X and circle B has a radius of 3X, to find the circumference we multiply the radius by two and then times PI, making Circle A have a circumference of 2XPi and circle B having a circumference of 6X PI, so it should be 3

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 <a href="#" class="seekto" data-time="664">11:04</a> makes more intuitive sense.

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. 😆

PS. Your explanation of sidereal day & year, and why the sidereal year is one day longer, was spot-on. However, the sidereal year is not a little less, but a little *more* than 365¼ days. It is about 365.2563 days. The difference is due to yet another circular motion the Earth performs - precession of its rotation axis, every ≈25,800 years. This is what makes the tropical year (year of the seasons, 365.24219 days, which is what our calendars all sync to), a little shorter than the sidereal year. The year of the seasons is measured from one March equinox to the next, which occurs a little before a complete (sidereal) revolution, because Earth's axis, and thus its equatorial plane, has wobbled a little bit, bringing the Sun back a bit early to the line of intersection of Earth's orbital plane and its equatorial plane, the event that defines the equinoxes. Fred

This is exactly why I have hated every multiple choice exam while at Uni. Which means almost every test. I paused the clip to doodle my way to an answer, saw the circumference ratio as 3, then examined the wording and thought "revolution" kind of means 1, then focused on the path travelled to get back to the "starting point" which meant the r/3+r circle, then got confused about wtf they were actually asking and gave up. Multiple choice tests are only as good as those who write them.

“Mess up this test as a teenager and your entire adult life is screwed” is such a top notch system.