Daily Archives: November 7, 2006

Election Day Celebration: Christian, Jew, and Muslim

This is the post that got me to thinking of binary search, the subject of several recent posts.

I went to the polls early today and while there I met a family acquaintance. Her daughter and my youngest daughter Jen are close friends. She was there as a election-day montior, to make sure the election was fair. And as I left I realized I had witnessed an improbable event, one that should make all of us rush to vote every chance we can, so we can enjoy that special privilege enjoyed by too few people in the world, and one for which hundreds and hundreds of thousands of our fellow citizens have died to maintain our right to exercise that privilege.

Here are some questions.

Q: Do you live in a country that has elections?

A: Yes. I live in the U.S.A.

Reality: most people don’t live in countries that have elections.

Q: Are the elections fair? [1]

A: Yes, they are. At least for the most part. They are fair by world standards, though the fairness of the 2000 Presidential election is still a matter of dispute.

Reality: Even if people live in countries that have elections, few of those elections are fair. Witness Iran.

Q: How do you know they are fair?

A: We have a good record. Moreover, we allow everyday citizens to serve as election monitors, to observe that the election laws are obeyed.

Reality: Few countries have such a good record as ours, 2000 notwithstanding.

Q: Did you know any of the election workers?

A: Yes. One of them is the mother of a good friend of my daughter.

Reality: Most people wouldn’t expect to know one of the election workers, though I live in a relatively small town and I often recognize the people who volunteer to help on election day. For example, when I voted in the primary a couple of months ago, one of the workers was an O’Donnell, brother to the twins whose birthday that occurred today; I mentioned them in one of today’s posts.

Election workers are paid for the work, but the pay is nominal. They are really volunteers, and I thank them for their service.

This ends the twenty-question, binary search part. So let me go on to describe some details that I think apply to only a handful of countries in the world. We have the great good fortune to live in one of them.

Q: When did that mother’s daughter become a friend of my daughter?

A: During high school. They had many similar interests, and a friendship developed.

Q: Are they the same religion?

A: No. My daughter is Jewish. Her friend is Muslim.

Q: What kind of Jew is your daughter?

A: Reform.

Q: What kind of Muslim is your daughter’s friend?

A: I don’t know. I know she is very observant. She even started wearing a white garment that covers most of her body several years ago. I asked her mother the name today, since she was wearing a similar garment. She said it is called an “abaya.”

Q: And they are friends?

A: Very much so. They keep in touch regularly.

Q: What does your daughter want to do?

A: She is a Yale graduate. She majored in Spanish. She is now a graduate student at Temple University in Philadelphia. She wants to be a bilingual Speech Pathologist, in English and Spanish.

Q: What does your daughter’s friend want to do?

A: She wants to be a teacher. She wants to teach English. She went to SUNY Stony Brook. She just finished teaching for a year in Brooklyn. She’s now married, and is with her husband at Penn State, where he is going for his doctorate in Statistics. She’s even written an article about her year teaching. It’s just been published in the “English Journal.” [2]

Q: Let me get this right. Your daughter is a Jew. Your daughter’s friend is Muslim. They have been close friends for several years. Both daughters want to be educators, and one of them wants to get a job as a high-school English teacher, and moreover expects to be able to wear an abaya in the classroom? Are you nuts?

A: No. I’m just very happy I live in the U.S.A. Where else could this happen? That’s also why I take such great pleasure in exercising my right to vote.

Binary search, decimal search. Who cares? What a great and wonderful country we live in.

Just think about it: A young woman who is a devout Muslim has as her heart’s desire to become a high-school teacher of English, to educate our future citizens about the joys of their language. And she has the confidence that she can find a school district that will let her do his while wearing an abaya.

By the way, when I mentioned this post was coming to my daughter Jen, and that the title would be “Muslim and Jew,” she reminded me that one of her roommates also wants to become a teacher, although at the university level, and in history. She is Catholic, hence the three religions that are named in the title of this post.

Three wonderful young women, friends to each other. All want to become educators. Each has a different religion.

Thank God I live in the U.S.A.

Notes:

1. Re fair elections, in one of his books Tom Friedman relates the following joke, I think it was about the former Syrian dictator Assad (The dead Assad. His son is also the current dictator of Syria, but he is still alive.). It’s about an election:

Sycophant: Beloved dictator. The election results are in. Everyone in the country voted for you except five people! Imagine that.

Dictator: Arrest those five people and shoot them.

2. You can find a link to the paper at National Council of Teachers of English. English Journal, Volume 96, Number 2, November 2006. The paper is titled “Speaking My Mind: Student Teaching at Ground Zero: One Muslim Woman’s Challenge” and was written by my daughter’s good friend, Zareen Niazi Atiyat. I tried to order a copy only to learn it was not available. I’m hoping that’s because many people have ordered a copy. If that’s not the case someone should provide the funding to make copies widely available. It’s a wonderful story.

On programming: Bugs

In the previous post I said 6 yes/no questions would suffice to identify one of the 52 cards in a deck of cards, and outlined an answer based on the deck having four suits, each with 13 cards.

But I was assuming you were holding a “standard” set of playing cards. My first question should have been, “Is this a pinochle deck?”

A pinochle deck has 52 cards in four suits. But there are duplicate cards. Only the 9, 10, jack, queen, king and ace are used. That gives four suits with 6 kinds, or ranks, of cards in each. It takes 2 questions for the suit, 3 for the rank, for a total of 5 questions.

In discussing the card-identification problem we have moved from mathematics to the world of programming. The process I described above for identifying the cards is called an algorithm. To program is largely to design and implement algorithms.

And as noted there is a fault in the algorithm I gave. There’s even a special word used by programmers to describe this: bug.

The origin of the term dates to Thomas Alva Edison, the famous inventor. The cited Wikipedia article includes the following quote from Edison:

Software bug

It has been just so in all of my inventions. The first step is an intuition, and comes with a burst, then difficulties arise—this thing gives out and [it is] then that “Bugs”—as such little faults and difficulties are called—show themselves and months of intense watching, study and labor are requisite before commercial success or failure is certainly reached.

I first learned of the assocation between Mr. Edison and “bug” back in the early 1970’s when the Supplement to the Oxford English Dictionary came out. I just tried to find the definition, only to learn you have to pay the OED folks $300/year for an individual account. However, I committed most of the reference to memory when I first read it, as it is such a charming story. As best as I can recall the first written use of “bug” in the sense we are using it here was in the Pall-Mall Gazette:

Mr. Edson, I was informed, was up late the previous night looking for a “bug” in his phonograph, an expression for a difficulty implying that some insect has secreted itself inside and is the cause of the difficulty.

Bugs are the bane of programmers. The good news for WordPress bloggers is that I haven’t run across many in my months of using WordPress, though I think I found one recently. Actually it’s more a design flaw than a bug. I posed a puzzler about it a few posts back. I’m still hoping the WordPress folks take a crack at guessing what it is.

More on binary search

When I first moved to NYC to start my graduate studies I lived in an apartment on the first floor of a building on the North side of West 95th between Broadway and Amsterdam.

A few months into my studies, during afternoon tea on the 13th floor of Warren Weaver Hall, home of the Courant Institute, I fell into a conversation with someone who worked there on the staff. [1] The conversation was as follows:

Me: Hi. … Where do you live?

He: I live on the West Side?

Me: Me too. What street?

He: West 95th.

Me: Me too. Which block?

He: Between Broadway and Amsterdam.

Me: Which building?

He: The one across from the parking garages.

Me: Me too. Which floor?

He: First.

Me: Which apartment?

He: 1A.

Me: I’m in 1D.

I was thankful the exchange ended there. I was starting to think he might be a room-mate I had never met. We had never met because we had different schedules. He worked regular hours, while most classes were held in the evening since so many students worked during the day and took classes part-time. We had different schedules.

The above exchange is an instance of the same technique used in binary searching.

Binary search is also the basis of the parlor game of twenty questions. Someone names something and you are given up to twenty yes/no questions to identify it. So you are exploring a space with 2 * 2 * … * 2 (2 times itself twenty times) or which is 2**12 (a**b stands or a multiplied by itself b times) * 2**8, or 4096 * 256, or about 1,000,000 possibilities.

Here is another example of binary search.

Question: You have a deck of playing cards. You tell me you thinking of one of them. How many yes/no questions does it take to identify which card you’re thinking of?

Answer: There are four suits, each with 13 cards. It takes two questions to find the suit, and 4 questions to find the card (3 questions suffice for 8 while 4 are needed for 16, and 13 is bigger than 8). So it takes 2*4, or at most 6 questions to identify the card.

Well not exactly. It’s a puzzle. Can you see the problem? Think about it before reading the next post, though I know because of the “future is now” effect that you most likely read the answer before you read the question.

Notes:

1. Let’s play Jeopardy, where I give you the answer and you give me the question:

Answer: Mathematics or Physics. Neither department would have the gall to omit an integer. Truth trumps superstition.

Question: Two college departments each have their own building, each with 13 stories. In one building the top floor is #14 while in the other it is #13. Which department is in the second building?

On mathematics: Binary search

As a programmer I live in a binary world. I sometimes count with two fingers, not ten, and to me the powers of 2 – 2, 4, 6, 8, 16, … – come into play often than the powers of ten – 10, 100, 1000 …

Actually there is a flaw in the above. I need just one finger to count in base two. With our ten fingers we can actually express 11 numbers since raising two closed fists denotes zero. (Whether this will require reprinting all our paper currency remains an open question.)

A powerful application of the power of two is the idea of “binary search.”

Suppose you tell me you are thinking of a number between 1 and 1000 and you want to know the smallest number of yes/no questions I can use to tell you that number. Let’s assume you know some mathematics and so are thinkiing of 314, the first few digits of “pi,” which is among other things the ratio of the area of a circle to its diameter. As the mathematically-trained programmer than I am, I ask the following series of questions: [1]

Me: Is it bigger than or equal to 500?

You: No.

(Now I know it’s less than or equal to 500, and so have eliminated half the possibilities.)

Me: Is it bigger than or equal to 250?

You: Yes

(Similarly, half of the remaining possibilities have been eliminated.)

Me: Is it bigger than or equal to 378? (378 = 250+128)

You: No.

Me: Is it bigger than or equal to 314? (320 = 250+64)

(I’ve just mentioned the number though I don’t yet know it’s the answer.)

You: Yes.

Me: Is it bigger than or equal to 432? (432 = 314 + 128)

You: No.

Me: Is it bigger than or equal to 378? (378 = 314 + 64)

You: No.

Me: Is it bigger than or equal to 346? (346 = 314 + 32)

You: Yes.

Me: Is it bigger than or equal to 318? (314+4)

You: No.

Me: Is it bigger than or equal to 316? (314+2)

You: No.

(Now I know it is 314 or 315.)

Me: Is it 314?

You: Yes.

The net of this is that I find a number using this method with a logarithmic number of questions. In case you don’t remember — or never learned about — logarithms, a short example should suffice. Consider the first four powers of ten:10, 100, 100, 10000. 100 is 10*10, or ten multiplied by ten twice, 1000 is ten multiplied by ten three times, and so on. The logarithms are defined by the number of multiplications and are thus: 1, 2, 3, 4. These are logarithms in base 10.

Logarithms also exist in a programmer’s binary world. The first few powers of two are: 2, 4, 8, 16. The corresponding logarithms, in base 2 and not base 10, are the same: 1, 2, 3, 4.

Before we steam on, since I’m probably giving you more mathematics than you expected, mathematics can also be a source of humor. I heard the following riddle from one of my favorite high-school teachers. His name was Goodsell Slocum and he taught mathematics. Here’s the riddle:

Question: Two squirrels are trapped in a cabin. The cabin is built with logs as is every piece of furniture in it. You come back the next day to find four squirrels. How is that possible?

Answer: They used the log tables to multiply!

To understand this you need to know that you can multiply two numbers by adding their logarithms. For example, 100*100, is 10000, because the log of 100 is 2 and so the log of the result is 4, which gives you 10000.

This is also the idea behind slide rules, but unless you are of a certain age you don’t even know what a slide rule is, so I won’t say about slide rules now.

Speaking about slide rules reminds me of another humorous story.

While driving daughter Jen to school a few years back I noticed she had one of those graphing calculators. She school had given her the use of one for one of her courses. I know it cost only about $50. I said that 30 years ago I had bought one of the first pocket calulators. It was made by Texas Instruments. I said I it only did the four basic functions — add, subtract, multiply and divide — and that it had cost me about $80, but it was still better than an adding machine. She said to me, “Dad, what’s an adding machine?” I said to myself, “Ah, youth.”

Back to mathematics.

You can prove that binary search works using a technique called “mathematical induction.” The idea is that you first prove something for one case and suggest a general formula. If you prove that if the formula works for the case n then it also works for n+1 then you have proven that the formula is always true. Let’s try it for binary search:

Theorem: Binary search takes log base-two steps.

Consider the case of just 0 and 1. This takes just one question: “Is it 0?”. So the formula works for 2 numbers since log-base-two of 2 is 1.

Assume for formula is true for n. The case n+1 consists of exactly twice as many numbers, so you just need one additional question: “Is the answer closer to zero than to the largest number” since this question divides the set of numbers into two equal sets, each of size n, and you can then use the method you know works for case n.

The method I used in the proof above, as well as binary search itself, illustrates another common mathematical technique that is often used in programming: Divide and Conquer. You can solve a big problem by dividing it into smaller problems.

So ends the first mathematics lesson. Now we’ll apprly our new knowledge to do some binary searching.

See you in the next post.

Notes.

1. My undergraduate B.S. degree and my M.S. degree were in mathematics; my Ph.D. is in computer science, and getting that also required taking a number of mathematics courses. I’ve forgotten most of what mathematics I learned but hopefully enough remains that I can put together a few posts on this topic.

On mathematics and programming: Donald E. Knuth

My colleague Steve O’Grady recently posted a link to the blog post Math For Programmers by Steve Yegge.

While thinking of a forthcoming blog earlier today I realized I could make use of “binary search,” itself an application of mathematics to programming, in a forthcoming post, and so planned to use that as the first post in an occasional series on this topic.

However, my previous post discussed the generation of random numbers, and so constitutes the first such occasional post.

Generating a list of “random” numbers is itself a very difficult problem. Don Knuth devotes 170 pages to this topic in Vol. II of his masterpiece multi-volume series “The Art of Programming.” These are the best books so far written about programming and mathematics and I doubt they will ever be equalled.

As it happens I had the good fortune to have Professor Knuth as my instructor in a course in Group Theory while a sophomore at Caltech, before he became a professor at Stanford. Indeed I recall a later lecture or talk when he announced he would be leaving Caltech to go to Stanford.

I also had him as a customer, as I served him many a meal in my days as a waiter at the Caltech Athaneum. He was one of the few people who regularly ate there on the weekends.

But my first and most vivid memory of him is of a light burning in a window.

The Caltech campus was quite small back in the 1960’s when I went there. Moreover, it was located in the middle of a residential neighborhood so there wasn’t much to do. As a result I spent a great deal of time walking around the campus.

When I did so late at night I almost always saw that the lights were on in one of the offices in the Sloan building, home of the Mathematics department. I was curious to see who was working so late so often so I wandered by one time and saw that the name on the door was “Knuth.” This was when I was a freshman, before I had him as a professor.

A few years later, when Vol. I of his masterpiece appeared, I knew what he had been working on.

Knuth never finished the series of books he planned, though he continues to work on them in his retirement.

He got sidetracked because of problems getting Vol II properly typeset. As a result he decided to write on his own software that would support the proper display of mathematical papers. In one of the great feats of programming he developed software not only to format such text, but even to format the characters and symbols that comprise the text.

The end result was an open-source program called Tex. It has been available for over two decades and today is used to format almost all the mathematics and scientific papers in the world.

Most notable is that it was all done by Knuth himself. It is freely available under the following license:

The files in this directory are master files maintained personally by
Donald E. Knuth. Nobody else is authorized to make any changes whatever
to them! If you modify the files for any purpose, you must give your
files a different name, so that installations of TeX throughout the world
will be 100% compatible when they use the official source files.

Please help preserve the integrity of TeX by reporting any violations of
these rules to the TeX User Group.

What is particularly striking is the remarkable stability of the code. There are few if any known bugs. Knuth has offered for a number of years a small payment to anyone who finds an error in his work. He’s had to write only a few such checks, and almost none have been cashed because to receive one is a high honor indeed. The value to the recipient of such a check comes not from the cashing it but from being able to post it on their wall.

There is a parallel to this in the past. I recall reading in that wonderful series of books, “The World of Mathematics,” that Gauss was notable in that not a single error had ever been found in his published papers.

On blogging: Sometimes the details don’t matter

In my previous post I gave several examples of writing made more effective by the inclusion of specific details.

Details can make your writing more effective, but they can also be distracting, and trying to provide too much detail may make it harder for you to express the meat of the post. That’s one of the reasons that Wikipedia is your friend. You can use links to Wikipedia articles that have details so your posts remain more concise.

And sometimes the details don’t matter.

Earlier this morning I took my dog out for his first walk of the day and also to retrieve today’s copy of the New York Times. I heard my neighbor Mr. O’Donnell say “Happy Birthday” to one of his daughters as she was about to get into her car. I shouted a “Happy Birthday” to her across the lawn. She mistook the message as a query and replied that she was 28 years old today, so I knew it one of the O’Donnell twins. They and my son Michael were classmates.

Later on in the morning on my way to vote I felt bad because I didn’t know her name and so was unable to use her name in my birthday greeting.

Then I realized it didn’t matter — she’s a twin, so she and her sister have the same birthday. “Happy Birthday O’Donnell Twins.”

Her’s another example I heard from Professor Max Goldstein at the Courant Institute almost forty years ago. Max was the director of the computer center at Courant and before coming to NYU had spent many years at Los Alamos.

While at Los Alamos in the 1950’s Max was part of the team that produced a book that contained 100,000 random numbers, back in the days when computers were not widely available and so there was a need for such a list for use by researchers. He helped supervise the typists and clerks as they assembled the document. Los Alamos, then and now, designs nuclear weapons, so the clerks and typists were well aware of the need for scrupulous accuracy in their work.

One day one of the typists came to Max and apologized, saying, “I just mis-entered several of the numbers. How do fix it?”

Max replied, “Never mind. It doesn’t matter.” She was astounded that a mistake didn’t matter, though of course in this particular case it really didn’t matter.

It was a random error, in a random list of numbers. Who would know?

In twenty-five words or less

Ever seen the words “In twenty-five words or less?” As a blogger, you have just about that number of words to grab the reader’s attention.

The preceding paragraph has exactly twenty-five words. If you are reading this one I have made my case.

Here’s a display of that paragraph that shows there are just 25 words:

Ever seen the words “In

twenty-five words or less?” As

a blogger, you have just

about that number of words

to grab the reader’s attention.

The most important words are the title. It is your first and only chance to get the reader’s attention. You want to give them a reason to read the first paragraph.

The purpose of the first paragraph is to give the reader a reason to go to read the second paragraph, and so forth.

Each of the initial paragraphs must be an attention-getter. Forceful, punchy writing is needed at the start to get the reader into the meat of your post, so they can see the thoughts you are trying to convey.

Having a strong beginning is also important in that sites such as deli.cio.us Dave and WordPress display only the first part of a post in their summary for it. See for example WordPress open-source posts. I’m very fond of the latter link in that for some days I have been the “featured” open-source WordPress blogger. “Thank you, WordPress.”

That’s why the newspaper is such a useful tool for a blogger. Each story is the work of reporters and editors who have spent years learning how to start of a story with forceful writing that attracts the reader’s attention, writing that makes you want to read on.

For example, here are the titles and first words from two stories on the first page of the Business section of today’s New York Times:

Committed to Coal, And in a Hurry, Too

Texas Utility Plans 11 Plants Even as Touch Emission Rules Seem Inevitable

By Matthew L. Wald

FAIRFIELD, Tex. — In a huge pit, gigantic buldozers and earth-moving machines are removing two layers of coal, the last shavings in a monumental task that has dug 200 feet down and expanded across 20 square miles over the last 35 years.

The coal feeds two plants nearby that help keep the lights on and the air-conditioners humming throughout Texas. But in doing so, the operation has released hundreds of millions of tons of carbon dioxide, the main contributor to global warming.

Note the details in the first paragraphs expressed in just three numbers — 200, 20, 35 — that tell you the article is about a big enterprise, in both space and time.

The second paragraph ties a large mining operation into its impact on the environment and global warming, so you know the article is not just about the scale of the mining.

Here’s another example:

Big Bonuses Seen Again For Wall St.

By Jenny Anderson

On Wall Street, the rich keep getting richer.

For a fourth consecutive year, year-end bonuses are forecast to be highly lucrative, with the payouts rising 10 percent to 15 percent from 2005, according to Alan Joohnson Associates, a leading compensation consultant.

Here the details come in the second paragraph. The first paragraph is just eight well-chosen words that make you want to read on. The article would be much weaker if the order of the paragraphs had been switched. That would have made the article appear to an editorial, not a report.

Here’s another example taken from the Op-Ed page:

Nicholas D. Kristoff

America’s Laziest Man?

Last year, Barry Diller took home a pay package worth $369 million, making him the highest-paid chief executive in America.

His shareholders didn’t do so well. Stock in the main company he runs, IAC/Interactive, declined 7.7 percent last year. For the three years ending in December 2005, the stock was up just 11 percent — compared with 49 percent for the. S & P. 500.

The title is just three words. Who wouldn’t want to know who was America’s laziest man? At the least you want to make sure the New York Times doesn’t think it’s you.

The second and third paragraphs offer some details. The rest of the piece is equally well-written. While Mr. Diller may be laughing all the way to the bank, I’m sure he will see this piece and then will know that a great many of his fellow citizens are also laughing — about him.

It’s even possible this post will be brought to Mr. Diller’s attention. One of my first posts was in the form of a public letter to a CEO. It was not to rebuke the CEO but to thank him. I often see in the WordPress Dashboard referrer’s list that someone is systematically looking for blog posts about that CEO.

Mr. Diller, if you see these words, then know that while you may be laughing all the way to the bank, I trust your shareholders are much less amused.

The first paragraph in this post has twenty-five words and the title has five, making for a total of 30 words. Let’s see if we can fit both the title and first paragraph into twenty-five words. This means we have to cut out five words:

In twenty-five words or less

Ever seen the words “In

twenty-five words or less?” You

have just about twenty-five words

to grab your reader’s attention.

On blogging: The newspaper is your friend

I wrote nine blog posts on Monday, November 6th (I’m starting this just before the end of the day so I’m counting it even though the posted date may show upas November 7th).

That’s a lot of posts for one day. I wrote the first just after midnight last night. All the rest were written after I read this morning’s edition of my local paper, The New York Times. The articles I read inspired several of the posts, or had a role in others. Let me explain.

A musical magic moment

Apology accepted

In Memory of Academician A. P. Ershov

This post about Andre Ershov was not directly inspired by the Times, but by an e-mail that came my way. However, I’ve been thinking for some time about writing some posts about my travels to Russian. I started thinking of doing this after reading a Tom Friedman column a few weeks back that had “Russia and China” in its title. I had planned to write a post on that, but there was lots of ground to cover, so I put it off. But that e-mail made me think about this a bit more, and I realized I could wrote a short post just about him. If I ever get around to finishing the “Russia/China” post then I may include part of this post in that.

Ernestine Gilbreth Carey, 98, Author of Childhood Memoir

I wouldn’t have written this post except that Ernistine’s father was Frank Gilbreth, and I realized this would provide a way to finish some unfinished business. I had written two posts about three software development models — DUD, RERO, BLOAT — but the first two appeared in an article about golf, as an aside. I’ve looking for a way to define all three models in a single post so I could talk about them in future posts and refer back to the defining post and so not have to refer back to an article about golf.

So I used Gilbreth as a lead to finish the business. Having the post based on the story about Ernestine gave the post a timely appearance when in fact that was not the case. It also provided a way to lead into the story, something much better than “Here is the post the pulls out two ideas I wrote in a post about golf.”

On designing software: DUD, RERO and BLOAT

On Journalism: Blogging, open-source, the internet, John von Neumann

I’d been meaning to write this post ever since I heard Professor Goldman talk, as I think this is a very important talk. But seeing two articles in the Business section that directly related to open-source made me realize — as I did in writing the Gilbreth post — that using those articles as leads could give the post a feeling of immediacy, that the post had been inspired by today’s news.

#110

On Blogging: Wikipedia is your friend

I had written so many posts recently that I have been think it would be useful to point out the utility. I did this today because I realized that if used the title “… wikipedia is your friend” then I could use the same title for “…newspaper is your friend.”

The newspaper can be a rich source of ideas for writing a blog post. Current events are obvious, and many bloggers just focus on them. They are usually the ones with a political ax to grind. However, a current event may suggest a way — as I hope I demonstrated above — to relate an area of great interest to you to that event, and so draw the reader’s attention.

I’ve taken inspiration for posts from every section of the paper: News, Local news, Arts, Opinion, Sports, and even Obituaries.

I mentioned once to my wife that I was writing a blog post about someone who had just died and she said that sounded a bit morbid. She could have put it another way, but that just gave me the title for a post, and so I’ll write a post on blogging based on obituaries.

Postscript added on 7 Nov 2006:

I forget to mention in the original post that I noticed a single word in an article in that issue of the New York Times that caused me to write the footnote that can now be found at the bottom of the page: A. J. Liebling

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