The SAT Question Everyone Got Wrong
December 3, 2023 1:47 PM Subscribe
It's a math question about meshed rotating circles. All of the provided answers were wrong due to a test error.
posted by CynicalKnight at 1:56 PM on December 3, 2023 [27 favorites]
posted by CynicalKnight at 1:56 PM on December 3, 2023 [27 favorites]
I'm at my job, which is in a large community college tutoring center. I spent most of my 5-hour shift with a student studying for an algebra final; now I have a bit more than an hour left, and no student currently.
I started this video, got to the problem, and thought, "OK, if every student got this wrong (and this quite literally probably includes me, as May 1, 1982 was the end of my junior year of high school) it will give me something meaty to dig into during my down time."
Being out of scratch paper, I put the problem up on the whiteboard. I rubbed my hands together gleefully. This was going to be so fun! It was going to be like that day a couple month ago when I decided to derive the quadratic formula!
I solved it in ten seconds.
I am very disappointed in 1982 me.
The problem, for those who don't want to or can't watch video, is a diagram of a small circle, A, touching a large circle, B. We are told that B's radius is 3 times A's radius. If we were to rotate A around B how many rotations of A would it take for A to get back to its starting point?
The choices are 3/2; 3; 6; 9/2; 9
It will be interesting now to watch the video, to what was tricky about it for students--or to find out that what I thought was an efficient solving job turns out to be me falling prey to one of the common errors the SAT likes to sprinkle into its answers.
posted by Well I never at 2:02 PM on December 3, 2023 [4 favorites]
I started this video, got to the problem, and thought, "OK, if every student got this wrong (and this quite literally probably includes me, as May 1, 1982 was the end of my junior year of high school) it will give me something meaty to dig into during my down time."
Being out of scratch paper, I put the problem up on the whiteboard. I rubbed my hands together gleefully. This was going to be so fun! It was going to be like that day a couple month ago when I decided to derive the quadratic formula!
I solved it in ten seconds.
I am very disappointed in 1982 me.
The problem, for those who don't want to or can't watch video, is a diagram of a small circle, A, touching a large circle, B. We are told that B's radius is 3 times A's radius. If we were to rotate A around B how many rotations of A would it take for A to get back to its starting point?
The choices are 3/2; 3; 6; 9/2; 9
It will be interesting now to watch the video, to what was tricky about it for students--or to find out that what I thought was an efficient solving job turns out to be me falling prey to one of the common errors the SAT likes to sprinkle into its answers.
posted by Well I never at 2:02 PM on December 3, 2023 [4 favorites]
Mod note: One deleted. Please remember that if you're commenting in a thread, you are expected to contribute to the overall discussion in a productive way. If you do not like the way a thread is structured, feel free to flag and move on.
posted by travelingthyme (staff) at 2:03 PM on December 3, 2023 [4 favorites]
posted by travelingthyme (staff) at 2:03 PM on December 3, 2023 [4 favorites]
It took me a far too long to grasp why the obvious answer wasn't actually correct. My initial intuition was that the obvious answer couldn't be the correct answer, but that's because it's a multiple choice test, not because I was thinking about the actual problem.
Then I was like "I know how gears work, so obvious answer is obviously correct." Only much later did I realize that planetary gears don't work the same way as ones that are fixed relative to each other with both spinning.
posted by wierdo at 2:04 PM on December 3, 2023 [1 favorite]
Then I was like "I know how gears work, so obvious answer is obviously correct." Only much later did I realize that planetary gears don't work the same way as ones that are fixed relative to each other with both spinning.
posted by wierdo at 2:04 PM on December 3, 2023 [1 favorite]
Oh, it's much cooler than I thought it was going to be! Love this coin rotation paradox thingy. I did in fact make a simple mistake in writing the diameter formulas for the two circles, which let to a false sense of having gotten it right. When I corrected my error, I got the answer the SAT makers thought was correct. Now to find out why we were all wrong.
This is so fun. Thanks for this, NotMyselfRightNow.
posted by Well I never at 2:10 PM on December 3, 2023 [4 favorites]
This is so fun. Thanks for this, NotMyselfRightNow.
posted by Well I never at 2:10 PM on December 3, 2023 [4 favorites]
Me: zero math skills.
Me after watching this video: What the Actual Fuck?
posted by chavenet at 2:22 PM on December 3, 2023 [17 favorites]
Me after watching this video: What the Actual Fuck?
posted by chavenet at 2:22 PM on December 3, 2023 [17 favorites]
It clicked for me when I noticed that's this is just like the difference between a solar day and a sidereal day.
How many times does the earth rotate in one year? It's not 365 times... it's 366 times.
...although we pretty much always say 365 times, since we usually count "sun directly overhead one day to sun directly overhead the next day" as one rotation. But that's just a wee bit more than a single rotation, as you can see if you watch the stars instead of the sun.
posted by clawsoon at 2:26 PM on December 3, 2023 [12 favorites]
How many times does the earth rotate in one year? It's not 365 times... it's 366 times.
...although we pretty much always say 365 times, since we usually count "sun directly overhead one day to sun directly overhead the next day" as one rotation. But that's just a wee bit more than a single rotation, as you can see if you watch the stars instead of the sun.
posted by clawsoon at 2:26 PM on December 3, 2023 [12 favorites]
This is a great example of a problem that is hard to wrap your head around if you try to do it only with numbers, but becomes easy once you actually draw a picture. If you draw the position of the smaller circle after each of its perimeter-length moves around the larger circle, including the radius that touches the larger circle, it quickly becomes clear that it's turning more than one rotation each time it does that! That's entirely lost if you're just comparing perimeters.
My 7yo is currently slogging through his teachers making him learn yet another way to add and subtract multi-digit numbers. He's way ahead of his math and he resents having to do these repetitive, multi-step techniques because he can just do the calculations in his head already. It's really tough to explain to a kid that having multiple ways to solve a problem, even if you already know the solution off the top of your head, is useful not because it helps you arrive at the solution, but because it gets you used to the idea that there are multiple approaches to a problem and it's important to have that larger toolkit. Once he's doing more geometry I'm going to have to pull this out of the closet as an example.
posted by phooky at 2:26 PM on December 3, 2023 [8 favorites]
My 7yo is currently slogging through his teachers making him learn yet another way to add and subtract multi-digit numbers. He's way ahead of his math and he resents having to do these repetitive, multi-step techniques because he can just do the calculations in his head already. It's really tough to explain to a kid that having multiple ways to solve a problem, even if you already know the solution off the top of your head, is useful not because it helps you arrive at the solution, but because it gets you used to the idea that there are multiple approaches to a problem and it's important to have that larger toolkit. Once he's doing more geometry I'm going to have to pull this out of the closet as an example.
posted by phooky at 2:26 PM on December 3, 2023 [8 favorites]
What the Actual Fuck?
Me, too. If we have a slow day at work thisweek, I'll ask one of the advanced math nerds in the tutoring center to explain it to me.
posted by Well I never at 2:27 PM on December 3, 2023 [2 favorites]
Me, too. If we have a slow day at work thisweek, I'll ask one of the advanced math nerds in the tutoring center to explain it to me.
posted by Well I never at 2:27 PM on December 3, 2023 [2 favorites]
tl;dr: Sidereal time. If folks are stumped, ask yourself a question: how many times would the coin rotate if it slid instead of rolled? What if it was simply orbiting?
posted by pwnguin at 2:27 PM on December 3, 2023 [3 favorites]
posted by pwnguin at 2:27 PM on December 3, 2023 [3 favorites]
This was surprisingly good. I was quite sure of my answer... just like the SAT maker... so there was some delicious WTF learning why it was wrong. Plus, sidereal days! And the math whiz kid who grew up to be a mathematician.
posted by zompist at 2:28 PM on December 3, 2023 [4 favorites]
posted by zompist at 2:28 PM on December 3, 2023 [4 favorites]
It clicked for me when I noticed that's this is just like the difference between a solar day and a sidereal day.
...and now that I'm watching the video, I see that they covered this.
posted by clawsoon at 2:34 PM on December 3, 2023 [4 favorites]
...and now that I'm watching the video, I see that they covered this.
posted by clawsoon at 2:34 PM on December 3, 2023 [4 favorites]
Follow-up question: If the earth were rotating in the opposite direction, but still revolving around the sun in the same direction, how many times would the earth rotate in one year?
posted by clawsoon at 2:38 PM on December 3, 2023 [1 favorite]
posted by clawsoon at 2:38 PM on December 3, 2023 [1 favorite]
This diagram, from the Wikipedia entry pwnguin linked to above, was really helpful for me in beginning to actually make sense of this.
posted by Well I never at 2:38 PM on December 3, 2023
posted by Well I never at 2:38 PM on December 3, 2023
It's amazing how much of the math world is interconnected.
The shape formed by the path of the centerpoint of the circle around the different shapes looks like basically the Minkowski sum of the two shapes. The subtraction version of this (the Minkowski difference) can be quite useful in collision detection algorithms for video games. (Here's one example.)
posted by Flaffigan at 2:38 PM on December 3, 2023 [3 favorites]
The shape formed by the path of the centerpoint of the circle around the different shapes looks like basically the Minkowski sum of the two shapes. The subtraction version of this (the Minkowski difference) can be quite useful in collision detection algorithms for video games. (Here's one example.)
posted by Flaffigan at 2:38 PM on December 3, 2023 [3 favorites]
It's an interesting video, especially the connection to sidereal time, but I can't be the only who was holding themselves back from yelling at their screen "just look at the distance travelled by the centre of the circle!" until he finally got to that point, in a complicated way.
posted by ssg at 3:17 PM on December 3, 2023
posted by ssg at 3:17 PM on December 3, 2023
I 3d-printed gears, one with 14 teeth, the other with 42. I meshed them, marked where they started, and rotated them. I marked every place the smaller gear came back to the big gear.
I have three marks, not four.
posted by scruss at 3:28 PM on December 3, 2023 [3 favorites]
I have three marks, not four.
posted by scruss at 3:28 PM on December 3, 2023 [3 favorites]
Thanks for sharing! I thought the paradox was interesting and it was also fun to see the recent interview with the guy who was one of the ones who pointed out the mistake when he was 15. In my opinion, if any topic warrants being explained in a video, this is certainly one.
posted by snofoam at 3:39 PM on December 3, 2023 [1 favorite]
posted by snofoam at 3:39 PM on December 3, 2023 [1 favorite]
I have three marks, not four.
That's covered in the video. You measured 3 solar days. But this all hinges on different interpretations of "rotation." If you count the number of times the small gear reference point touches the other gear, you get N rotations. If you count the number of times the small gear is oriented in the absolute starting position (ie the line from center to the reference point is pointing due east) you get N+1. The extra rotation is spread around the entire orbit.
posted by pwnguin at 3:46 PM on December 3, 2023 [14 favorites]
That's covered in the video. You measured 3 solar days. But this all hinges on different interpretations of "rotation." If you count the number of times the small gear reference point touches the other gear, you get N rotations. If you count the number of times the small gear is oriented in the absolute starting position (ie the line from center to the reference point is pointing due east) you get N+1. The extra rotation is spread around the entire orbit.
posted by pwnguin at 3:46 PM on December 3, 2023 [14 favorites]
Yeah, scruss’s experiment is equivalent to the interpretations discussed starting at timestamp 5:51 in the video.
posted by mbrubeck at 3:51 PM on December 3, 2023
posted by mbrubeck at 3:51 PM on December 3, 2023
every single student got wrong.
Here's the thing though: who realizes this. Are the markers of the test not just told that 3 is the answer and therefore every student who chose it is "correct"? At what point do the test-markers realize the error? Is it when one of the 4 "complainers" complains?
And, if presented with multiple choice and the implication that one of the answers is correct and you choose the same answer the test-creator thought correct, are you not correct, given the parameters of the test?
posted by dobbs at 3:54 PM on December 3, 2023 [5 favorites]
Here's the thing though: who realizes this. Are the markers of the test not just told that 3 is the answer and therefore every student who chose it is "correct"? At what point do the test-markers realize the error? Is it when one of the 4 "complainers" complains?
And, if presented with multiple choice and the implication that one of the answers is correct and you choose the same answer the test-creator thought correct, are you not correct, given the parameters of the test?
posted by dobbs at 3:54 PM on December 3, 2023 [5 favorites]
"...and you choose the same answer the test-creator thought correct, are you not correct, given the parameters of the test?"
300,000 wrongs don't make a right.
posted by mule98J at 4:02 PM on December 3, 2023 [6 favorites]
300,000 wrongs don't make a right.
posted by mule98J at 4:02 PM on December 3, 2023 [6 favorites]
I am entertained by how much I found myself offended by your suggestion, dobbs. Thanks for that. But no, consensus doesn't imply correctness.
posted by tigrrrlily at 4:04 PM on December 3, 2023 [4 favorites]
posted by tigrrrlily at 4:04 PM on December 3, 2023 [4 favorites]
And, if presented with multiple choice and the implication that one of the answers is correct and you choose the same answer the test-creator thought correct, are you not correct, given the parameters of the test?
Two of the students who complained had given the answer they thought the test was looking for. I think they're the ones who account for the stated possibility of a student losing 10 points on the test after the re-grading: instead of getting x/y questions right, they would then get (x-1)/y-1) and the test would be scored accordingly. If they simply gave the point to every student while keeping y=number of questions the same, scores could only go up.
I am pretty sure that at this time, you still lost a fraction of a point for a wrong answer (to discourage guessing) but not for a blank. So the recalculating was probably pretty complicated and I would totally watch a video explaining it.
This year the SAT is being administered entirely online for the first time. It was still a paper test in the 2022-23 school year. How I know: I tutored SAT prep during the past year. I took the GRE online more than 20 years ago, and when I started tutoring the SAT in 2021, I was really surprised it was still a paper test.
posted by Well I never at 4:08 PM on December 3, 2023 [2 favorites]
Two of the students who complained had given the answer they thought the test was looking for. I think they're the ones who account for the stated possibility of a student losing 10 points on the test after the re-grading: instead of getting x/y questions right, they would then get (x-1)/y-1) and the test would be scored accordingly. If they simply gave the point to every student while keeping y=number of questions the same, scores could only go up.
I am pretty sure that at this time, you still lost a fraction of a point for a wrong answer (to discourage guessing) but not for a blank. So the recalculating was probably pretty complicated and I would totally watch a video explaining it.
This year the SAT is being administered entirely online for the first time. It was still a paper test in the 2022-23 school year. How I know: I tutored SAT prep during the past year. I took the GRE online more than 20 years ago, and when I started tutoring the SAT in 2021, I was really surprised it was still a paper test.
posted by Well I never at 4:08 PM on December 3, 2023 [2 favorites]
I'm pretty good at math and got this wrong. It is as much about wording as it is about considering frame of reference — these things can trip up almost anyone.
posted by They sucked his brains out! at 4:30 PM on December 3, 2023 [1 favorite]
posted by They sucked his brains out! at 4:30 PM on December 3, 2023 [1 favorite]
My maths skills are terrible, but it seemed obvious to me that none of the answers were correct just by looking at the relative size of the circles. Admittedly, I assumed the improper fractions were red herrings and ignored them, but I knew the answer was not 3, 6 or 9. I probably would have answered 3 because it's the number that seemed the 'rightest' and both 6 and 9 were just obviously wrong.
The explanation was really interesting :-)
posted by dg at 4:58 PM on December 3, 2023
The explanation was really interesting :-)
posted by dg at 4:58 PM on December 3, 2023
→
The gearbox in your car is so fucked if one of the gears decides to take an extra rotation relative to the number of teeth
posted by scruss at 5:01 PM on December 3, 2023 [3 favorites]
But this all hinges on different interpretations of "rotation."
The gearbox in your car is so fucked if one of the gears decides to take an extra rotation relative to the number of teeth
posted by scruss at 5:01 PM on December 3, 2023 [3 favorites]
It clicked for me when I noticed that's this is just like the difference between a solar day and a sidereal day.
For me, it clicked when he showed the circle going around a triangle. I think also doing an illustration of a circle going up one side of a line, rotating over it, down the other side, and then rotating under it could also show why the +1 is always there.
posted by tclark at 5:03 PM on December 3, 2023 [3 favorites]
For me, it clicked when he showed the circle going around a triangle. I think also doing an illustration of a circle going up one side of a line, rotating over it, down the other side, and then rotating under it could also show why the +1 is always there.
posted by tclark at 5:03 PM on December 3, 2023 [3 favorites]
Huh. I visually guesstimated the little circumference and walked that around the big circle in my head and it looked like about 4, so I picked 9/2 as that was closest to 4.
The actual answer was fascinating.
posted by jellywerker at 5:39 PM on December 3, 2023
The actual answer was fascinating.
posted by jellywerker at 5:39 PM on December 3, 2023
scruss: I 3d-printed gears, one with 14 teeth, the other with 42. I meshed them, marked where they started, and rotated them. I marked every place the smaller gear came back to the big gear.
...and, if you held the larger gear still, each mark that you made would have happened after 1.333... rotations of the smaller gear.
To illustrate, draw an arrow on the table from the center of the smaller gear to the point at which you make each mark. If your starting arrow is 0 degrees, the next arrow will be at 120 degrees (after rotating 540 degrees), followed by 240 degrees for the next arrow (after rotating another 540 degrees), then back to 0 degrees.
When you go from 0 degrees at the start of a rotation to 120 degrees at the end of a rotation, you know that you can't have gone exactly 1 rotation.
posted by clawsoon at 6:25 PM on December 3, 2023 [2 favorites]
...and, if you held the larger gear still, each mark that you made would have happened after 1.333... rotations of the smaller gear.
To illustrate, draw an arrow on the table from the center of the smaller gear to the point at which you make each mark. If your starting arrow is 0 degrees, the next arrow will be at 120 degrees (after rotating 540 degrees), followed by 240 degrees for the next arrow (after rotating another 540 degrees), then back to 0 degrees.
When you go from 0 degrees at the start of a rotation to 120 degrees at the end of a rotation, you know that you can't have gone exactly 1 rotation.
posted by clawsoon at 6:25 PM on December 3, 2023 [2 favorites]
> If you draw the position of the smaller circle after each of its perimeter-length moves around the larger circle, including the radius that touches the larger circle, it quickly becomes clear
what!
> how many times would the coin rotate if it slid instead of rolled? What if it was simply orbiting?
whaaat!!!!!!!
> Me: zero math skills.
> Me after watching this video: What the Actual Fuck?
Completely accurate, you have spoken the truth of my heart, chavenet. But I have an addendum.
Me after reading this thread: OH MY GOD WHY DO I SUDDENLY UNDERSTAND THIS OBVIOUS NONSENSE it should not make sense make it stop
posted by MiraK at 6:27 PM on December 3, 2023 [6 favorites]
what!
> how many times would the coin rotate if it slid instead of rolled? What if it was simply orbiting?
whaaat!!!!!!!
> Me: zero math skills.
> Me after watching this video: What the Actual Fuck?
Completely accurate, you have spoken the truth of my heart, chavenet. But I have an addendum.
Me after reading this thread: OH MY GOD WHY DO I SUDDENLY UNDERSTAND THIS OBVIOUS NONSENSE it should not make sense make it stop
posted by MiraK at 6:27 PM on December 3, 2023 [6 favorites]
For me, it clicked when he showed the circle going around a triangle. I think also doing an illustration of a circle going up one side of a line, rotating over it, down the other side, and then rotating under it could also show why the +1 is always there.
Did he do the one where you reduce the thing you're rotating around to a single point? Somebody mentioned somewhere that that was what helped them envision it. Instead of going around a bigger gear, you just make the circle go around a dot.
posted by clawsoon at 6:30 PM on December 3, 2023 [4 favorites]
Did he do the one where you reduce the thing you're rotating around to a single point? Somebody mentioned somewhere that that was what helped them envision it. Instead of going around a bigger gear, you just make the circle go around a dot.
posted by clawsoon at 6:30 PM on December 3, 2023 [4 favorites]
Another approach: Look at what happens when circle B is reduced to a point -- that is, a degenerate circle of radius zero and circumference zero. So now the radius of circle B is zero times the radius of circle A. But when we roll circle A around it, it's pretty clear that circle A isn't rotating zero times.
posted by baf at 6:44 PM on December 3, 2023 [5 favorites]
posted by baf at 6:44 PM on December 3, 2023 [5 favorites]
The proper solution is to take the ACT.
posted by JohnnyGunn at 6:54 PM on December 3, 2023 [1 favorite]
posted by JohnnyGunn at 6:54 PM on December 3, 2023 [1 favorite]
That was interesting! I'm a math teacher and I instantly got the answer the test-takers thought it was, using the ratio of the circumfereces discussed in the video. But really interesting to read how many ways the question could be interpreted and to think about reference points. Also, the visuals on the video are really, really good.
posted by subdee at 7:37 PM on December 3, 2023 [1 favorite]
posted by subdee at 7:37 PM on December 3, 2023 [1 favorite]
i got this on my second guess after falling for the obvious '3' honeypot and then exploiting the metaknowledge that none of the answers were correct. the intuition that got me there is that the centre of circle a has to travel around a circle with a radius (starting from the centre of circle b) of 1+1/3 = 4/3 to get back to its initial position. the rest of the proof is left as an exercise for the reader.
apposite Calvin & Hobbes
posted by logicpunk at 8:07 PM on December 3, 2023 [7 favorites]
apposite Calvin & Hobbes
posted by logicpunk at 8:07 PM on December 3, 2023 [7 favorites]
after continuing to watch, that's the same solution the big math genius got so who doesn't apply themselves now, huh, Ms. Trappoletti?
posted by logicpunk at 8:17 PM on December 3, 2023 [2 favorites]
posted by logicpunk at 8:17 PM on December 3, 2023 [2 favorites]
I would have gotten that question right, i.e. counted as correct by the SAT test.
But I doubt I would have noticed that all the answers were wrong. To reach that conclusion would probably have required more time and thought than I'd devote to a single question. You rule out a couple answers, one of the remaining ones looks correct, fill in the circle and move on to the next question.
TBH, even when I had tons of leftover time at the end, I did not go back and reexamine difficult questions, etc. The SAT just doesn't reward that kind of overthinking (at least back in the late 90s).
posted by ryanrs at 9:20 PM on December 3, 2023 [2 favorites]
But I doubt I would have noticed that all the answers were wrong. To reach that conclusion would probably have required more time and thought than I'd devote to a single question. You rule out a couple answers, one of the remaining ones looks correct, fill in the circle and move on to the next question.
TBH, even when I had tons of leftover time at the end, I did not go back and reexamine difficult questions, etc. The SAT just doesn't reward that kind of overthinking (at least back in the late 90s).
posted by ryanrs at 9:20 PM on December 3, 2023 [2 favorites]
The SAT was the only thing which got me into big state university. My high school GPA was a slacker's 2.96 (or something like that), but I scored 1320 on the SAT, thanks to good logic skills and excellent reading skills...all of which are long gone at this point.
posted by maxwelton at 9:58 PM on December 3, 2023 [2 favorites]
posted by maxwelton at 9:58 PM on December 3, 2023 [2 favorites]
Imagine the small circle rotating around the larger one.
By the terms of the question, it’s clear that one rotation of the small circle has taken place when the marked radius is once again pointing straight down.
But now imagine what the diagram will look like when the small circle has traveled 1/3 of the way around he big one.
At that point the marked radius is clearly NOT pointing straight down, it’s pointing toward the center of the big circle! Therefore the small circle must have rotated MORE THAN 360° at that point. 120° more, in fact.
Repeat that cycle twice more in order to restore the original condition.
For each of the 3 cycles, the small circle has rotated once + 120°
So for all three together, the small circle has rotated 3 times + 360°, which is the same as rotating 4 times.
posted by jamjam at 11:20 PM on December 3, 2023 [2 favorites]
By the terms of the question, it’s clear that one rotation of the small circle has taken place when the marked radius is once again pointing straight down.
But now imagine what the diagram will look like when the small circle has traveled 1/3 of the way around he big one.
At that point the marked radius is clearly NOT pointing straight down, it’s pointing toward the center of the big circle! Therefore the small circle must have rotated MORE THAN 360° at that point. 120° more, in fact.
Repeat that cycle twice more in order to restore the original condition.
For each of the 3 cycles, the small circle has rotated once + 120°
So for all three together, the small circle has rotated 3 times + 360°, which is the same as rotating 4 times.
posted by jamjam at 11:20 PM on December 3, 2023 [2 favorites]
I should have said 'it's clear that one rotation has taken place when the marked radius is once again pointing straight right'.
posted by jamjam at 11:37 PM on December 3, 2023
posted by jamjam at 11:37 PM on December 3, 2023
On a. 3 = ✓ “It ain't what you don't know that gets you into trouble. It's what you know for sure that just ain't so.” Mark Twain not.
On sidereal, check out Matt Parker on eclipses, Saros and Sidereal, Anomalistic , Synodic, Draconian months?
posted by BobTheScientist at 1:26 AM on December 4, 2023
On sidereal, check out Matt Parker on eclipses, Saros and Sidereal, Anomalistic , Synodic, Draconian months?
posted by BobTheScientist at 1:26 AM on December 4, 2023
300,000 wrongs don't make a right.
Consider that 300,000 wrongs never make a right, but that 300,001 do.
(Someone check my math on that please)
posted by ensign_ricky at 5:53 AM on December 4, 2023 [2 favorites]
Consider that 300,000 wrongs never make a right, but that 300,001 do.
(Someone check my math on that please)
posted by ensign_ricky at 5:53 AM on December 4, 2023 [2 favorites]
“Did he do the one where you reduce the thing you're rotating around to a single point? Somebody mentioned somewhere that that was what helped them envision it. Instead of going around a bigger gear, you just make the circle go around a dot.”
Yeah, the other day after I watched the video (and fell for the obvious-but-wrong answer), the "shrink the inner circle to a point" variation which disproves the simple circumference reasoning came to me as my "eureka" insight. I came into the thread thinking I might be the first to mention this, but I should have known that other mefites would beat me to it.
This is a good example of how sometimes a little knowledge can be worse than none — without a formula at hand which appears, wrongly, to provide a simple calculation route to the answer, an entirely ignorant person must either infer it empirically or deduce it from first principles.
This is why, in math and science, it's sometimes very productive to reason starting from further back toward foundations. Many of the tools, techniques, and abstractions we typically rely upon trade convenience for comprehension.
And that, in turn, is related closely to how we frequently discover when teaching something, we didn't understand it as well as we thought.
posted by Ivan Fyodorovich at 6:32 AM on December 4, 2023
Yeah, the other day after I watched the video (and fell for the obvious-but-wrong answer), the "shrink the inner circle to a point" variation which disproves the simple circumference reasoning came to me as my "eureka" insight. I came into the thread thinking I might be the first to mention this, but I should have known that other mefites would beat me to it.
This is a good example of how sometimes a little knowledge can be worse than none — without a formula at hand which appears, wrongly, to provide a simple calculation route to the answer, an entirely ignorant person must either infer it empirically or deduce it from first principles.
This is why, in math and science, it's sometimes very productive to reason starting from further back toward foundations. Many of the tools, techniques, and abstractions we typically rely upon trade convenience for comprehension.
And that, in turn, is related closely to how we frequently discover when teaching something, we didn't understand it as well as we thought.
posted by Ivan Fyodorovich at 6:32 AM on December 4, 2023
Mind completely blown, until he got to the explanations.
posted by beagle at 6:45 AM on December 4, 2023
posted by beagle at 6:45 AM on December 4, 2023
OH MY GOD WHY DO I SUDDENLY UNDERSTAND THIS OBVIOUS NONSENSE it should not make sense make it stop
Unexpected mathematical enlightenment is another side effect of exposure to Neil Peart practice sessions. It's like secondhand smoking; there's no cure. The only way to stabilise the condition is assiduous avoidance of letting Peart become a gateway to harder drummers.
posted by flabdablet at 7:42 AM on December 4, 2023 [1 favorite]
Unexpected mathematical enlightenment is another side effect of exposure to Neil Peart practice sessions. It's like secondhand smoking; there's no cure. The only way to stabilise the condition is assiduous avoidance of letting Peart become a gateway to harder drummers.
posted by flabdablet at 7:42 AM on December 4, 2023 [1 favorite]
When I was like 9 or 10 I remember figuring out the sidereal time thing by myself and challenging my very smart dad about the revolution of the earth being a few minutes whatever shy of 24 hours and him just NOT getting it, and I kind of feel like him now watching this video. Like I've studied too much math in the subsequent years to see the answer as anything but 3 (though TBH I like the snarky, pedantic answer of "1" as well).
posted by St. Oops at 8:09 AM on December 4, 2023
posted by St. Oops at 8:09 AM on December 4, 2023
To my Geometry/Algebra/Calculus Teachers in High School Who Sagely Told Me that I Would Use Math in Real Life: WELL PLAYED.
posted by tafetta, darling! at 8:50 AM on December 4, 2023 [2 favorites]
posted by tafetta, darling! at 8:50 AM on December 4, 2023 [2 favorites]
I saw it by using circles of equal size.
When you start at the top and make it to the bottom, the orbiting circle is touching the inner circle on the bottom with its top.
The outer circle started touching on its bottom. After half a circle, it has to be touching on its top. And after half a circle, the inner must be touching on its bottom.
So it is right side up. And half way around.
So it managed to rotate 360 degrees, but is only half way done.
So two equal circles, the outer one rotates twice!
From that visualization I worked out the "extra rotatation" rule.
posted by NotAYakk at 11:34 AM on December 4, 2023 [1 favorite]
When you start at the top and make it to the bottom, the orbiting circle is touching the inner circle on the bottom with its top.
The outer circle started touching on its bottom. After half a circle, it has to be touching on its top. And after half a circle, the inner must be touching on its bottom.
So it is right side up. And half way around.
So it managed to rotate 360 degrees, but is only half way done.
So two equal circles, the outer one rotates twice!
From that visualization I worked out the "extra rotatation" rule.
posted by NotAYakk at 11:34 AM on December 4, 2023 [1 favorite]
For a wheel rolling without slipping around the outside of a convex polygon, things are very simple. At each vertex, the wheel must pivot through an angle equal to the external angle of the vertex. (The center of the circle sweeps out an arc around the corner, adding to the distance travelled). To go around the shape, then, we just add up the external angles. Regardless of shape, it's 360-degrees for a closed polygon. So the number of rotations required to roll around the outside is p /(2pi*r)+1, where p is the perimeter of the shape, and the total distance travelled by the center of the circle is p+2*pi*r. This holds in the limit the path is a circle, in which case we get r_b/r_a + 1 for the number of rotations, and r_b+r_a for the distance travelled by the center.
Rolling around the "inside" (or even on the outside of non-convex shapes) is slightly trickier. At a vertex with interior angle theta, the wheel skips ahead a distance equal to 2*r*cot(theta/2)--this the length of the path the wheel never touches as it transitions from one edge to the next. For a regular polygon with n sides, each of length l, we have p=n*l and this means the number of rotations as it goes all the way around the inside is p/(2pi*r) - n/pi*tan(pi/n). So the "deficit" after going around completely is different for shapes with different numbers of sides. The second term converges to 1 in the limit n -> infinity, corresponding to a circle, and the numerator of the first term is just the perimeter, so we get r_b/r_a-1 for the number of rotations (as the video explains).
Additionally--if I've done this right--we can consider a wheel rolling along a sawtooth/zig-zag path where each turn is ~98.6 degrees or 1.72 radians, after every even number of turns the wheel's total rotation (in radians) is equal to the path distance travelled divided by the radius (as in the straight-line case), whereas after each odd turn, it's either ahead (or behind) by 1.72 radians.
posted by dsword at 12:05 PM on December 4, 2023 [1 favorite]
Rolling around the "inside" (or even on the outside of non-convex shapes) is slightly trickier. At a vertex with interior angle theta, the wheel skips ahead a distance equal to 2*r*cot(theta/2)--this the length of the path the wheel never touches as it transitions from one edge to the next. For a regular polygon with n sides, each of length l, we have p=n*l and this means the number of rotations as it goes all the way around the inside is p/(2pi*r) - n/pi*tan(pi/n). So the "deficit" after going around completely is different for shapes with different numbers of sides. The second term converges to 1 in the limit n -> infinity, corresponding to a circle, and the numerator of the first term is just the perimeter, so we get r_b/r_a-1 for the number of rotations (as the video explains).
Additionally--if I've done this right--we can consider a wheel rolling along a sawtooth/zig-zag path where each turn is ~98.6 degrees or 1.72 radians, after every even number of turns the wheel's total rotation (in radians) is equal to the path distance travelled divided by the radius (as in the straight-line case), whereas after each odd turn, it's either ahead (or behind) by 1.72 radians.
posted by dsword at 12:05 PM on December 4, 2023 [1 favorite]
I think it's 3, will read the above comments and watch later.
It's a little disappointing that the best article ever I'm aware
of on the SATs is now almost 40 years old:
David Owen on the SATs [PDF]
I recently rewrote his SASAT at the end of that Harper's article, and after a bit of hesitation got 5/5. For those who like to know before they click, it's a set of reading-passage questions taken from a recent SAT (meaning like 40 years ago recent), but without the actual reading passage.
posted by morspin at 12:38 PM on December 4, 2023
It's a little disappointing that the best article ever I'm aware
of on the SATs is now almost 40 years old:
David Owen on the SATs [PDF]
I recently rewrote his SASAT at the end of that Harper's article, and after a bit of hesitation got 5/5. For those who like to know before they click, it's a set of reading-passage questions taken from a recent SAT (meaning like 40 years ago recent), but without the actual reading passage.
posted by morspin at 12:38 PM on December 4, 2023
My first thought was that if *nobody* got this right it means that the correct answer wasn't among the choices provided because the probability of guessing would get some *correct* answers as far as the test administrators were concerned.
Then I went to the obvious answer of 3 based on the ratios of radii and quickly determined that it was a matter of frame of reference because from the global frame after the smaller circle had done one full rotation the point on the smaller circle in contact with the larger circle would not have touched the larger circle yet and therefore the smaller circle would have to rotate 3 plus a bit more.
Then I got bored and went to watch TV not wanting to bring out scratch paper. I was going to break out the trig and go into radians and such but as it works out.... it's for the problem at hand 3 plus 3 times 1/3 equals 4.
I might have got this at 12 years old or at least known the answers were all wrong.
posted by zengargoyle at 5:59 PM on December 4, 2023 [1 favorite]
Then I went to the obvious answer of 3 based on the ratios of radii and quickly determined that it was a matter of frame of reference because from the global frame after the smaller circle had done one full rotation the point on the smaller circle in contact with the larger circle would not have touched the larger circle yet and therefore the smaller circle would have to rotate 3 plus a bit more.
Then I got bored and went to watch TV not wanting to bring out scratch paper. I was going to break out the trig and go into radians and such but as it works out.... it's for the problem at hand 3 plus 3 times 1/3 equals 4.
I might have got this at 12 years old or at least known the answers were all wrong.
posted by zengargoyle at 5:59 PM on December 4, 2023 [1 favorite]
Still not getting this, and the video is too long and intrusive for me to finish (please don't say "just watch it": that's equivalent to "just hit yourself in the face repeatedly with a shovel" for me).
So if I have meshing spur gears of 16 and 48 teeth. If I let them run in the simplest configuration, the small gear turns 3 times for every turn of the big gear [video, silent].
But if I have a fixed observer at the centre of the big gear, they will see the small cog rotating around the big one four times [video, silent].
So where do the extra 16 teeth come from when the big gear is fixed?
posted by scruss at 8:22 PM on December 4, 2023 [1 favorite]
So if I have meshing spur gears of 16 and 48 teeth. If I let them run in the simplest configuration, the small gear turns 3 times for every turn of the big gear [video, silent].
But if I have a fixed observer at the centre of the big gear, they will see the small cog rotating around the big one four times [video, silent].
So where do the extra 16 teeth come from when the big gear is fixed?
posted by scruss at 8:22 PM on December 4, 2023 [1 favorite]
Suppose you stick the two gears together with superglue, making them completely fixed relative to each other.
Then drag the small one around the outside of the big one, like in your second video, except now you need to let the big gear turn with it (because they are stuck together).
When you get back to the starting configuration, both gears have gone through one revolution, even though zero of their teeth have traveled through each other. This single revolution is due entirely to the circular shape of the small gear’s path.
Thus we have one extra revolution but zero “extra teeth.”
posted by mbrubeck at 8:54 PM on December 4, 2023 [1 favorite]
Then drag the small one around the outside of the big one, like in your second video, except now you need to let the big gear turn with it (because they are stuck together).
When you get back to the starting configuration, both gears have gone through one revolution, even though zero of their teeth have traveled through each other. This single revolution is due entirely to the circular shape of the small gear’s path.
Thus we have one extra revolution but zero “extra teeth.”
posted by mbrubeck at 8:54 PM on December 4, 2023 [1 favorite]
So where do the extra 16 teeth come from when the big gear is fixed?
That's a nice little paradox, scruss.
Consider meshing teeth laid out flat as opposed to teeth arranged on a circular gear.
Isn’t there just a little bit more room for an extra increment of pivot for each of the outer gear teeth as they travel around the fixed gear as opposed to the teeth laid out in a straight line? Because the teeth on the fixed gear each stick out a bit beyond all the others?
It’s kind of like pivoting around the vertices of a polygon as the wheel goes around the fixed gear such as dsword is talking about.
posted by jamjam at 8:58 PM on December 4, 2023
That's a nice little paradox, scruss.
Consider meshing teeth laid out flat as opposed to teeth arranged on a circular gear.
Isn’t there just a little bit more room for an extra increment of pivot for each of the outer gear teeth as they travel around the fixed gear as opposed to the teeth laid out in a straight line? Because the teeth on the fixed gear each stick out a bit beyond all the others?
It’s kind of like pivoting around the vertices of a polygon as the wheel goes around the fixed gear such as dsword is talking about.
posted by jamjam at 8:58 PM on December 4, 2023
Even if you take the limit as the number of teeth go to infinity and the gears become perfect circles, there is still an infinitesimal amount of pivot for each point on a circle that does not exist on a straight line.
posted by jamjam at 9:06 PM on December 4, 2023
posted by jamjam at 9:06 PM on December 4, 2023
“So where do the extra 16 teeth come from when the big gear is fixed?”
Straighten the circumference of the central circle out into a line. Roll the other circle upon it from one end to the other. In this example, that circle will rotate three times.
Pin that circle to where it now sits on the end of the line.
Now twist the line back into a circle (with the other circle on its exterior, pinned in place).
That pinned circle, when you do this, will rotate exactly one additional time, despite there being no additional movement along that length.
This is why your counting teeth thing doesn't work. The additional rotation is implicit in the circularity of the path, unrelated to any specific distance traveled upon it.
Also, to repeat: likewise, shrinking the central circle into a point, which by definition has zero "circumference", nevertheless results in the outer circle rotating exactly once.
Using two circles is a bit of misdirection in that it encourages you to utilize the familiar formula at hand. If, instead, it was just some wiggly line with some given length l, you'd likely just divide l by 2πr ... and then you'd probably immediately notice that the mere distance won't tell you the number of rotations because the path itself is "rotating". (Here you should stop and consider what the frame of reference we're interested in when we use the word "rotate".)
Think about how you might characterize a spiral of length l. It could be a loose spiral or a tight spiral, but in any case have a fixed length. To characterize that "tightness", you'll think about how the tangent changes along the curve. Because there's no limit on the tightness of a spiral, there's no upper limit on the rotations along such a spiral path, assuming a non-zero length.
tldr; the "extra" rotation arises as a result of the path itself rotating precisely 360 degrees.
posted by Ivan Fyodorovich at 10:26 PM on December 4, 2023 [2 favorites]
Straighten the circumference of the central circle out into a line. Roll the other circle upon it from one end to the other. In this example, that circle will rotate three times.
Pin that circle to where it now sits on the end of the line.
Now twist the line back into a circle (with the other circle on its exterior, pinned in place).
That pinned circle, when you do this, will rotate exactly one additional time, despite there being no additional movement along that length.
This is why your counting teeth thing doesn't work. The additional rotation is implicit in the circularity of the path, unrelated to any specific distance traveled upon it.
Also, to repeat: likewise, shrinking the central circle into a point, which by definition has zero "circumference", nevertheless results in the outer circle rotating exactly once.
Using two circles is a bit of misdirection in that it encourages you to utilize the familiar formula at hand. If, instead, it was just some wiggly line with some given length l, you'd likely just divide l by 2πr ... and then you'd probably immediately notice that the mere distance won't tell you the number of rotations because the path itself is "rotating". (Here you should stop and consider what the frame of reference we're interested in when we use the word "rotate".)
Think about how you might characterize a spiral of length l. It could be a loose spiral or a tight spiral, but in any case have a fixed length. To characterize that "tightness", you'll think about how the tangent changes along the curve. Because there's no limit on the tightness of a spiral, there's no upper limit on the rotations along such a spiral path, assuming a non-zero length.
tldr; the "extra" rotation arises as a result of the path itself rotating precisely 360 degrees.
posted by Ivan Fyodorovich at 10:26 PM on December 4, 2023 [2 favorites]
where do the extra 16 teeth come from when the big gear is fixed?
Well the big gear is fixed, so they have to come from the small one.
posted by flabdablet at 10:30 PM on December 4, 2023
Well the big gear is fixed, so they have to come from the small one.
posted by flabdablet at 10:30 PM on December 4, 2023
But if I have a fixed observer at the centre of the big gear, they will see the small cog rotating around the big one four times [video, silent].
When you stop the big gear, the little gear becomes a rotation vampire and sucks up the extra rotation that the big gear is no longer doing.
posted by clawsoon at 3:01 AM on December 5, 2023
When you stop the big gear, the little gear becomes a rotation vampire and sucks up the extra rotation that the big gear is no longer doing.
posted by clawsoon at 3:01 AM on December 5, 2023
At what speed and in which direction would the big gear have to turn for the small gear to circle the big gear once but still only rotate 3 times?
posted by clawsoon at 5:02 AM on December 5, 2023
posted by clawsoon at 5:02 AM on December 5, 2023
If a planet gear with P teeth is rolling all the way around a fixed sun gear with S teeth while staying in mesh with it, then the planet will perform S/P revolutions due to having rolled past S teeth, plus one more due to having completed an orbit of the sun, for a total of (S/P)+1 revolutions.
If we want the planet to revolve only S/P times instead, we can't magic away the revolution due to orbit, so we just have to arrange for the planet to roll past P fewer teeth per orbit instead. That will need the sun gear to be turning at +P/S revolutions per orbit. So, +1/3 revolutions per orbit for the S/P=3 case.
As a reality check, consider the case where P and S are equal, as in the rolling pennies version of this setup. In order for the outer penny to complete an orbit of the inner penny while revolving only once, the outer penny can't be allowed to roll along any of the inner penny's circumference; instead, the contact point can be welded and the inner penny made to revolve one whole time per outer penny orbit.
Taking that even further, consider what would need to happen for a planet gear to complete an orbit around the sun gear while not revolving at all. In that case we need the planet to roll backwards by P teeth on its way around, which it will do if the sun gear turns at (S+P)/S = (P/S)+1 revolutions per orbit.
This is just the formula for the number of revolutions per orbit that a planet gear makes around a fixed sun but with P and S swapped, which makes sense considering that it's now the planet gear that can't rotate.
Here's that exact setup put to practical use in order to avoid infringing the patent on the crank.
posted by flabdablet at 10:43 AM on December 5, 2023 [2 favorites]
If we want the planet to revolve only S/P times instead, we can't magic away the revolution due to orbit, so we just have to arrange for the planet to roll past P fewer teeth per orbit instead. That will need the sun gear to be turning at +P/S revolutions per orbit. So, +1/3 revolutions per orbit for the S/P=3 case.
As a reality check, consider the case where P and S are equal, as in the rolling pennies version of this setup. In order for the outer penny to complete an orbit of the inner penny while revolving only once, the outer penny can't be allowed to roll along any of the inner penny's circumference; instead, the contact point can be welded and the inner penny made to revolve one whole time per outer penny orbit.
Taking that even further, consider what would need to happen for a planet gear to complete an orbit around the sun gear while not revolving at all. In that case we need the planet to roll backwards by P teeth on its way around, which it will do if the sun gear turns at (S+P)/S = (P/S)+1 revolutions per orbit.
This is just the formula for the number of revolutions per orbit that a planet gear makes around a fixed sun but with P and S swapped, which makes sense considering that it's now the planet gear that can't rotate.
Here's that exact setup put to practical use in order to avoid infringing the patent on the crank.
posted by flabdablet at 10:43 AM on December 5, 2023 [2 favorites]
I find it interesting that this feels way less puzzling when thinking about polygons. Take two identical rectangular objects like playing cards or bar coasters. Place them flat on a table with the bottom of one against the top of the other, and rotate the top one about the other such that they always remain connected. It clearly takes 2 full rotations to go around once, even though the perimeters are identical and each side of the rotated card only touches the other once. Now take some scissors and start cutting the cards into identical hexagons and do the same... Same result. Now start rounding the corners, eventually creating identical circles. Why would it suddenly take 1 fewer rotations simply because you rounded the corners? (In fact, I would guess that your cards or coasters probably had rounded corners to begin with!)
If the SAT had posed a question like this without the correct answer, I'm sure far, far more people would have noticed. For some reason circles are unintuitive for almost everybody. (Myself included!)
posted by dsword at 12:48 PM on December 5, 2023 [3 favorites]
If the SAT had posed a question like this without the correct answer, I'm sure far, far more people would have noticed. For some reason circles are unintuitive for almost everybody. (Myself included!)
posted by dsword at 12:48 PM on December 5, 2023 [3 favorites]
If you count the number of times the small gear is oriented in the absolute starting position (ie the line from center to the reference point is pointing due east) you get N+1.
Yeah this is how it made sense to me too - using the test/video example, there's a difference between how many times the letter "A" in the smaller circle is pointing up vs how many times the tip of the letter "A" touches the edge of the B circle.
posted by creatrixtiara at 4:02 PM on December 5, 2023
Yeah this is how it made sense to me too - using the test/video example, there's a difference between how many times the letter "A" in the smaller circle is pointing up vs how many times the tip of the letter "A" touches the edge of the B circle.
posted by creatrixtiara at 4:02 PM on December 5, 2023
The discussion of sidereal vs solar days in the video is also very, um, confusing. (Read: completely wrong). Think about, say, the moon, which is tidally locked with the Earth. On the moon, "Earth days" are infinite and sidereal days are roughly equal to the lunar period. It goes through 1 rotation according to an external observer as it revolves around the Earth, not 2. Or consider satellites with retrograde rotational directions. Satellites do not roll without slipping and there is no definite relationship between rotations and revolutions. (The circumference of the Earth is nowhere near 1/365 its orbital path.)
posted by dsword at 7:05 PM on December 5, 2023
posted by dsword at 7:05 PM on December 5, 2023
Wait... I think I misunderstood what he was trying to say about the relationship.
posted by dsword at 7:15 PM on December 5, 2023
posted by dsword at 7:15 PM on December 5, 2023
Satellites do not roll without slipping and there is no definite relationship between rotations and revolutions.
The relationship that exists is the same as in all the other examples, though: The number of rotations of a satellite relative to the fixed stars differs by one from its number of rotations relative to the center of the earth, per revolution it makes around the earth.
posted by clawsoon at 7:16 PM on December 5, 2023 [2 favorites]
The relationship that exists is the same as in all the other examples, though: The number of rotations of a satellite relative to the fixed stars differs by one from its number of rotations relative to the center of the earth, per revolution it makes around the earth.
posted by clawsoon at 7:16 PM on December 5, 2023 [2 favorites]
So where do the extra 16 teeth come from when the big gear is fixed
Tooth fairy?
posted by dg at 7:47 PM on December 5, 2023 [4 favorites]
Tooth fairy?
posted by dg at 7:47 PM on December 5, 2023 [4 favorites]
Agreed about the relative number of rotations. What I found confusing is that the notion of sidereal vs solar days makes sense even for highly elliptical orbits, where the difference can vary significantly over the course of a year, and even parabolic/hyperbolic trajectories, where there's no revolution whatsoever, or even if you have no gravity at all and everything just follows straight paths... i.e. this doesn't strike me as useful for timekeeping at all, which is what the video seemed to imply.
posted by dsword at 7:52 PM on December 5, 2023
posted by dsword at 7:52 PM on December 5, 2023
This only just came up in my embarrassingly long Watch Later queue. It felt like a practical joke, especially the part where from the opposite frame of reference the wrong answer was correct, until Jungreis explained it.
posted by ob1quixote at 7:15 AM on December 6, 2023
posted by ob1quixote at 7:15 AM on December 6, 2023
« Older Happy International Day of Persons with... | It's forever, this time I know Newer »
This thread has been archived and is closed to new comments
A: 3/2
B: 3
C: 6
D: 9/2
E: 9
The figure shows two circles A and B, touching at a single point on their edge.
Honestly, I watched this the other day and its an interesting watch.
posted by biffa at 1:55 PM on December 3, 2023 [8 favorites]