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.