My sister told me this physics problem, and I have no idea how to solve it. It goes like this. There are two identical balls, one lying a distance x from the edge of a frictionless table, and the other being held up in the air, a distance x from the edge of the table. (See diagram above.) They are connected by a massless rope.

At time t=0, the ball hanging in the air is dropped.

Do you see what’s going to happen? That ball will start to fall down and approach the table, while the ball sitting on the frictionless table will be pulled toward the edge.

Two questions, both of which I can’t answer:

(1) Which will reach the vertical plane of the edge of the table first? (In other words, will the ball on the frictionless table fall off the table before the falling ball hits the side of the table? Or vice versa? See my diagram below to see what the estimated path of the falling ball will look like. It curves towards the table, but also the distance from the corner of the table to the falling ball will increase as more time passes.)

(2) What is the equation for the path of the falling ball? (Again, see diagram below.)

It seems I was scooped.

In my ongoing obsession with the law and math (see post I and post II here), I learned that for many years, all the supreme court justices shake hands before each meeting.

The “Conference handshake” has been a tradition since the days of Chief Justice Melville W. Fuller in the late 19th century. When the Justices assemble to go on the Bench each day and at the beginning of the private conferences at which they discuss decisions, each Justice shakes hands with each of the other eight. Chief Justice Fuller instituted the practice as a reminder that difference of opinion on the Court did not preclude overall harmony of purpose. (here)

It still goes on today. There are 9 justices. A natural question: how many handshakes?

The “handshake problem” is a common problem in math classes. It’s equivalent to the problem of calculating the number of diagonals of a polygon. (Each vertex is a person; each line — including the edges of the polygon itself — represents a handshake.) It’s equivalent to adding the numbers .

I was thinking it would be a good hook for a class where this problem was presented. Better than:

Assume we have n people at a dinner party who all don’t know each other. They all introduce themselves to (yawn) each other… (yawn)… and shake hands. How many (yawn) handshakes are there?

In the same sense that that’s not a true-to-life problem, neither is the supreme court handshake problem. But it’s a hook.

Of course, I looked it up to see if anyone else had thought of it (likely) and guess what? As I said: I was scooped.

I was so enthralled with Judge Sand’s city of Yonkers case that I couldn’t help but do a little more research on him. And lo and behold, nary a page in from a google search, I found another case that deals with math!

The setup

Henry Barschall, working for the American Physical Society and the American Institute of Physics, investigated the cost of science journal subscriptions for libraries. Even then in 1986 (as we hear is still the case now), science journals were increasing their prices — for a variety of reasons. And science journals were on average more expensive than journals in other disciplines (e.g. in 1985, the average yearly subscription for a science journal was $328, while for philosophy, it was only $47 — here).

The survey was done and ready to be published in Physics Today. Barschall outlined his methodology here. The intial results which were published in 1986 are below:

Note that for the physics journals, the American Physical Society and the American Institute of Physics were the most cost effective (0.7 cents/1000 characters), while Gordon and Breach were the least cost effective (31 cents/1000 characters). For this case, it’s important to note who sponsored the study! (The same organizations who came out the “winners.”)

An expanded survey, published in 1988, took into account not only the price, but the “impact” (or importance) of the journal… The metric for the impact was the average number of papers which cited an article in the journal two years after its publication. The more “impact” the journal had, according to the study, the more it mitigated a higher cost (hence: a ratio was created).

Again, APS and AIP were near the top of the list, while Gordon and Breach were way down at the bottom, with a cost/impact ratio more than twice the next publisher.

Gordon and Breach sued the APS and AIP. Judge Leonard Sand heard the case, and summarized it as follows:

This action, a dispute between publishers of scientific journals, raises an issue of first impression: whether a non-profit publisher may be sued for false advertising under the Lanham Act for publishing comparative surveys of scientific journals which, through the employment of a misleading rating system, rate its own publications as superior.

[…]

These articles, G&B alleges, constituted the start of a “continuous promotional campaign” waged by AIP and APS against them with the aid of Barschall’s survey results. First, G&B alleges, AIP and APS distributed “preprints” of the 1988 survey results to librarians (the primary purchasers of scientific journals) at a conference in June 1988. Since that time, G&B alleges, defendants have continued to disseminate the results of Barschall’s surveys to prospective purchasers through press releases, letters to the editor, “electronic mailings,” and meetings with librarians (“secondary uses” of the articles). G&B states that its attempts to reach an accommodation with defendants have been fruitless, and that it has been forced to resort to the courts. After filing a series of legal actions in Europe claiming unfair competition, and failing to receive satisfaction in Swiss, German, and French courts, G&B brought this lawsuit in September 1993.

G&B contends that the articles are promotional materials cloaked in the deceptive guise of “neutral” academic inquiry, and thus constitute misleading advertising under the Lanham Act, 15 U.S.C. § 1125(a), and comparable provisions of New York General Business Law. Barschall’s studies, G&B contends, far from being neutral, in fact constitute a “cynical promotional campaign” by AIP and APS to deceive librarians and other consumers of scientific journals into thinking that their journals have been “scientifically” proven to be of superior value.

Judge Sand’s judicial opinions, the trial transcript (really interesting stuff! I swear!), Barchall’s papers, and other related material have been put on this site at Stanford for you to read.

Eventually in 1997 (yes, justice takes a long time to render), Judge Sand made his final ruling:

Barschall’s methodology has been demonstrated to establish reliably precisely the proposition for which defendants cited it — that defendants’ physics journals, as measured by cost per character and by cost per character divided by impact factor, are substantially more cost-effective than those published by plaintiffs. Plaintiffs have proved only the unremarkable proposition that a librarian would be ill-advised to rely on Barschall’s study to the exclusion of all other considerations in making purchasing decisions. This consideration in no way makes Barschall’s study or defendants’ descriptions thereof false, and accordingly judgment is granted to defendants.

And the case is over!

Could this be useful in a math class?

I don’t know. I honestly can’t see myself using it next year, except maybe for a quick classroom discussion — a few “do nows” where each day I introduce more and more of the case and the evidence, and we talk about the math issues it raises…

1. Before presenting the data, it could lead to a discussion of how one might measure the cost-effectiveness of journals, and different ways students could imagine calculating them.
2. Does Barschall’s method of measuring impact make sense? Does taking a ratio of cost per thousand characters to impact work? Why divide? (That will be hard for my students to get.)
3. A baby project: students could go to the school library and get data about various online databases that the library subscribes to, and try to determine whether subscribing is cost-effective for the school. Or at least, the class could try to brainstorm out what sorts of data we would want (e.g. the cost of the database annually, whether — and if so, how quickly — the cost of the database has been increasing, the number of times the databases are accessed each day, the amount of information the databases contain, etc.)But honestly, this seems like it isn’t really worth it. It would make for an interesting independent study, though.
4. A discussion about whether the survey was ethical or not, merely by virtue of the fact that those producing it were players in the game (non-profit players, but players none-the-less). [A similar discussion would be whether drug companies should produce their own studies of drug effectiveness.]
5. Ideas for how one might present the data visually?
6. Should a court (even with scientific experts being called in) be the final arbiter in whether a mathematical study/survey was sound or not? If not, who should be?

THE PROBLEM IS HERE!

In the Quach family, there is an age-old tradition that has been intact since the 1500s. In this tradition, every member of the Quach family receives one lottery ticket for his/her first 25 birthdays.

Based on empirical data collected over the years, it was determined by the family Mathematician that the probability of a Quach family member winning the lottery at least once during that 25 year stretch is 211/243.

Assuming the above statement is true, what is the probability of a Quach family member winning the lottery at least once during any 5 year interval? (Assume that lottery wins are completely independent of each other).

So the probability that a member of the Quach family does not win the lottery in 25 years is 32/243. Since winning each year is independent of any other year, we know:

P(not winning any year)=P(not winning 1st year)*P(not winning 2nd year)*…*P(not winning 25th year)

And since the probability of not winning each year is the same, we get:

P(not winning any year)=P(not winning in nth year)^25
32/243=P(not winning in nth year)^25
P(not winning in nth year)=(32/243)^(1/25)

Now we need to find the probability of winning at least once in a five year interval, which is actually 1-P(not winning in 5 year interval).

P(not winning in 5 year interval)=P(not winning in 1st year)*P(not winning in 2nd year)*…*P(not winning in 5th year)

P(not winning in 5 year interval)=P(not winning in nth year)^5

P(not winning in 5 year interval)=(32/243)^(1/5)=2/3

Since we want to find P(winning at least once in 5 year interval), we know that that will have to be 1/3.

I want me some of them odds.