Monday, December 27, 2010

A couple of puzzles

Since we just had a geometric puzzle, let's go numerical this time:

1. A probability puzzle

By the time the movie started, there were 91 men and 121 women in the theater. If the first moviegoer to enter the theater was a man wearing glasses, and the second was a woman not wearing glasses, then what is the probability that equal percentages of men and women in the theater were wearing glasses?

Solution here.

2. Fun(?) with fractions

I saw a survey in the newspaper once in which 33% of the people surveyed said "no" to something, while 67% said "yes." Whenever I see something like that, I always think, "Huh - I wonder if they only asked three people."

Well, suppose you saw it reported that 34% of people surveyed answered "no" to something, while 66% said "yes." What is the smallest number of people who could have participated in this survey? Assume that the people reporting these results have rounded their figures to the nearest whole percentage point.

Solution here.


I'll end with a graph that shows what the solution to puzzle #2 would be for any possible pair of percentages adding to 100%. Don't study the graph too carefully if you are going to solve the puzzle yourself!

Tuesday, December 21, 2010

Merry Christmas! (Solution to the Optimization Game)

December 21st! Well, it's winter, and once again America turns its collective thoughts to the Winter Biathlon.

Hah - just kidding.

But I confess I was thinking of the winter biathlon the other day, the reason being that it's an optimization problem of sorts. Winter biathlon is the sport in which an athlete's score depends on speed (cross-country skiing) as well as accuracy (rifle marksmanship). So, just as in other optimization problems, you won't do your best by maximizing over each variable separately. If you ski too fast, your heart rate will spike and your marksmanship will suffer; but skiing too slowly, or taking too much time with your aim, will hurt your skiing time as well. An optimum approach balances competing factors.

The same is true for the optimization game I posted a while back. Recall that in order to play the game, you choose four distinct points on the unit circle. Your score is the area of your quadrilateral, plus the area of the largest triangle that may be formed by deleting one of your points.

Supposing for a moment that you wanted to maximize the area of the quadrilateral, then you would choose four points to form a square, for a score of 3. But this would mean settling for a fairly small triangle (area 1). Alternatively, if you wanted to maximize the area of the triangle, then you would choose three of the points to form an equilateral triangle; but this would mean settling for a fairly small quadrilateral. So, just as in the winter biathlon, the optimum approach requires that we sacrifice a little of the quadrilateral score and a little of the triangle score for the good of the sum.

My own intuitive solution to the area puzzle was as follows. Begin with a square, oriented as a diamond (with points at the four cardinal points North, South, East and West). Given four points in this configuration, the score is 3. Now imagine grabbing hold of the points at East and West, and nudging them both slightly northward. Will the score increase or decrease?

* Because the original square was a maximal-area quadrilateral, when you nudge the two points northward, the area of the quadrilateral will not change, to first order. (We say that the area is "stationary.")

* Meanwhile, the area of the triangle based at the south pole will increase to first order, because the altitude of the triangle will increase to first order, while the base remains unchanged to first order. (The tangent to the circle is vertical at the East and West cardinal points.)

* So altogether, when we nudge the East and West points slightly northward, our score increases, to first order. Hence, this nudging is a good way to improve on the square configuration.

* Of course, if we nudge the points too far toward the north pole, then our score will suffer, because ultimately the score approaches zero as the points reach the north pole. Thus, there is a local optimum configuration, in the shape of a kite, that improves upon the square configuration.

A little geometry, together with some first-semester calculus, suffices to find the optimal shape in this family:

This shape scores about 3.1488, or to be exact

Anyway, this is as much thinking as I did before posting the puzzle. I was confident that the kite was best, but I didn't want to prove it because I hoped it would be at least theoretically possible for someone to beat my score. But I'm afraid that the intuitive solution presented above turns out to be the best. A sketch for a workmanlike proof follows.

In the meantime, Merry Christmas all! Best wishes for a healthy and happy 2011.


Consider any four distinct points on the unit circle. Label the points P,A,B,C as follows: first choose a point so that the remaining three points form a triangle of maximal area in the configuration; label the chosen point B. Then label the points adjacent to B by A and C in such a way that A,B,C are traversed counter-clockwise around the circle. Label the remaining point P. If necessary, rotate the configuration so that P has coordinates (1,0) - here shown at the south pole - understanding the unit circle to be x^2 + y^2 = 1. Then points A,B,C have coordinates given respectively by (cos a, sin a), (cos b, sin b), (cos c, sin c), where 0 < a < b < c < 2 pi:

By construction, triangle PAC is a triangle of maximal area in the configuration, so the score for the configuration can be expressed as twice the area of triangle PAC plus the area of triangle CBA. Using vector cross products, the score can be expressed in terms of a,b,c as

2S = 2 sin(a) - 2 sin(c) - sin(a-c) + sin(c-b) + sin(b-a).

At this point, it is a relatively straightforward exercise in third-semester calculus to show that the optimal configuration, unique up to rigid motions of the circle, is the kite we arrived at by intuition.

Monday, December 13, 2010

I am for you, and you are for me, not only for our own sake, but for others’ sakes

I came across an article in Slate today written by an old acquaintance of mine, the constitutional scholar Kenji Yoshino. The topic of Yoshino's piece is a newly published scholarly paper entitled "What Is Marriage?" which argues that the state need not, and indeed should not, recognize same-sex marriage. You can find the paper here. In his piece for Slate, Yoshino argues that the authors' position actually does more to cheapen the idea of marriage than to protect it. This made me curious to read the paper itself. (Warning: there is frank language in what follows.)

The paper begins by arguing that
some sexual relationships are instances of a distinctive kind of relationship - call it real marriage - that has its own value and structure, whether the state recognizes it or not, and is not changed by laws based on a false conception of it.
The authors then set out to discover what this real marriage is. The major premise is this:
As many people acknowledge, marriage involves: first, a comprehensive union of spouses; second, a special link to children; and third, norms of permanence, monogamy, and exclusivity.
This sounds reasonable, at least to me. But I started scratching my head when the authors began to develop these principles. Here is the implication they draw from the principle that marriage necessarily involves a comprehensive union of spouses:
Because our bodies are truly aspects of us as persons, any union of two people that did not involve organic bodily union would not be comprehensive—it would leave out an important part of each person’s being.
I think what they are trying to say is what Whitman said when he wrote "Yet all were lacking, if sex were lacking." One can agree with this, it seems to me, and still deny that anything has been proved in this part of the paper. The authors have merely clarified their own favored meaning for the term "comprehensive." Others might consider a union "comprehensive" if it involves profound and lasting feelings of love and trust. (The authors consider such people "revisionists.")

The authors' point is also confusing because it fails to attend to time. It cannot be uncommon for married couples in their fifties, or even in their forties, to all but set aside their 'organic bodily unionizing.' We still consider them really married. Perhaps the conclusion the authors wanted to draw was that "any union of two people that did not, at some point in the union's history, occasionally involve organic bodily union, would not be comprehensive." (Or is real marriage a time-dependent concept, like some kind of indicator light on the headboard that illuminates when you're having sex?)

The authors continue:
This necessity of bodily union can be seen most clearly by imagining the alternatives. Suppose that Michael and Michelle build their relationship not on sexual exclusivity, but on tennis exclusivity. They pledge to play tennis with each other, and only with each other, until death do them part. Are they thereby married? No. Substitute for tennis any nonsexual activity at all, and they still aren’t married: Sexual exclusivity—exclusivity with respect to a specific kind of bodily union—is required.
I find this confusing, because the authors were supposed to be talking about comprehensiveness (the first principle), yet they've helped themselves to exclusivity (the third principle). And they also seem to be confusing necessity with sufficiency. What I mean by that is, exclusivity with respect to a specific kind of bodily union is required...OK, say for a moment that we agree with that. But does exclusivity with respect to a specific kind of bodily union suffice? Presumably not - there is that second principle yet to attend to, the one about the "special link to children." Yet if exclusivity with respect to a specific kind of bodily union is not sufficient, then why are we denigrating tennis as not being sufficient? Perhaps we should be asking whether tennis is necessary to be married? Tennis with the kids, maybe? By now I'm confused enough to believe anything.

Then we get to the good parts. Assuming I have parsed all of the euphemisms correctly, I think the authors next argue that 'organic bodily union' only occurs when a woman accepts a man's penis into her vagina. ("organic bodily unity is achieved when a man and woman coordinate to perform an act of the kind that causes conception.") Interesting. Let's recap the argument then:

1. A real marriage requires a penis to enter a vagina. (At least once? On date nights? The authors are unclear.)

2. Mathematically then, it follows that a real marriage requires an odd number of penises and an odd number of vaginas.

3. But in a typical same-sex relationship, there are an even number of penises or an even number of vaginas.

4. Therefore, same-sex relationships cannot be real marriages. QED

The paper goes on for quite a while, and frankly I got tired of reading it. But it seems as if the main argument is really the one about penises entering vaginas. Thinking about penises entering vaginas, say the authors, helps us to make sense of real marriage as a harmonious complex of organic bodily unions, special links to children, and norms of permanence and exclusivity.

OK - if you say so. Or not. Again, I'm not sure what has been proved in this paper. It seems to be a great, big, superheated version of a bumper sticker I saw once: "God Made Adam and Eve, Not Adam and Steve!" Anyway, I'm reminded of another work of philosophy I read not long ago: Frankfurt's On Bullshit.

Tuesday, November 30, 2010

Science Reporting and Evidence-Based Journalism

I tore an article out of The American Scholar about a year ago and crumpled it into my pocket to think about later. The pages turned up again this weekend when my wife was sorting through one of the rat’s nests of loose papers I maintain around the house. The article is an extract of a speech by a Washington Post reporter named David Brown. The speech is available here:

Brown talks about ways in which science journalism could be a lot better than it is today. He argues that science journalism, done better, could be a model for improving journalism as a whole. The implication is that our democracy could certainly use some better journalism.

You'd probably be using your time more wisely if you were to read the speech instead of reading this blog post, but I'll carry on for a moment under the assumption that you have time for both.

According to Brown, the most important thing science journalism has that makes it a good model for journalism as a whole is evidence. When you read a story, says Brown, "[N]otice how much space is devoted to describing the evidence for what is purportedly new in this news, and how much is devoted to someone telling you what to think about it. Ask yourself whether there is enough information in the story to permit you to reach your own opinion about its newsworthiness. I think you’ll be surprised. If there isn’t enough information to give you, the reader, a fighting chance to decide for yourself whether something is important, then somebody isn’t doing his or her job."

I decided to apply this test to a short piece of science reporting, which I reproduce here in its entirety:

Lithium, Water, and Suicide

Lithium is a staple prescription for bipolar depression and suicidal tendencies. But it is also a naturally occurring element, with traces found in most of the world's drinking water, and that raises a question: Do levels of lithium in tap water correlate with suicide rates? A recent study by a team of Japanese doctors says yes.

Published May 1 in the British Journal of Psychiatry, the study reported that in 18 municipalities in southwest Japan, towns with relatively low levels of lithium saw higher suicide rates than towns with relatively high levels. The lithium content ranged from 0.7 to 59 micrograms, much lower than the 200 to 400 milligrams usually prescribed to bipolar patients (and much, much lower than the toxicity threshold). Nevertheless, the researchers speculated that even very low levels of lithium can have a cumulative, prophylactic effect on mood swings that might induce suicidal thoughts, completely separate from the effect large doses have on mood disorders. The implication, says lead researcher Takeshi Terao in response to an e-mail inquiry, is that "adding lithium to drinking water may be useful to prevent suicide."

But even Terao admits that further study is needed. Among other concerns, not enough is known about the long-term effects of even low levels of lithium, according to Dr. Allan H. Young, the director of the Institute of Mental Health at the University of British Columbia. Still, given the immense social costs of suicide, the team in Japan concluded in a follow-up paper that adding lithium to drinking water offers "an easy, cheap and substantial strategy for worldwide suicide prevention."
Although this article is better than most in that it includes several pieces of numerical information, it still leaves us pretty hungry for the one piece of information we need: namely, how big was the observed effect?

Reporters might not always realize that publication in a prestigious journal is no guarantee that the study in question matters. Sometimes the study only proves a theoretical point. Sure, maybe the study offers convincing evidence that eating more X will raise your risk of getting Y. But what if it only raises the risk from one in a million to one in a thousand? This is hardly newsworthy. Yet it might very well get published in the newspapers anyway. The headline? "Eating X Makes You A Thousand Times More Likely to Get Y!"

Brown makes this same point when he says,
Science stories, and especially medical stories, have a really good shot of getting out on Page 1. They are inherently interesting and they appeal to what might be termed, somewhat cynically, as the narcissism of the reader. But that often isn’t enough to get them on the front page. To get there, the story must emphasize novelty, potency, and certainty in a way that, as a general rule, rarely exists in a piece of scientific research. That truth is why so many medical stories only mention the [relative] magnitude of change that occurs with a new diagnostic test or treatment, and not the absolute change it brings about.

To be fair, reporters are not entirely to blame. They don't manufacture all of the buzz. The research university itself, ever conscious of building its brand, might even be suggesting the overblown headline in its own press release. More fundamentally, a scientific journal article is itself a rarified form of reporting, and the scientist's job security depends no less than the journalist's upon having something sensational to report. The scientist certainly has no incentive to downplay his own results - not in the research writeup itself, nor later when the reporter calls for a quote.

In any event, by withholding the evidence about the magnitude of the effect observed, "Lithium, Water, and Suicide" fails Brown's basic test of science journalism. It doesn't give us the information we need to decide whether the research in question matters or not. The closest the reporter gets to the notion of effect size or importance is the final "kicker" that adding lithium to drinking water offers "an easy, cheap and substantial strategy for worldwide suicide prevention." But that quote by itself isn't evidence.

I liked Brown's speech, but I think that giving a speech about how science journalism should be better is not actually going to make it any better. Or anyway, so I thought when I read "Lithium, Water, and Suicide." That's because it was in the same issue of The American Scholar in which Brown's speech itself appeared.


In case you're curious to find out more about the lithium research, the article is available online. Also, in a later issue of the journal, there is some interesting discussion of the shortcomings of the study. The authors do not dispute the shortcomings, rightly pointing out that the study was meant to be suggestive, and that a lot of work remains to be done before the link between suicide and lithium in tap water can be taken as established. (Note in particular that lithium intake from food is not negligible in comparison with lithium intake from tap water.) And even if the link were established as a matter of basic science, a host of other questions arise when we consider adding lithium to tap water as a public health strategy.

I wanted to find out the rationale for the statement that lithium offers an "easy, cheap and substantial strategy for worldwide suicide prevention," so I did a little digging. If you Google this phrase, you'll see what a buzz the lithium story created a year ago. It even made it onto the 9th annual New York Times "Year in Ideas" list. What you won't find so easily is the follow-up scientific paper that contains this phrase. But I finally tracked it down and paid $31.50 for a copy. The citation is Terao et al., "Even very low but sustained lithium intake can prevent suicide in the general population?" Medical Hypotheses 73 (2009), 811-812.

The striking thing about this paper is that it does little better than the American Scholar article did in explaining the absolute magnitude of the effect in question. Yes, it quotes the slope of the regression line in the original research. But the raw data for that regression were massaged in various ways, and the y-axis of the regression model is a confusing measure called "suicide standardized mortality ratio." So we are never actually told how many suicides might be prevented each year based on a given treatment model. We never find out what that model would cost to implement. We never find out what the public health benefit would be in terms of years of life saved or lost wages regained. Even order of magnitude estimates would have helped. But as it is, if we want to decide for ourselves whether the results matter, then there's almost nothing to go on. And after all, if the scientists themselves aren't providing the necessary information, then how can we expect the reporters to give it to us?

Saturday, September 18, 2010

In My Ample Spare Time

These two animations might bring back a few memories for my former students....

     Click for first animation

     Click for second animation

     Click for third animation

(Higher-resolution .mov files available on request.)

Sunday, August 8, 2010

Rare photos from the 1930s and '40s

Some news outlets are publicizing a rare collection of color photographs from the Depression-era Farm Security Administration (think Dorothea Lange). A breathtaking series of photographs is here. Below, a few that I grabbed quickly.

Friday, July 30, 2010

An optimization game

Here's a game I thought of the other day while driving in my car:

Choose four distinct points on a circle of radius 1 in such a way as to maximize your score. Your score is the area of your quadrilateral plus the area of the largest triangle that may be formed by deleting one of your points.

As a warm-up, you might show that if you choose the points to form a square, then your score is 3.

My high score is shown below. Can you beat it?

(Numerically, this is between 3.14880129427 and 3.14880129428.)

Monday, January 18, 2010

The Fortune-Tellers

I asked a fortune-teller my future. In answer she said: "The next fortune-teller you ask will mislead you."

I asked a second fortune-teller my future. She answered: "The next fortune-teller you ask will tell you true."

A third answered me this: "The next fortune-teller you ask will lead you astray."

The fourth would not read my future, but said: "The first fortune-teller you sought answered false."

Wednesday, January 13, 2010

Ginger Beef

They say supermarkets are a good place to meet people, and I've certainly found that to be true. When people see you trundling down the aisle with two little redheads in tow, they're liable to drop their rutabagas and march right over. Shopping takes a little longer than it used to, but obviously I love it. Why shouldn't everybody dote on them like their daddy does?

But I wish I had a nickel for every time somebody has said, "Did you know redheads are going extinct?" Then at least I'd have enough money to do what I'm about to do, which is to offer a $100 Reward for the first person who sends me a reputable scholarly publication that draws such a conclusion. I don't think any such research exists.
Mail contest entries to Offer void where prohibited; I will be the final authority as to the meaning of the terms "reputable", "scholarly", "publication", and "draws such a conclusion."
According to How Stuff Works, the coming extinction of redheads is a myth that goes around the web periodically. But that hasn't stopped everybody from quoting myth as fact. (See here for example.)


The other thing people often tell me is that redheads feel pain more acutely than other people do. This is more than a myth. "Increased Sensitivity to Thermal Pain and Reduced Subcutaneous Lidocaine Efficacy in Redheads" (Liem et al., Anesthesiology, 102(3), 509-514, March 2005) was a study of 30 female redheads and 30 brunettes. The redheads reported pain from cold temperatures at about 43 degrees Fahrenheit, whereas brunettes reported pain at a significantly (and significantly) lower temperature, about 32 degrees Fahrenheit. (Some data here.)

Interesting results, although my own elder daughter seems pretty impervious to the cold.

And I'd say the little one seems pretty robust too.
Subject reported feelings of euphoria upon application of the stimulus.