This is an 18 minute video about an SAT problem from 1982. If you're not sure you want to commit to it, here's a contemporaneous NYT article:<p><a href="https://www.nytimes.com/1982/05/25/us/error-found-in-sat-question.html?unlocked_article_code=1.JE0.bxTM._IG_RekctyzU&smid=url-share" rel="nofollow">https://www.nytimes.com/1982/05/25/us/error-found-in-sat-que...</a><p>(Gift link.)<p>And a recent Scientific American article:<p><a href="https://www.scientificamerican.com/article/the-sat-problem-that-everybody-got-wrong/" rel="nofollow">https://www.scientificamerican.com/article/the-sat-problem-t...</a><p>(Both of these are linked from the video description itself.)<p>I'll bet Marilyn vos Savant wouldn't have got this one wrong[1]. :-)<p>[1]: <a href="https://web.archive.org/web/20130121183432/http://marilynvossavant.com/game-show-problem/" rel="nofollow">https://web.archive.org/web/20130121183432/http://marilynvos...</a>
What makes sense to me is to think about something that DOESN'T roll.<p>Suppose I start in Greenwich, walk - without rolling - down the prime meridian to the south pole, up the international date line to the north pole, and back down the prime meridian to Greenwich.<p>How many rotations do I go through? One. I get a full rotation because I've followed the earth's curvature all the way around the globe once, even though I'm walking straight without rolling.<p>So the answer is "how many rotations due to rolling" plus "one bonus rotation for passing around the curvature of the circle."
The guy who figured this out (who Veritasium interviewed) is crazy smart.<p>Also, a lot of kids math problems (middle school and below) are super vague. I get that they're designed to teach a concept, but they could do it in a more exact/precise (idk what the word is) way.
I’m no genius, but I used to score well on such tests as a youngster. I was handed an IQ test circa ~1996 and got about two thirds through it before I found a question where none of the responses were correct. I brought it to the teacher’s attention. “There is no way you could possibly be right, this test was reviewed first!”<p>Anyway, blame it on photocopy errors or whatever, but the answers were wrong. Still mad about it.<p>I took another one ~2007 when I was looking to be a marketing person for a construction equipment rental company. This one was about 30 questions over 60 minutes. I got to question 26, which was some extraordinarily complicated question about how many cuts would it take to chop down a large board into the pieces you needed. After I wasted several minutes methodically writing this out, I realized the true test was realizing it was a time waster question one should skip. 27-30 were far simpler and then the buzzer ran out on me.
This was also the infamous amc 2015 "clockblock" question<p><a href="https://artofproblemsolving.com/wiki/index.php/2015_AMC_10A_Problems/Problem_14" rel="nofollow">https://artofproblemsolving.com/wiki/index.php/2015_AMC_10A_...</a>
Why should we take the meaning of revolution in this problem not as the contacting point to meet the circle again, but just as the pointing the same direction? I think this is the source of confusion here.<p>I'm not the native speaker of English and I might take the meaning of "revolution" very absurdly here. Just curious and I have to ask this. :-)
None of the explanations gave me an intuition for it except the circle rolling down a straight line with the same length as its circumference.<p>A roll down the line will rotate the circle once. Coming back the same. But rolling around one corner will add half a circumference, and another half for rolling around the other. So you get 2 * 0.5 extra circumferences, and so + 1 C. Somehow that helps with the other polynomials for me too. Super cool.
I think the explanation is kind of messy tbh.<p>People are going to have one of two interpretations. If a coin rolls around another coin, is the distance traveled equal to the edge along the the static coin? Or is it the distance traveled by the center of the rotating coin?<p>If it’s defined by the former, the answer is 3 because the circumference is 3x as long as that of the traveling coin. If it’s defined by the latter then it’s 4 because the circumference of the base coin buffered out by the radius of the traveling coin is 4x as long that of the traveling coin.<p>I vote 3 personally
The coin around coin problem has its own wikipedia page, although some people want to delete it:<p><a href="https://en.wikipedia.org/wiki/Coin_rotation_paradox" rel="nofollow">https://en.wikipedia.org/wiki/Coin_rotation_paradox</a>
One interesting thing that's not in the video: there is a real application beyond the astronomy one, a scenario where you would want to roll one object around another object, and care about how many rotations it makes.<p><a href="https://en.wikipedia.org/wiki/Epicyclic_gearing" rel="nofollow">https://en.wikipedia.org/wiki/Epicyclic_gearing</a><p>These are used for car differentials, bicycle gear hubs, pencil sharpeners, and many other applications.
The application to sidereal time is where this really gets interesting. It’s far from a trivial question of semantics or a mathematical trick. It’s something astronomers have known about for ages, since the amount of time it takes for the Earth to make one rotation is different from the perspective of distant stars than from the perspective of the sun.
My answer to this after reading the question was 1, because by definition it is one revolution. So I thought that was trick, and the answer essentially is none of the above. But there is a lot more nuance here than just that with multiple definitions of revolution versus rotations, and sidereal time is where this answer can get weird.
There is an intuitive way to see why this is true.<p>Picture a circle rolling around the outside of another. Easy to picture, right? Even if it’s not clear how many times it rotates, you can still visualize it.<p>Now imagine the outer circle has to spin around the <i>inside</i> of the first circle.<p>Stack two quarters, and try to spin the top quarter around the inside of the bottom quarter.<p>You can’t do it. There’s no room. If they’re identical size, the top quarter can’t spin "around" the inner edge of the bottom quarter.<p>Now put a penny on top of a quarter. Spin <i>that</i> around. Not along the outside, but along the inside of the quarter. You’ll see the penny is tracing a very tight circle. There’s not much room.<p>Now put the penny next to the quarter and spin it around the outer edge of the quarter. It goes easily. Plenty of room.<p>If you do the naive calculation, you’d get the same answer for both cases. But clearly there’s a difference.<p>The difference is more apparent if you imagine the coins had little teeth, like gears. If you go around the outside, it’ll go just fine. But if you hollow out the quarter and put teeth along the inside, and try to spin the penny around in that, you’ll find that it doesn’t make anywhere close to a full rotation each time you go around.
The way I think of this is: If instead of having a small circle rolling around a larger one, think of a circle with ANY non-zero radius rolling around one with a zero (or really really small) radius. You’ll always end up with at least one rotation, regardless of how small the other circle is. So +1 makes sense.
Question: the outside circle is a tire made of rubber, which wears down at a known rate.<p>Which tire wears down more:<p>A tire that rolls around a circle of circumference D.<p>A tire that rolls down a flat surface, total distance D?
I'm surprised that only 3 people reported the error.<p>While problem is not trivial, my initial reaction was "it can't be 3, it's too obvious". And it seems that a lot of people in the comments here do get to the right answer on their own.<p>So, why not hundreds of reports?
Second part of the video is way more interesting, where he discusses how this toy problem actually relates to how we define what a year is depending on whether we look at the stars or at what's on Earth.
Interesting: I took the SAT in 1982 and don’t remember this problem. They don’t give you your % right, just percentile and a mysterious score between 200 and 800 for some reason. Does 800 mean 100%?
Tldr it was a four choice multi choice question but the all the options were wrong because the people setting the test got it wrong.<p>So I'm not sure everyone got it wrong, just that they weren't confident enough in a rushed test to cross out A,B,C and D and write in the correct answer!