The SAT Question NO ONE Got Right

it was 10 30 seconds 10 minutes remaining no matter how hard you study no matter if you have your trusty ti-89 titanium with you what a calculator test taking feels like one of the most stressful things we do in life and you can get really really stressed out on a stressful test if you encounter a question that you just know you know the answer to and yet incorrect you get it wrong anyway this happened a few decades ago at the exact same time to hundreds of thousands of students all on the very same test do you think you can do any better what's that no it's not your fault it's fine it's never your fault now entering the facility in 1982 millions of students across the united states were preparing to take the sat a standardized test that's very important for college admissions if you're not familiar it's a fairly rigorous assessment of reading writing and arithmetic that weighs heavily upon the souls of students looking to get into a good college or university but in 1982 in the math portion of one of the test versions as reported by math books and youtube videos with millions of views they encountered this specific problem which reads imagine two circles circle a and circle b circle a and b are touching but circle a has one third the radius of circle b now imagine circle a traveling all the way around circle b once how many times does circle a revolve now if you've taken high school mathematics you should be able to take a crack at this question so why don't you take a crack at this question just grab your own ti-89 titanium minimum mine still turns on from college i didn't script this part and give it a shot i'm gonna i'm gonna be here and oh i still got i still got my stuff on it [Music] what's that no it's not your fault ah [ __ ] what no it's never your fault i'm the one pressing the button oh hey arya didn't tell me how long she was going to give you to take what answer did you get well if you're like me you got the answer as three or choice b and you may have even then confidently tweeted about it at somebody oh look at me it's three by the way look at my head but in fact three is not the correct answer and it's not just me who got it wrong not a single one of 300 000 students that encountered this problem got this question right how is that possible well it turns out that the correct answer to this problem wasn't even on the test a bit anticlimactic i know but before we figure out what the right answer is and should be i got a problem with this problem this famous question as depicted in the most popular videos about it is deceptively ambiguous think about it what does one revolution in this case really mean does it mean how many times we count the letter a and circle a returning to the orientation we usually see the letter a in or does it mean considering the coin's perspective and asking ourselves how many times would the coins say it revolved around different interpretations here will lead to different answers so let's do the math if one circle is going to roll against another their circumferences are going to be interacting a circle circumference is pi times its diameter and diameter is twice the radius so what is each circle's circumference going to be in terms of the radius if the radius of circle b is always going to be three times larger doing the digits and canceling out terms we find that the circumference of circle b is also going to be three times larger than a's and so to travel all the way around the circle circle a must then revolve through three of its circumference or three times but i just said that three isn't the right answer so what's going on this is the ambiguity problem i mentioned before maybe instead of tracking circumferences we instead need to track the orientation of circle a from our point of view which would mean tracking the center of circle a so that's how letter a inside of it revolves well if we want to do that yes i am doing all this on a ti-89 titanium fantastic calculator well if we want to do the math on this we could trace out the new circle that the center of circle a makes the radius of this new circle would be the radius of circle a plus the radius of circle b now if we do the digits and the math again like we did before with this new information what am i what am i what am i khan academy in here if we do this math again we get 4 and indeed 4 is the correct answer to the original s a t question but this just seems deliberately confusing why would such a poorly worded question be on such an important test there is a lot more mathematics to unpack here that we don't have time to but i at least want to give you some additional circle mathematics that you can see at home with your very eyes now as you can see on the screen behind me if you trace out the line a point on a circle makes as it revolves around another equally sized circle you get a lovely little heart shape called a cardioid like a heart now it just so happens that if you take something that moves in straight lines like light and you reflect that off the inside of a circle you can get a cardioid too watch this take your phone's flashlight and shine it inside of the cup do you see the cardioid that forms light rays are bouncing all around but there is some boundary where many of the rays of light are tangent where they touch but don't pass this boundary where light bunches up so to speak and gets brighter creates what's called a caustic and it forms as long as you shine light from the edge of the coffee cup towards the inside a caustic is a very complicated mathematical concept to be sure but you may not know that you're already familiar with definitely the most beautiful example you can find a certain optic that happens when you shine sunlight through droplets of water sprinkled through the air that's right a rainbow is another example of a caustic rainbows coins coffee cups mathematics it's everywhere there's nothing in it to find out why such a poorly worded question would be on such a critical assessment of students like any good seeker of the truth we should go back to the source now at the time in 1982 this test mistake was big news 300 000 test scores had to be recalculated it was such big news that the new york times picked it up in the new york times article which you are seeing now they say pretty much everything we've said so far except for one big change it printed the question from nearly 40 years ago in full and the original question says after how many revolutions will the center of circle a reach its starting point the center of circle a we just did that math and we understand the answer you know it's almost it's almost like the makers of popular math books and youtube videos on this same question made a mistake by not including the original question in its entirety thereby making the video itself almost deliberately misleading for the viewer as to how tricky the actual question really is for views imagine that let's check our answers anyway i'm going to use a handy online tool i found to make things easier i've set the radii to their appropriate ratio let's focus on the center of the smaller circle first and see how many times the radius dimension there the line points to the left the equivalent of the letter a going back to its starting position okay here we go one two three four see okay this time we're going to focus in on the coins perspective and count how many times the dot on the edge of the smaller circle touches the larger circle's surface here we go one two three see the same situation has two different answers when the question is not properly specified you see by not including the original question in full the popular retellings of this problem make the problem much harder than it ever actually was by allowing readers and viewers to fall prey to what's called the coin rotation paradox the coin rotation paradox is a counter-intuitive conundrum similar to this sat question that you can perform at home all you need are two equally sized coins i'm using quarters here because i'm stupid rich so here's the question i'm presenting to you with my very veiny hands how many revolutions will one coin make around the other well your brain reflexively tells you just one because the circumferences are the same but let's watch washington's nose here let's see it when it's in the same orientation facing this way that it is now to minimize slipping as best you can there you go one revolution and it's in the same orientation so to complete it the essence of this coin paradox is that even though the circumferences are the same no slip that no slipping even though the circumferences are the same it takes two full revolutions this result does seem odd at first but it's the same as tracking the center of circle a as it travels around circle b and we know and understand that math already now let's track a spot on the coin's edge when does washington's nose touch the right side of the coin again let's try it from its own perspective minimize slipping perfect perfect absolutely perfect i think you see where this is going if you do it right there it only takes one revolution just one the coin rotation paradox isn't really a paradox of course it's just a result that goes against our knee-jerk reasoning because of a lack of specificity and our brains just naturally default to one point of view our own over another i think it's kind of ironic that popular retellings of this s.a.t question fall prey to the very same illusion that makes this question so hard in the first place just goes to show you it pays to do your homework time's up please turn in your tests wish me luck until next time so if supernova b explodes next to zeta b with a velocity don't worry about it now exiting the facility thank you so much to the very nerdy staff at the facility for their direct and substantial support in creating this video today especially i want to call out research assistant twilight katana and visiting scholar robert blackburn if you want to join the facility right now and get yourself a nice sweet lab coat and join us you can go to patreon.com kyle hill join the patreon and the discord right now over 1100 nerds are giving me episode ideas seeing merch ideas early getting episodes early and organizing their own game nights coming up with canon giving me episode ideas it's great and if you support the facility just enough you get your name on arya here each week and as you can see there's there's more and more of you so i don't know how to pass the if it sounded like i was calling out people who make these kind of trite math books and perhaps youtube videos that aren't as specific as they can be that's because i am i'm calling them out specificity matters you can't just have one or seven or twenty thousand twenty thousand what you must be specific that's how we learn that's how we catalog things you know who taught me this ti-89 ti-89 and this is true is the only person's name that i remember from college i literally cannot remember a single person's name that's where my head's at from all of us here at the facility thanks for watching that's right i sound like this i'm a calculator

Channel: Kyle Hill

Views: 871,861

Rating: 4.8963909 out of 5

Keywords: because science, engineering, kyle hill, learning, math, physics, science, stem, the facility, sat, test, homework, kyle hill channel

Id: _zCDbB9wvrc

Length: 13min 7sec (787 seconds)

Published: Thu Jul 16 2020

Reddit Comments

[u/mcgravier]

The answer '4' is incorrect as well.

Since the question doesn't specify what kind of revolutions we're talking about, we have to count both revolutions around centre of circle A and centre of circle B. Circle A will do 4 revolutions around it's own centre and one large revolution around centre of circle B.

So the answer is 5.