Nope, this post is not about what's wrong with the Scholastic Aptitude Test, more commonly known as the SAT! It's about a specific problem that appeared on the SAT in 1982, in which everyone, including the test writers, answered the problem incorrectly. The question is shown below:
Okay, what is your answer? If you answered "B" you are in great company. If the radius of the smaller circle, Circle A, is 1/3 the radius of the larger circle, Circle B, then it only makes intuitive sense that Circle A will rotate 3 times as it moves around the circumference of Circle B. Unfortunately, that's not the right answer! The correct answer is that Circle A will rotate 4 times, which is not even an option listed in the question above - which is why no one answered it correctly, even if you knew the right answer! Apparently three individuals (out of around 300,000 total exam participants) wrote back to the College Board, the organization that conducts the SAT exam, pointing out the error. The question had to be thrown out, and the entire exam rescored that year!
Just to prove it to yourself, take a printout of the diagram and cut the diagram into 2 separate circles of different sizes. Place them like the question shows and carefully try to rotate Circle A around Circle B and count the number of revolutions it has done around it. It would be 4, not 3! Better yet, start with two ordinary quarters, as shown in the video here. Rotate one quarter around the circumference of the other quarter - but do so slowly, so that you can see how many rotations the quarter completes as it passes around the other one. As you will see for yourself, the quarter rotates twice as it passes around the other quarter, even thought both quarters are exactly the same size!
What's even crazier (again, watch the video here) is that if you convert the circumference of Circle A in the original problem to a straight line and then rotate Circle B along the straight line, it does so three times! In other words, there is something about rotating around a circle that changes the number of rotations. Lastly, try this next one. If you view the number of rotations of Circle A around Circle B from the perspective of Circle B (i.e. imagine you are standing on the surface of Circle B) or even vice versa, you will note that Circle A rotates just three times instead of four!
The mathematical proof of the answer to this problem, at least as I understand it, is that from start to end, the center of the moving coin (Circle A in the original problem) travels a circular path. The circumference of the stationary coin (Circle B in the original problem) and the path of the center of the moving coin (Circle A) form two concentric circles. The radius of the outer circle is the sum of the two coins' radii. In other words, the circumference of the path of the moving center of Circle A is equal to 2π multiplied by the radius of Circle A plus the radius of Circle B. Watch the video here again for a pictorial explanation of this same point (and see also a similar problem known as Aristotle's Wheel Paradox).
If you really want to be gobsmacked, consider how this coin rotation paradox applies to the number of days it takes for the Earth to rotate around the sun (with day, of course, defined as the rotation of the Earth). Again, when viewed from an external perspective (i.e. that of a distant observer in outer space), it looks like it actually takes 366.25 days for the Earth to rotate around the sun, instead of what we consider a calendar year as 365.25 days (the former length of time is called a sidereal year).
My point here is to use mathematics to demonstrate one of the defining characteristics of a High Reliability Organization (HRO), the principle of "Reluctance to Simplify". Leaders in High Reliability Organizations know that the simplest explanation is not always the correct one. They take the next necessary step to dig deeper into the problem, in order to come up with the right solution (not the wrong one).
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