Avoiding Sum Cubes

One of the most interesting and challenging open problems in combinatorics on words is to decide whether there exists an infinite word over a finite subset of N, the non-negative integers, with the property that it contains no two consecutive blocks of the same length and the same sum (a "sum square").

For example, 01231301020103102310313231301020131013230 is a word of length 41 with this property, but if you append any one of {0,1,2,3} to it, it no longer does. Appending 0 gives the sum square 00; appending 1 gives the sum square (3231301020)(1310132301); appending 2 gives the sum square (103132313010)(201310132302); appending 3 gives the sum square (132)(303). So this word cannot be extended to an infinite word avoiding sum squares; the longest such is of length 50. Of course, there could be an infinite word avoiding sum squares over some other subset of N; no one currently knows.

This problem was originally stated by Pirillo and Varricchio in 1994, and independently by Halbeisen and Hungerbühler in 2000.

Today we posted a preprint in the arxiv that solves a related unsolved problem. Instead of avoiding sum squares, we show that we can avoid sum cubes: three consecutive blocks of the same length and same sum.

The construction is actually quite simple: the infinite word in question is the fixed point of the morphism
0 → 03
1 → 43
3 → 1
4 → 01,
and can be obtained by repeatedly applying this morphism starting with 0. Here are the first 50 terms:
03143011034343031011011031430343430343430314301103 .
I found this morphism several years ago.

However, proving that this word has the desired property is not simple. The proof was recently achieved by Luke Schaeffer, using ideas of James Currie at the University of Winnipeg and Julien Cassaigne at the Institute de Mathématiques at Luminy in France.

More Lousy Reporting from the Record

Yet another crappy article in my local paper, the Waterloo Region Record. This time it's about the Waterloo Region school board's decision to end the practice of distributing Gideon bibles in school.

From the headline ("Bible ban in schools ignores its influence") to the content, the article is misleading and inaccurate. Nothing was "banned" by the Board's decision. Students are free to bring bibles to school, and bibles aren't being removed from libraries. The only thing that was changed by the decision is that an explicitly evangelical organization will no longer be allowed special dispensation from a public school board to distribute its sacred text to a captive audience of 5th graders, a right granted to no other organization and no other religion.

The author's article, Liz Monteiro, didn't interview a single person in favor of the decision. False claims by Cindy Watson, the school board trustee who voted in favor of continuing the practice of distributing bibles, that it is "not proselytizing", were allowed to go unchallenged. David Seljak, who is usually sensible, is quoted as saying "To eliminate the study of religion from our curriculum is an exercise on mythmaking that borders on propaganda." Only problem? Nothing about "eliminat[ing] the study of religion" was at issue in the School Board's vote.

Andrew Mills, a youth pastor, is allowed to make the remarkable claim that "It's a false dichotomy to think faith is opposed to learning." Well, let's go to Wilmot Centre Missionary Church and see how many books on evolutionary biology (not creationist books) are in their library. Of course faith is opposed to learning. By its very definition, faith leads to beliefs that cannot be questioned and cannot be swayed by evidence.

All in all, more lousy reporting from the Record.

More Egnorance

Just as I suspected, Egnor's blog provides even more hilarity.

Here he claims:

'Separation of church and state' is not in the Constitution and is not Constitutional Law.

Well, the words "separation of church and state" are not in the US Constitution, but neither are the words "right to a fair trial". Yet I doubt Egnor would make the same claim about the right to a fair trial.

It is a clear falsehood to imply that the concept of separation of church and state is not in the Constitution. Of course it is - right there in the First Amendment - and no reputable lawyer claims otherwise. It is just plain weird how someone can proclaim his religion with such pride, and yet violate its tenets so casually.

I really wonder why becoming unhinged about evolution means you also become unhinged about global warming, separation of church and state, etc.

Waterloo Region School Board Ends Gideon Distribution

For years, the Gideons have distributed bibles to grade 5 students in local schools here in Waterloo. No other religious group was afforded this access, a practice which is clearly discriminatory.

Back in November, the issue came up in the Waterloo Region School Board, and the board then voted to continue the practice.

Kudos to Ted Martin and Kathleen Woodcock, who were the only members of the School Board sensible enough to vote no.

But then - horror of horrors - it was suggested that groups other than Christians be allowed to distribute their religious propaganda, too.

Suddenly the Board had a change of heart, and voted 8-3 to end the practice.

Here are the members who voted to continue it:

* Cindy Watson
* Harold Paisley
* Colin Harrington

Shame on them.

Mary is the Ideal Christian?

Oh, look: the brilliant brain surgeon Michael Egnor has a blog, which is called (I kid you not), "Egnorance".

This is destined to be an endless fountain of unintended amusement.

Already we have the renowned Dr. Egnor claiming that Mary is "the original Christian disciple, and a model and a mother for all of us".

Let's see: Mary

* had affair with some guy not her husband
* got pregnant by him
* lied about it
* convinces gullible husband that it was actually some god who raped her

Yes, I'd say that she certainly is a good role model for theists.

T-Shirt Idea

Wikipedia says, "To understand recursion, you must first understand recursion." ThinkGeek even sells a t-shirt with that slogan on it.

But I think a much better version of this would be:

"To understand recursion, you must first understand recursion - 1".

Based on this, I submitted the following idea to ThinkGeek:



Who knows, maybe they'll make it into a t-shirt.

Yet Another Boring Attack on Materialism

Here.

Kastrup's observation is trite and unoriginal.

Darwin himself remarked, "The horrid doubt always arises whether the convictions of man's mind, which has developed from the mind of the lower animals, are of any value or at all trustworthy. Would anyone trust the conviction of a monkey's mind, if there are any convictions in such a mind?"

And, of course, we know from extensive research that our perceptions are often wrong, and wrong in predictable ways.

But none of this provides the "profound contradiction" Kastrup thinks he has discovered. Yes, there are problems in perception and cognition, but the map between the real world and our mental model can't be wildly wrong. If it were, organisms with a better model would have outcompeted us. And the scientific method itself provides self-checking through replicability and peer review.

The bottom line: our own perceptions, and those of the instruments we devise, are all we have. So we use them, and in doing so, we try to be on our guard for mistakes in reason and perception caused by our own biology.

Doug Groothuis endorses Bachmann

Let's see: she's

- a pathological liar
- dumber than a bag of nails
- a nasty bigot

Despite those wonderful qualities, Michele Bachmann is still the ideal Presidential candidate in the opinion of Doug Groothuis.

(He doesn't allow any comments on that post - no wonder.)

Phillies Games, Old and New

We attended two games in Pittsburgh this past weekend. Pittsburgh is 5 hours away from where I live, but it's still the closest National League city to us. Since son #2 is a big Phillies fan, it's a reasonable choice.

We weren't alone. I'd guess that, conservatively, about a third of the people attending the games were Phillies fans. Red shirts were everywhere.

When I was a kid, nobody wore t-shirts or jerseys with the names of their favorite players. Now everybody does. And when I was a kid, everybody bought and waved pennants with the name of the team on them. Now nobody does.

I still remember my first game really clearly. It was on July 9 1967 at Connie Mack Stadium in Philadelphia. We got there early, sitting in our great front-row seats that cost only $3.25 each. During "pepper" before the game, a ball bounced near the little railing, and my father reached down and picked it up! I had my first real official major league baseball, and I never got another ball again since then.

Back in '67 the Phillies had some good players (Johnny Callison, Richie Allen, Cookie Rojas, Jim Bunning) but overall they were pretty mediocre. But on July 9, against the Cardinals, they produced practically the best first game a ten-year-old could see. In the 8th inning, Richie Allen hit one of the longest home runs ever at Connie Mack Stadium to tie the game. And then, in the bottom of the 10th, Tony Gonzalez pinch-hit the winning home run. Perfect!

But back to 2011. PNC Park is a really nice place to watch a game. Even the cheap seats were pretty good, and there is a spectacular view of the Pittsburgh skyline in the outfield.

We got there early for both games. On Friday we hung out in left field and watched batting practice. Ryan Madson, who has a bit of a reputation for being a jerk, confirmed it for us when he deliberately teased fans by pretending to throw them a ball, and then throwing it on the grass instead and laughing.

I'm glad to say that not all the players are like that. On Sunday we got there early to watch batting practice, and hung out by the right field line. There were a lot of pitchers shagging fly balls, including J. C. Romero, Kyle Kendrick, José Contreras, Danys Báez, and Cliff Lee. Danys Báez came over and tossed son #2 a ball, and then later J. C. Romero signed it.



But then Romero did something really classy: there was another kid with a Cliff Lee jersey that he wanted Lee to sign, but Lee was standing about 60 feet away. So Romero took the jersey over to Lee, had Lee sign it, and then returned it to the kid. Now there's a player who knows how to treat fans.

The Phillies lost the first game, but won the second. We got to see Chase Utley make a terrific play in the 7th to save the game, and a great double play by Wilson Valdez. We also got to see Halladay slide at home to score a run, something he apparently hasn't done since high school.

All in all, a fun time in Pittsburgh. Son #2 thanks J. C. Romero!

The Information - by James Gleick

I'm currently reading The Information: A History, A Theory, A Flood by James Gleick (famed for being the author of Chaos). It's not bad at all; in fact, it's pretty good. For the moment, I'll be content to make the following observation:

My colleagues Ming Li and Bin Ma (Ming is the author of An Introduction to Kolmogorov Complexity and Its Applications) get a nice mention on page 320, as does Charlie Bennett's theory of logical depth.

Intelligent design advocates will gnash their teeth to see that their hated Richard Dawkins gets ten full pages, and his book The Selfish Gene is described as "brilliant and transformative" -- which, of course, it was.

They'll also be surprised to see that their own "Isaac Newton of information theory" doesn't get a single mention. Not a word.

This all goes to show that Gleick actually knows something about the subject and is not fooled by the bleatings of the religious.

My Talk for the Wilf Conference

Laurier (the other university in Waterloo, Ontario) is hosting a conference this May 26-29 in honor of Herb Wilf's 80th birthday. Wilf (b. June 13 1931) is a very influential mathematician, known for his work on combinatorics and combinatorial algorithms. He also founded the Electronic Journal of Combinatorics.

Of course, it's better in person, but if you can't make my talk at 10:40 AM on May 27 2011 in Room N 1001, Faculty of Science Building, WLU, then you can see the slides here.

I DIdn't Buy This in the Video Store

Because at the price of $9.97, it was about $20 more than it's worth.

I love the part about learning about the Grand Canyon from creationist geologists who are "the scientists who know it best".

What's next, a video where you can learn information theory from Bill Dembski? Or journalism from Denyse O'Leary?

Catenaries on "Car Talk"

Here's a puzzle from the radio show, Car Talk, with the misspelling corrected:

"There are two telephone poles. Each one is 100-feet tall. They are parallel and an unknown distance apart.

"We're going to attach a 150-foot rope from the very top of one of the poles to the top of the other. This rope will, of course, droop down somewhat. That drooping rope is called a catenary, from the Latin word for chain.

"The question is: What must be the distance between the two poles, so that the lowest point of the catenary is 25-feet above the ground?"

I confess, I worked out the equation for a catenary to try to answer this, but that's a waste of time if you think about the problem a little.

The answer, if you need it, is here.

Craig: If God Kills Kids, It's OK

Fundamentalists say the darndest things!

But they're not always pretty. Here we have the truly appalling spectacle of William Lane Craig justifying genocide. It's OK, he says, if God does it.

And - believe it or not - Craig is actually a respected Christian philosopher. Doesn't it make you wonder about what one would have to do to lose respect?

Craig is fond of syllogisms, so here's one just for him:

1. All sane beings agree that genocide is wrong.

2. The Christian god thinks genocide is just peachy.

3. Therefore...

Can Irrational Numbers be Represented in a Computer?

I often see the following claim, or variants of it:

"In a computer (as we understand computers, at least), irrational numbers can't be fully accurately represented."

This one happens to be from Bradley Monton, Seeking God in Science: An Atheist Defends Intelligent Design, Broadview Press, 2009, p. 128, but Monton isn't alone.

This is a common misunderstanding that, I think, has two bases. First, for people without much mathematical training, a real number is inextricably linked with its base-10 (or perhaps base-2) representation. For them, the number π is 3.141592653 ..., as opposed to being defined by an integral or infinite series. Second, people without computational training often don't understand that some infinite objects -- including some base-10 or base-2 representations of irrational real numbers -- are routinely represented as finite objects in computers.

Let's take √2 as an example. Although it is irrational, symbolic algebra systems such as Maple and Mathematica routinely represent √2 "exactly" and allow manipulations with it. For example, if you type

> x := sqrt(2);
> x^2;

into Maple, it will happily return the "exact" answer 2. And, similarly, other arithmetic operations involving √2 will give the "exact" answers:

> expand((123+45*sqrt(2))^3);

3355317 + 2224665 √2;

Similarly, we can manipulate other famous irrational numbers such as e, π, etc, and often get "exact" results. The point is that many of the irrational numbers that people actually care about need not be stored in terms of their base-10 or base-2 representation.

But even if we insist that yes, they must be stored in this way, there can still be finite representations. For example, consider the base-10 number defined by having a "1" in the i'th digit if i is a perfect square (like 1,4,9,16, etc.), and 0 otherwise. Now it is easy to see that this number is irrational. But we can still store it in a finite way (for example, as a function that returns the i'th digit on input i), and do at least some simple manipulations on this representation. For numbers whose base-k representation is encoded by a finite automaton, for example, we can do addition, even if the carries come from "infinitely far" to the right. (This is not trivial.)

In fact, any algebraic number, and many others, can be stored as a finite program that on input i returns the i'th digit. In some cases we can even do this by a very simple recurrence. For example, Graham and Pollak, in a 1970 paper, gave a simple explicit recursion to compute the i'th bit of √2 .

The misguided claim I started with can be fixed. If instead we say, "The base-2 representation of most irrational real numbers is not compressible to a finite representation, so the entire base-2 representation of most irrational real numbers cannot be stored in a finite computer", we would be correct. Here "most" means "all but a set of measure zero". But this is not particularly interesting, since for actual computation we are rarely interested in "most" irrational numbers -- we are interested in the computable ones.

Review of Monton's "Seeking God in Science"

Bradley Monton is a philosophy professor at UC Boulder and a self-proclaimed atheist. He's written a little book (147 pages for the main text, not counting the preface, endnotes, index, etc.) entitled Seeking God in Science: An Atheist Defends Intelligent Design, which I finally had a chance to read. It consists of four chapters:

1. What is intelligent design, and why might an atheist believe in it?
   
2. Why it is legitimate to treat intelligent design as science
   
3. Some somewhat plausible intelligent design arguments
   
4. Should intelligent design be taught in school?
   

I'm afraid this book is not very good. Monton comes off as rather naive (displaying little understanding of the abundant and documented dishonesty in the ID movement) and ignorant of science, the history of intelligent design creationism, and its role in the creationism-evolution wars.

The first chapter of the book is devoted to one of my least favorite philosophical games: trying to create a definition for a concept that covers all possible cases, by starting with a definition and iteratively refining it. He spends 25 pages (pages 16-40) playing this game with the concept of "intelligent design" itself, in a tedious and unenlightening way (for example, he even addresses the possibility that God is biologically related to humans!) and here is what he comes up with (italics in original):

"The theory of intelligent design holds that certain global features of the universe provide evidence for the existence of an intelligent cause, or that certain biologically innate features of living things provide evidence for the doctrine that the features are the result of the intentional actions of an intelligent cause which is not biologically related to the living things, and provide evidence against the doctrine that the features are the result of an undirected process such as natural selection."

Now I don't particularly like this game (although it has a long history -- philosophers have enjoyed applying it to "chair", for example), because for almost any definition proposed it is easy to come up with some outlandish counterexample. Still, as a mathematician, I enjoy and admire precision, so perhaps it's not a game completely without value. But after reading his definition I could only mutter, All that work! - and he still has an imprecise and unusable mess.

Unusable, since key terms like "intelligent cause" and "undirected process" are not defined or made rigorous. Could it be, as other commentators have already observed, that when we try to define "intelligent cause" we discover that natural selection itself could be considered intelligent by our criteria? Could it be that intelligence is a continuous measure, not a discrete quality, so that speaking of an "intelligent cause" is essentially meaningless unless the amount of intelligence is quantified?

Mess, because by calling intelligent design a "theory", Monton begs the question.

Imprecise, because this definition doesn't cover much of what the intelligent design advocates themselves discuss. For example, in Dembski's book No Free Lunch, he spends a good 10 pages discussing the case of Nicholas Caputo, an election official accused of rigging elections. Dembski implies that his intelligent design methodology can help resolve the case of whether Caputo cheated. But this case has nothing to do with a "global feature" of the universe or a "biologically innate" feature of living things.

Monton seems rather naive about the intelligent design movement. For example, on page 12, he claims, "As a matter of public policy, the Discovery Institute opposes any effort to require the teaching of intelligent design by school districts or state boards of education." But this claim could only be made by someone who doesn't understand (a) that the Discovery Institute has a long history of dissembling and (b) that intelligent design, as practiced by its leading proponents (Behe, Dembski, Meyer) is largely a negative program of casting doubt on the theory of evolution, or examining its supposed deficiencies. Therefore, Discovery Institute programs like "Teach the Controversy" and "Critical Analysis of Evolution" are, in fact, just covers for getting intelligent design into the classroom. This is abundantly clear to most people who have studied the intelligent design movement in any depth.

Part of the book is devoted to analyzing the views of pro-science philosophers, such as Taner Edis, Massimo Pigliucci, and Robert Pennock. Needless to say, Monton thinks they have it wrong in many ways; they are "sloppy" and "confused". But much of his criticism seems misplaced. For example, he gives the following advice to Barbara Forrest: she "focuses too much on attacking the proponents of intelligent design for the supposed cultural beliefs they have, instead of attacking the arguments for intelligent design that the proponents of intelligent design give". But Forrest has never said that intelligent design advocates are wrong because of their cultural beliefs; rather, she has fearlessly and tirelessly explored the goals and strategies of intelligent designers, as well as the sociological and political connections between intelligent design creationism and the religious right. Monton is apparently unconcerned with these details, and that's his right. But then his criticism amounts to "I don't share your interests", and that's rather pathetic.

Monton is a fan of Laudan, citing the following passage approvingly: "If we would stand up and be counted on the side of reason, we ought to drop terms like "pseudo-science" and "unscientific" from our vocabulary; they are just hollow phrases which do only emotive work for us." I strongly disagree. As I mentioned already, almost any definition or classification is subject to exceptions, but it's still useful to be able to say something is a chair or not a chair, even if we cannot always agree about the boundaries. Science, as a social process, has a number of characteristics, and it is perfectly legitimate and useful to point out that creationism and its modern variant, intelligent design, fail to share many of these characteristics.

There are signs that although he thinks intelligent design merits a book-length treatment, Monton hasn't really grappled with the issues. For example, on page 17, he cites a beehive as an example of a feature of the universe that "indisputably exist[s] as a result of an intelligent cause" and then, in a footnote, says that "It was surprising to me that some readers objected to this line of thought, saying that ... bees ... aren't intelligent." Well, I'd guess that this surprise comes largely from the fact that Monton hasn't really thought deeply about what intelligence is. We now know a lot about algorithms and naturally-occurring tools to perform computational processes, but Monton doesn't seem to know anything about it. But then he has some real misconceptions about mathematics and computing, claiming that computers can't represent irrational numbers. (I've addressed this misconception here.)

Several times in the book, Monton refers to the Newtonian account of physics and argues it has been "refuted". On page 50, he uses it to argue that false scientific theories can still count as science (so if intelligent design has been refuted, it could still be considered science). On page 152, he uses it to argue that false scientific theories are routinely taught in high school science (so intelligent design, even if false, could still be caught). But this black-white classification of theories as either "false" or "not false" doesn't even come close to capturing the status of Newtonian physics. Yes, it doesn't give the right answers for particles moving at high velocity, for example. But I can't think of a single scientific theory that unfailingly predicts the outcome of every single experiment. It is more correct, I think, to view theories and equations as our models of reality and to have a good idea of their shortcomings and applicability. No one uses special relativity to solve simple problems in kinematics; they use Newton and they don't apologize for it. If we classify theories purely as "true" or "false" then we lose the nuance that some "false" theories are pretty damn good and others are worthless.

Chapter 3 summarizes some of the arguments of intelligent design advocates, such as alleged "fine-tuning", the origin of the universe, the origin of life, irreducible complexity, and the simulation argument. There is not really much analysis that is new here, but I found his discussion of the simulation argument the most interesting part of the book.

Chapter 4 addresses the question of whether intelligent design should be taught in school. By "taught in school", Monton means "taught in public high-school science classes" (although he takes two whole pages to explain this - an example of how clunky the writing is). One of the objections Monton addresses is "we wouldn't be teaching a real controversy", and he answers this by citing Michael Behe as an example of a real scientist who disagrees with the scientific consensus. Ergo, there is a real controversy. But if Monton's definition of "real controversy" is "one scientist disagrees" or even "a handful of scientists disagree", then there is a "real controversy" about relativity, heliocentrism, and the germ theory of disease. Indeed, it would be hard to come up with a scientific theory for which there is no controversy in Monton's sense. Monton's position is absurd. There are controversies, and then there are controversies; it's not a black-and-white term. The "controversy" over evolution is exactly like that over relativity: a very small number of experts in the field, and a larger number of cranks, disagree with current consensus. That doesn't mean their objections merit coverage in science class. I'm not opposed to teaching controversies, but let's teach some real ones.

Finally, I'd say that the book, and Monton himself, seems curiously disengaged from the extensive mainstream criticism of intelligent design. To give one illustration, he doesn't cite much of the literature arguing against the claims of intelligent design advocates. Nowhere will you find any mention of, for example, the fine article of Pallen and Matzke (published in 2006 in Nature Reviews Microbiology) -- although other articles of Matzke are cited -- or the article of Wilkins and Elsberry (published in 2001 in Biology and Philosophy). He lists two conferences where he's presented his work, and both of them were hosted by the "Society of Christian Philosophers". Four people are listed as endorsers on the back of his book, and three of them are non-biologist critics of evolution (Berlinski, Dembski, Groothuis). And Monton has a blog, but he doesn't allow any comments on it. I can't help but think Monton's book would have been much better if he had made more attempts to be engaged with those who disagree with him.

More Muddled Thinking about the Brain

If you can stand it, read this article by Raymond Tallis, entitled "What Neuroscience Cannot Tell Us About Ourselves".

There are a lot of confident assertions, all presented without any real evidence:

• "It is unlikely that the gap between neuroscientific stories of human behavior and the standard humanistic or common-sense narratives will be closed"
  
  
• "But there is nothing in the activity of the visual cortex, consisting of nerve impulses that are no more than material events in a material object, which could make that activity be about the things that you see."
  
  
• "neural activity is not about anything and so can be neither correct nor mistaken"
  
  
• "A consistent materialism should not allow for the possibility of memory, of the sense of the past"
  

And Tallis seems to have no understanding of what "information" is.

But most of the article seems to be gobbledygook. If Tallis were to try to come up with a rigorous definition of words like "about", he might make some progress. By his argument, it makes no sense to count tree rings to determine a tree's age, since tree rings are not "about" the age of the tree.

As Arthur C. Clarke noted, "When a distinguished but elderly scientist states that something is possible, he is almost certainly right; when he states that something is impossible, he is probably wrong."

Psychic Fail

A photo from New Orleans.

If I were a real psychic, I wouldn't pay for "24 hr. surveillance", as the sign at the left indicates. Instead, I'd just send security guards around at the time I knew people were about to break in.

Neil Postman - Perpetually Clueless

Yesterday I attended a baseball game in Buffalo, NY: the Bisons versus the Lehigh Valley Iron Pigs. There was, as usual, a big video scoreboard in centerfield.

So this clueless quote from Neil Postman, recently posted on Doug Groothuis's blog, is appropriate:

"Media may now be serving as a surrogate for reality, and a preferred one at that. At stadiums throughout the country, huge TV screens have been installed so that spectators can experience the game through TV because TV is better than being there, even when you are there."

Postman has got to be one of the most overrated media commentators ever.

As I commented to my young son, in many ways, watching baseball at a stadium is a much better experience today than it was when I attended my first game in 1967.

It's not that "TV is better than being there" or that it is "a surrogate for reality", as Postman claims - whatever that means.

It's that a big screen offers more information than the scoreboard of 1967: you get the batter's average and other statistics; you get a reminder of what happened earlier in the game when that batter was up; and you get instant replays of interesting plays you might not have fully appreciated or understood the first time around.

Postman never really understood media; his observations are generally self-important, trite, and ignorant.

"Be it Ever So Humble" - a short story by Louise Shallit

Here's a short story entitled "Be it Ever So Humble" that appeared in Story magazine, Vol. 27, No. 115 (September-October 1945) by my mother, Louise Lee Outlaw. I think this may have been the first short story she published in a national magazine, although she had been writing for newspapers since 1938. Although it doesn't display her at the height of her powers, there are some very nice lines.

As an interesting footnote, another story in the same issue was written by Joseph Heller, who, of course, later became famous as the author of Catch-22.