Thursday, November 26, 2015

My Year's Best List 2015

The best things that I read, watched, listened to, and otherwise ingested in 2015!

(Previous lists: 2014, 2013.)

Best Books—Fiction

John Banville, The Sea (2005)

Grieving the death of his wife, a man moves to the seaside town where he spent summers as a child. It was there, too, as a child that he met a little girl and fell in love with her. That sets the psychological stage for The Sea, a melancholy and uncanny novel with prose so magnificent that I often gasped or even blurted out "WOW" while reading it.

Every review of the book that I saw online had spoilers, so I won't link to one. Not that it's a novel of suspense or anything—I just tend to think that reviewers tell too much.

Buy it online: The Sea

Elena Ferrante, My Brilliant Friend (2012)

Characters and setting are wonderfully realized in this coming-of-age novel about female friendship set in 1950s Naples. Ferrante has an excellent sense for plot, and meanwhile the prose is good caliber and does what it needs to do. If I could say so without in any way suggesting that the book is a pastiche, I would describe My Brilliant Friend as great Italian cinema committed to paper.

This is the first book in a series of four "Neapolitan novels"—you can read about the series and its famously reclusive author at the New York Review of Books (contains spoilers).

Or, you can just jump right in and buy it online: My Brilliant Friend.

Karl Ove Knausgaard, My Struggle, Book 4 (English edition 2015)

My Struggle is an international literary sensation and a milestone in artistic realism. Knausgaard's hybrid novel-memoir plays back seemingly every event, idea, and urge of the author's first forty-three years of life. That ought to be boring, but if you go into a bookstore and begin reading this book on any randomly chosen page, fifteen minutes later I think you'll still be standing there, pulled along by the ceaseless current of prose.

Four volumes of My Struggle have been translated into English so far. I began the series with Book 4, which was a good starting point; Jeffrey Eugenides calls Book 4 "the fleetest, funniest and, in keeping with its adolescent protagonist, most sophomoric of the volumes translated into English thus far."

Buy it online: My Struggle, Book 4.

Best Book—Nonfiction

Marie Kondo, The Life-Changing Magic of Tidying Up (2014). Japanese lifestyle expert Marie Kondo shares profound insights about clutter and categorization, and she teaches a practical plan for radically changing the complexion of your household. Kondo's pronouncements are amusingly extreme, and she can sound like a cult leader; but if it's a cult, then count me in. Buy it online: The Life-Changing Magic of Tidying up.

Best Short Stories

Joe Dunthorne, "The Line." Winter 2014 Paris Review.  Buy the issue online.

James Lasdun, "Feathered Glory." Spring 2015 Paris Review. Read a teaser excerpt here, and buy the issue online.

Best Poem

Nick Twemlow, "Attributed to the Harrow Painter." Summer 2015 Paris Review. Buy the issue online.

Best Essay

Edward Hoagland, "Walking the Dead Diamond River" (1973). Ted Hoagland is a master of the essay form, and this work displays all of his hallmarks: carefully controlled tone, often wryly elegiac; finely crafted sentences and noteworthy diction; subtle development from paragraph to paragraph, following a logic that is associative and peripatetic yet never meandering; and a wealth of reportage about colorful characters and lore plus loads of physical observation and detail. The essay is reprinted in the 2014 compendium On Nature. Borrow it from a library or, if you have a large library of your own at home, buy it online: On Nature.

Best Long-Form Journalism 

Michael Finkel, "The Strange & Curious Tale of the Last True Hermit."
For nearly thirty years, a phantom haunted the woods of Central Maine. Unseen and unknown, he lived in secret, creeping into homes in the dead of night and surviving on what he could steal. To the spooked locals, he became a legend—or maybe a myth. They wondered how he could possibly be real. Until one day last year, the hermit came out of the forest.
This story appeared on Conor Friedersdorf's list in 2015. Read it online.

Best Play

Fool for Love at the Samuel J. Friedman Theater

Sam Shepard's Fool for Love is an emotionally charged one-act play about two tortured lovers, set in a cheap motel room on the edge of the Mojave desert. This year's Friedman Theater production was Fool for Love's first outing on Broadway (Shepard wrote the play in 1983).

If you read the play, you'll notice that the stage directions at the beginning are peculiarly specific; the producers clearly took that cue and invested enormously in designing the excellent set, lighting, and sound. It would be easy for the actors in such a violent play to overdo their lines and movements, but Sam Rockwell as Eddie was controlled without sacrificing physical potency, and Nina Ariande gave a staggering performance as May. There comes a moment in the play when May collapses to the floor, crying raggedly, unable to speak, suffering a pain so exposed and so pure that I wept along with her. Here is the NY Times review. UPDATE: And here is a New Yorker review of the play combined with an essay on Shepard's life and work.

Best Movies

(Links point to reviews.)

The Conversation – Coppola at his peak...'70s art-house thriller still looks & sounds great.
Fury Road – IMAX auteur-cinema, where have you been all these years??
It Follows – Joins Halloween in the hall of Midwestern hormonal horror.
John Wick – Stylish shoot-em-up with elements of fantasy
Once Upon a Time in the West – Sergio Leone's operatic apotheosis rendered additional Westerns forever unnecessary.

Deserving of special mention is Satyajit Ray's heartbreakingly beautiful 1955 masterpiece, Pather Panchali, in a newly restored print that I was able to see this year at New York's Film Forum.

Biggest disappointment: American Sniper. The film had some excellent aspects, but in key places it suffered from poor artistic taste, and I couldn't escape the conclusion that it was made to serve social purposes as much as artistic ones. (Here is an article analyzing people's reactions to the film.)

Snootiest Cocktail

"Last Caress," Hotel Surrey. Made with, get this, chartreuse snow, foraged immature berries of juniperus virginiana, and house-label champagne. Louis the XVI's favorite slushie flavor, basically. 

Best Meal in an Airport

Saison Bistro, Newark Airport, Terminal C. Over the years I've seen restaurants come and go in this location, including a steakhouse and a Soul Food place. Currently it's a French bistro, and I had a very good piece of salmon there in early 2015. (In fact, it appears to be an Alain Ducasse project.) They have iPads for ordering and you know that never works, but the waitstaff quickly remedied the situation and I made it to my gate in plenty of time.

Best Solo Drive

In July I was doing some work in Boise, so for a change of pace I extended my visit and drove around Idaho, a trip that took me to the impressive Snake River Plain, the eerie landscapes of Craters of the Moon National Monument, and the dramatic Sawtooth Mountains.

Best Diner

Louisa's Place, San Luis Obispo, California. Downtown SLO is a lot more posh than it used to be, with clothing boutiques and single-source coffee purveyors everywhere you look. But walk into Louisa's and you step into a more comfortable past. My brothers, my sisters, and I all grew up in our parents' diner, and being in Louisa's felt to me like being at home. I stopped at Louisa's for breakfast during another great solo drive, one that started in Big Sur and ended in Orange County.

Best Art

On Kawara, "Silence," at the Guggenheim Museum. Kawara (1933–2014) was a conceptual artist, and although his date paintings get most of the attention, his obsessive array of codebooks, postcards, maps, and catalogs are just as important. His decades-long body of work coheres as a whole; Kawara appears to have made his own life into a single staggering artwork. Here is the Guardian UK review of the Guggenheim show, and here is the NY Times review.

Marc Yankus, "The Space Between" (photography, 2014; combination chemical-digital process).  Arresting, gorgeous, and lonely, these images of buildings are simultaneously dreamlike and concrete. (The artist cites de Chirico as an influence.) Online: The Space Between.

Best of the Year—Period. 

Dianne Reeves, Valentine's Day Concert, Jazz at Lincoln Center

Jazz vocalist Dianne Reeves has a godlike musical talent and a generous musical soul. When I wasn't grinning like an idiot at this concert, I was wiping away tears.

Reeves has been tilting toward pop forms in recent years, and on Valentine's Day she sang fabulous versions of "Waiting in Vain" by Bob Marley and "Dreams" by Fleetwood Mac. Another song in a world-music vein transcended the genre, hopping mysteriously from continent to continent before coming in for a smooth landing in South America.

The diva still unleashes power on standards! Her "Don't Explain" was especially memorable— in fact, overwhelming. She sang the song without a microphone (she has no trouble being heard without one), and she slowly paced the stage as a woman stunned, vulnerable, tragically naked. It is, after all, a brutal song.

Here is the NY Times review of the show.

You can check out her latest album on iTunes.

I'll bring this year's list to a close with some vintage Reeves. Here she is singing (the verb seems insufficient) the jazz standard "You Go to My Head." Enjoy!

Friday, November 20, 2015

Sharing Some Math Problems

I thought it might be nice to share some of the problems I've written for Saturday School, just in case other kids might have fun with them too.

The problems are at The grade level of the topics is roughly from Kindergarten to grade 2.

Happy mathing! If you see any errors, please let me know in the comments. (For the sake of context, I should note that I made each of these pages quickly on a Saturday morning.)

Update 11/16/2016: URL corrected

Wednesday, November 18, 2015

Lewis Carrol's Jest

Earlier I wrote about Lewis Carroll's book of "pillow problems." The other night I was reading problem #72 and did a double-take:
A bag contains 2 counters, as to which nothing is known except that each is either black or white. Ascertain their colours without taking them out of the bag.
I thought this must be a trick question. My answer was along the lines of "Ask someone else to take the counters out of the bag for you." But turning to the solution, I found this calculation:

"Whaaaat? That's absurd," I thought. "Both of the original counters might have been white, or both might have been black—a calculation can't retroactively change the counters' colors!"

I thought this must be a jest on Carroll's part. But then again, Carroll took seriously his role as a public communicator of mathematics, and so he would never leave a misleading calculation on the page for people to absorb as if it were true. At the same time, Carroll himself couldn't be mistaken about such a simple problem, especially since some other problems in the book show Carroll creating probability paradoxes and resolving them correctly.

Then I noticed the words "THE END." I thought, It would be just like him to end with a laugh. But still I didn't quite believe that he would commit a fallacy to paper, even for a joke. On the other hand, fallacious proofs like 0 = 1 are a commonplace in mathematical recreations....

I returned to the introduction, which I hadn't read in several months, and there I found this paragraph:
If any of my readers should feel inclined to reproach me with having worked too uniformly in the region of Common-place, and with never having ventured to wander out of the beaten tracks, I can proudly point to my one Problem in 'Transcendental Probabilities'—a subject in which, I believe, very little has yet been done by even the most enterprising of mathematical explorers. To the casual reader it may seem abnormal, and even paradoxical; but I would have such a reader ask himself, candidly, the question "Is not Life itself a Paradox?"
OK, there we go—found the wink-wink! With problem number 72, Carroll was taking tongue in cheek and showing the reader a tempting piece of sophistry.

Next morning, I searched online for discussions of the problem. One is here, but the content is not available for free and I didn't read it. The problem was also mentioned by the great Martin Gardner (1914–2010) in his book The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Puzzles, Games, and Word Plays. (I don't own this book, but in case anybody reading this post is responsible for putting together a Christmas list....)
The book's last problem, number 72, has been the subject of much controversy. ... The proof is so obviously false that it is hard to comprehend how several top mathematicians could have taken it seriously and cited it as an example of how little Carroll understood probability theory!
Gardner goes on to explain how in the book's Introduction Carroll "gives the hoax away." Gardner however doesn't seem to spend any time exploring the paradox. If I come up with any additional thoughts about it, I'll update this post with them. In the meantime, I leave you in the capable hands of another admirer of Lewis Carroll.

Monday, November 16, 2015

Fishing Vacation (A Word Puzzle)

"I don't know if I've ever been more content (or more sunburned) than in East Cape, Baja Peninsula. That was the best of our vacations. The boat weightless on the waves, a sixpack on ice. With all we were catching in the afternoons, even a medium-size roosterfish or bonefish wasn't worth reeling in."

Circle the number words ZERO through TEN in this story.

My kids enjoyed doing this one. A PDF suitable for printing is at

Sunday, November 15, 2015

Why Won't Your Calculator Tell You What 3 Divided By 0 Equals?


Suppose somebody types 3 ÷ 0 into a calculator, hits the = button, and gets an error message. If the person asks you what is going on, what do you say?

I think in these situations we mostly tend to say, "You can't divide by zero." That isn't wrong, but I suddenly find it odd, because it sounds more like physics than mathematics. You can't travel faster than light, you can't build a perpetual motion machine, you can't unscramble an egg, and you can't divide by zero. One of these things is not like the others, yet the formulation doesn't reflect that difference. A scientist could doubt whether it is really impossible to travel faster than light—all scientific knowledge is provisional—but we can say dogmatically that nobody will ever build a calculator that gives a numerical answer for 3 ÷ 0.

Instead of saying "You can't," in a case like 3 ÷ 0, I think it would be better to say to the student,
"There's no such number."
It isn't so much that "you can't divide 3 by 0" as that there's no such number as 3 ÷ 0.

Why is there no such number as 3 ÷ 0? Because remember what 3 ÷ 0 actually means: it means the number that, when multiplied by 0, gives 3.

Think about it! There is no such number!

Any number multiplied by 0 always gives 0, never 3.

"You can't do that" also sounds like an arbitrary rule to remember...just another of the endless rules adults are always hitting kids with. "There's no such number," on the other hand, is a statement of mathematical fact.

Saying "You can't" invites the reply, "Why can't I?" A good answer to which would be: "There's no such number." One might as well skip a step.

It certainly seems absurd to share 3 cookies among no people, or to ask how many servings of size zero are in 3 cups of ice cream...but what is really absurd about the first situation for example is not the vaguely puzzling idea of sharing among no people; it's the concretely outrageous idea that the result of sharing 3 cookies among no people would be shares of half a cookie! Or seven cookies, or any other number of cookies. Likewise, there could not be eleven 0-cup servings in 3 cups of ice cream. No number could be the value of 3 ÷ 0.

So tell me again, why does 3 ÷ 0 give Err on the calculator? Could it be that the school bought cheap calculators to save money? Would a more expensive calculator have told us the number? No. Even at Chichi Academy they get an error message. There is no such number, so the calculator has to display something else.

"Undefined" is the word that careful math educators tend to use for division by zero. And that word is accurate...but saying that "Division by zero is undefined" strikes me as little better or even no better than saying that "You can't divide by zero." After all, what is the force of "undefined," in a discourse where precious little ever gets defined in the first place? I think it means nothing to a student and it gets internally stored in student memory as: "You can't." Anyway, in a case like 3 ÷ 0, "undefined" is shorthand for "there is no such number"! Instead of using shorthand, one could make a statement that has content.


"There's no such number" describes a case like 3 ÷ 0 where the dividend is nonzero. What about the doubly bizarre case of 0 ÷ 0? This, recall, would have to mean the number that, when multiplied by 0, equals 0. But every number, when multiplied by 0, equals 0. So there is a case to be made that any number could be the value of 0 ÷ 0. In a sense, this is the opposite situation to 3 ÷ 0, where there was no such number.

Could there be fifty 0-cup servings of ice cream in 0 cups of ice cream? Sure, why not! Heck, let it be a thousand 0-cup servings. You can feed any number of people with no ice cream, as long as everybody would be satisfied with getting none.

If 0 cookies are shared equally among 0 people, could each person get a dozen cookies? It wouldn't violate any laws of physics!


My own calculator, Mathematica, is pretty chichi, and unsurprisingly it gives two different results for 3 ÷ 0 and 0 ÷ 0, along with some error messages:

Mathematica uses the non-numeric quantity "ComplexInfinity" for various purposes in the analysis of functions. If you ask Mathematica for the value of 0 times ComplexInfinity, you won't get a numeric answer—you'll get "Indeterminate," another non-numeric quantity. So in Mathematica, 0 × (3 ÷ 0) ≠ 3.


I'll close this post with two other cases where "There's no such number" seems like a better answer than "You can't do that":

  • Instead of saying "You can't take the logarithm of a negative number," you could say, "There's no such number as Log(−8)." Why? Because remember what Log(−8) means: it means the number that, when you raise 10 to that power, you get −8. But try a few cases and you'll see that there is no such number...10 raised to any real power is always a positive number, never −8. There's no such number as Log(−8).
  • Instead of saying "You can't take the square root of a negative number," you could say, "There's no such number as Sqrt(−1)." Why? Because remember what Sqrt(x) means: it means the nonnegative number that, when you square it, you get x. But try a few cases and you'll see that there is no nonnegative number that squares to give you −1. There's no such number as Sqrt(−1).

In these two examples, "number" means "real number." High school students learn that there are two complex numbers whose square is −1, and college students in math or physics learn that there are infinitely many complex numbers that, when you raise 10 to that power, you get −8.

Can complex numbers also help us divide by zero? No. Just like the real numbers, the complex numbers are algebraically a field, which implies that even in the complex numbers, there's no such number as 3 ÷ 0.

Wednesday, November 4, 2015

Slot Machine Word Puzzle

Form a long word by choosing one short word from each wheel and then stringing together your three choices.

(You might imagine that the three wheels are the guts of a slot machine—you'll hit the jackpot if the three short words that end up in the display window join up to form a long word.)

Ten words may be formed in all.

Monday, November 2, 2015

How Do You Put 29 Marbles Into 4 Vases? Not With Division.

Elmer has 29 marbles and 4 vases. He puts the same number of marbles in each vase. How many marbles does he put in each vase? How many marbles are left over?

I saw a problem like this in a math textbook, and the intended answer was, "7 marbles in each vase, with 1 left over."

I wondered why the answer couldn't be "1 marble in each vase, with 25 marbles left over." After all, \(4\times 1 + 25\) equals \(29\).

Or "2 marbles in each vase, with 21 marbles left over."   \(4\times 2 + 21\) equals \(29\), too.

If the intent of the problem was to have a unique answer, then it ought to have said something like this:

Elmer had 29 marbles and 4 vases. He put the same number of marbles in each vase. What is the largest number of marbles each vase could have received? In that case, how many marbles would be left over?

Such problems are often presented to students as division problems; students are taught to record the answer as \(29 \div 4 = 7 R 1\), or perhaps \(29 \div 4 \rightarrow 7 R 1\). Notice that neither of these jottings is an equation: in the first case because "7 R 1" isn't a number, and in the second case because \(\rightarrow\) isn't an equals sign. Couldn't students be taught to represent a situation like this using an honest-to-goodness true equation, namely \(29 = 4\times 7 + 1\)?

The equation \(29 = 4\times 7 + 1\) shows clearly what is going on: the 29 marbles can be grouped into 4 equal groups of 7, with 1 left over.

I know it is difficult to break with a tradition like "\(29 \div 4 = 7 R 1\)." Maybe there are even reasons for that tradition that make it inevitable. Perhaps, in that case, students might be taught to represent the problem sometimes using an actual equation like \(29 = 4\times 7 + 1\).

It does bother me, though, this approach of presenting problems as division problems when division doesn't solve them. I mean, the answer to Elmer's problem isn't \(29 \div 4\), so in what sense is this a division problem?

One might wish to say that the reason this is a division problem is that "we are sharing equally." Well, sort of yes, sort of no. We start out with 29 marbles, and in fact we don't share those 29 marbles equally. We share 28 of them equally. And we throw the other one away.

Here's another way to think about it. Take the (true, actual) equation \(29 = 4\times 7 + 1\), divide both sides by 4, and simplify. The result is

\(29\div 4 = 7 + \frac{1}{4}\).

But what is the meaning of "seven and a quarter marbles"? That notion doesn't make sense in the context of the problem.

Division breaks a quantity into equal parts. That's what it does. If the stated number of objects in your problem can't be broken into the stated number of equal parts without the situation turning silly, then that could be a hint that your problem isn't really a division problem.


I thought I'd keep this space for additional thoughts that might accumulate over time.

If Elmer's problem isn't a division problem per se, then we could still see it as a problem with division in it. Along those lines, here is another way to represent the problem and the answer: \((29 - 1)\div 4 = 7\). This equation is another nice account of what's going on: it's an (actual, true) equation that correctly uses division, and moreover it uses subtraction to show the "throwing away" that I alluded to above. 29 marbles, throw away 1, share the rest equally among 4 vases, yields 7 marbles in each vase.

UPDATE 11/14/15 - A one-page PDF of this post, formatted nicely for printing, is available at