SAT Redo

These days I don’t get much free time to check out the math-sci youtubes, but every so often Veritasium or some other offers a click bait title I cannot resist. There was this on on the infamous SAT coin rolling problem. “The SAT Question Everyone Got Wrong”

That’s intriguing because you know there’s a deeper story behind it than merely “it was just a hard problem”.

SAT Administration

The issue that prompted this blog today is the almost throw-away remark Veritasium made near the end on the inability of the SAT to re-score all the tests, citing it’d cost too much money that “came out of the pockets of the test takers.”

I thought:

1. Why would they charge people for a re-score, if the question was known to be bogus? It’s an ethics issue, not a finance issue, and SAT board got the ethics wrong.
2. The cost is in real terms – the examiners could be doing something better, or could they? It is still a public good to be fixing badly scored tests.
3. The monetary cost is then irrelevant, the government just marks up the bank accounts of the examiners who have to re-score all the tests. Fair compensation for these workers, presumably they were not the ones writing the bad question.

Hopefully someone at Veritasium takes note of my comment:

@15:25 SAT officials made another huge error of far more overriding importance in fact an epic tragedy. The cost of redoing the scoring does not “come out of the pockets of test takers”. That’s a macroeconomic myth. There is in fact no such thing as “tax payer funding”. No one can pay a dime in tax until the issuer of the currency first issues it, or licences chartered banks as agents of the state to issue credit (which is not net currency creation, has to be paid back with interest). Tax payers cannot (legally) fund the government in monetary terms, that’d mean they’re engaging in counterfeit. Tax payers and others “fund” the government in real terms by exchanging their goods or labour in return to that which is accepted for payment of taxes, fees, levies and other imposed liabilities (aka. US dollars, or yen, pounds, euro, peso, etc. the numeraire for whatever country imposes these liabilities by force of the state).
      Only the US government can issue USD by fiat, and they tell you what it’s worth, and the “cost” is a vote in Congress for the authorization, the operation is simply taking a computer at the central bank and typing in numbers to mark up the bank accounts of the employees re-marking the SAT test. That’s a real cost in terms of those employees could be doing something else more useful perhaps, but it is not a monetary cost for any tax payer, in fact the issued wage for the government employees gets spent (presumably) thus helping someone else pay their taxes.
      The reason why this error is an epic tragedy is that it applies not just to lack of willingness to re-score SAT tests, but to the entire macroeconomy. The false thinking about “tax payer funding” is a cause of neoliberal austerity policy and massive wastes of human lives in unemployment — forever lost output — exceeding the costs of all major wars in human history.

OK, that the macro.

Sounds like SAT is begin phased out. Probably good riddance. Entrance qualifications for university programs do need to be in place, limited teaching resources and all that, but with MIT, Standford, Harvard and other universities freely able to publish lectures and course materials online, no paper printing cost and whatnot, do we really need to restrict university placements? Perhaps we do? After all, chatGPT is showing there is still a role for teachers, it requires honest human work to assess students for certifications if they’re able to plagiarize chatGPT.

If universities are going to continue as our main source of higher education and qualification, then standard need to be met. But information technology is constantly lowering the real costs involved, and universities should be lowering the cost of education, not increasing it! This is a public funding issue.

Also I’m talking about inflation adjusted costs, so the real costs.

If the real resources we once had are still available they have to be realistically in total a lot cheaper now, so university fees should be practically zero. The real cost to enter a university and do all the formalities of accreditation should be the intellectual cost. You have to prove you are worthy.

On that score, I advocate giving students all the time they need. There is no need for a rush through college. If they fail a course once, provided they wish to continue and have the entrance requirements, let them keep trying.

The Coin Puzzle

Roll a circle of radius $R/3$ around a bigger circle of radius $R$. How many times will the smaller circle rotate around it’s center?

Apparently a 1980’s SAT test did not offer the correct answer in the multichoice.

Most students would say it rotates $3$ times around, since the smaller coin circumference being $2\pi R/3$ unrolls $3$ times get once around the larger coin of circumference $2\pi R$.

However, this is not the case.

While the story about the circumferences is correct, it does not dictate the full rotation of the small coin around it’s center, that’s because in addition to unrolling it’s circumference around the bigger coin, the smaller coin is also “in orbit” (so-to-speak) around the larger coin, and that orbit accounts for one extra rotation.

Hence the total number of rotations of the small coin is $3+1 = 4$.

Generalizations

The cool thing in the Veritasium episode is the generalization to rolling a circle around any shape. Triangle, square, pentagon, or even a line. Scientific American had an article going into the issue for celestial mechanics and why astronomers need to use sidereal time, not solar time.

For only one of these will the small circle rotate around it’s center the naive expected perimeter divided by small circle circumference number of times. Which one?

Answer: The line of course. (No additional orbit.)

What about rolling to coin inside another shape?

In this case the inner “orbit” is subtractive, you have to subtract one circumference. So the result is the ratio of the shape’s perimeter to the small coin circumference minus $1$.

A simple proof was offered by a mathematician who was one of those students from 1982.

Proof using uniform speed

Parameterize the path smoothly in time, so a constant speed around. Say it takes $T=2\pi R$ seconds.

The center of the small coin moves around the perimeter $p_c = 2\pi(R+R/n)$ at a uniform speed for it’s circular orbit, which is, $$ v_c = \frac{p_c}{T} = \frac{2\pi (n+1)R/n}{2\pi R} = \frac{n+1}{n}. $$

Count the number of revolutions of this small coin by counting how fast the contact point must be moving. Call this speed $v_p$ for point-of-contact. The number of revolutions will be the angular velocity $v_p/(R/n)$ divided by $2\pi$ times the time taken, $T$. N_\text{revs} = \frac{\omega 2\pi R}{2\pi} = v_p n $$ However, we know what $v_p$ must be if there is no slipping of the coin as it rolls around the perimeter, it must be exactly the same as the speed of the coin center. Why is that so? It is simply because the coin is rigid. Imagine shrinking the coin radius to $0$, the centre would then be the contact point. It has to be moving at the same speed. Another way to see the same result is to calculate it directly.

Thus $v_p = v_c$, hence, $$ N_\text{revs} = \frac{n+1}{n} n = n+1. $$

To prove the result for the case of rolling the coin around the inside of a shape, just start with the centre perimeter being $p_c = R - R/n$.