Saturday, December 27, 2014

Hangman Challenge

My holiday gift to readers this year is a simple Hangman game:

___  ___  ___  ___

If you'd like to play, email me your first move. I'll reply and we'll begin a thread that way.

There's no gallows here, and there's no limit to the number of guesses you can take. Just keep guessing letters until you guess the word. Try to solve the puzzle using as few guesses as possible.

Later I'll post the mystery word, and I'll say a few words about the principles I used to select it.

Until then, enjoy the holidays! All my best for a healthy and happy 2015.

Saturday, December 20, 2014

Change for a Dollar

So it turns out that there are 293 ways to make change for a dollar using pennies, nickels, dimes, quarters, half-dollars, or dollar coins. If you limit yourself to the more common coins (pennies, nickels, dimes, or quarters), then there are 242 ways to make change for a dollar. I copied all of the combinations from Frank Morgan's post on this topic and transferred them to this online spreadsheet, so you can play with the data if you like.

Having all of the combinations handy on a computer makes possible some wacky puzzle ideas, like this one:

A customer went to the bank and gave the teller \$242 in dollar bills. The customer said, "Give me change for each one of these dollar bills, please—pennies, nickels, dimes, or quarters—and furthermore, I want no two of these dollar bills to be changed the same way." The teller obliged, and soon the customer had a large pile of coins in front of him. "On second thought," said the customer, concerned about the weight of the coins, "let's change as many of these pennies as we can for dollar bills." The teller did so. "And you know what?" the customer said. "Let's also make dollar bills out of as many of these nickels as we can." This was done. "OK," the customer said, "let's do the same for the dimes." When that was done, the customer said, "What the heck, let's change as many of these quarters as we can for dollar bills too." After this was done, the customer had \$241 in dollar bills, plus change for a dollar, and he walked happily out of the bank. How many pennies, nickels, dimes, and quarters was the customer finally left with?

I don't know if there is any realistic way to find the answer without using a computer. Anybody who's sufficiently spreadsheet-savvy should feel free to post it in the comments!

Wednesday, December 10, 2014

Lane Change (A Brachistochrone Problem)

Imagine driving your car along the freeway. Initially in the right lane, you decide to get into the left lane. Given that you could lose control of the car if its acceleration vector ever exceeds a given threshold, what is the shortest time in which you can effect the lane change?

(At the end of the maneuver, you must be moving forward at the speed you started with.)

The natural unit of time in this problem is \(\sqrt{\frac{2w}{a_0}}\), where \(w\) is the lane width and \(a_0\) is the acceleration threshold. Expressed in these units, I can make the lane change in \(\sqrt{2}\) ticks of the clock:

During the first half of the motion, the acceleration vector points due left; during the second half, the acceleration vector points due right. The forward velocity component thus remains constant over time.

Formula 1 drivers might maneuver in something like this fashion, but regular drivers probably make a tradeoff between the duration of the maneuver and the difficulty of executing it.

The following graph illustrates some of the difficulty. It shows the component of the acceleration vector along the instantaneous velocity vector (what I call \(a_\parallel\) in my textbook):

Perhaps you can see from the graph what an aggressive sequence this is!

Finally, here is the resulting speed curve:

Monday, December 8, 2014

Speaking of Andromeda...check out this cool image of it

This image shows what Andromeda would look like in the evening sky if it were brighter:

Surprising, no? Here is a larger version of the image. Astronomer Phil Plait has further details about the image at Slate.

Sunday, December 7, 2014

Left Turn at Andromeda, Cont'd

The title of my last post was a reference to the old Bugs Bunny cartoons in which Bugs says, "I knew I shoulda taken that left turn at Albuquerque." Here are some further thoughts on the two problems.

First, the problem of getting back to the origin with a velocity vector rotated 90 degrees. How long will it take, using a constant-thrust accelerator?

Measured in appropriate units of time, the 45-45-90 solution takes around 3.68 ticks of the clock:

Those slowdowns at the corners are painful to watch; one can do better. The best I'm able to do is \(\sqrt{10}\approx 3.16\) ticks of the clock:

Here is a side-by-side comparison:

As for the second problem, making the left turn without having to return to the origin first, the best I can do is \(\sqrt{4+2\sqrt{2}} \approx 2.61\) ticks of the clock:

If anyone can do either problem faster, let me know!

In the meantime, here's a supercut of Bugs Bunny:

Saturday, November 29, 2014

Shoulda made that left turn at Andromeda

Here's another brachistochrone problem I was playing with:

You're coasting along in your space ship, when suddenly you miss a left turn. If you begin applying a constant-acceleration thruster immediately, how long will it take you to navigate your way back to the point where you missed the turn, reaching it with the same speed you had when you passed through it the first time, but with the direction of motion turned by 90 degrees from what it was originally?

One way to do it is to follow a 45-45-90 triangle:

Can it be done faster?

Second: What if you aren't required to revisit the origin after missing the turn, but instead all you have to do is get yourself traveling on the y-axis with vertical velocity at the original speed of motion?

Friday, November 21, 2014

My Year's Best List 2014

The best things that I read, watched, listened to, and otherwise ingested in 2014!

(Last year's list here.)

Best Books—Fiction

This year's list includes books from three different centuries:

Tolstoy, Anna Karenina (1877)

I'd been expecting Anna Karenina to feel like work, but to the contrary I was immediately absorbed thanks to the novel's plot, prose, psychological insight, and historical sweep.

Two practical notes:

  • When I was deciding which edition to buy, I decided not to obsess over the translation. Instead, recognizing that I was going to be lugging this book around for a while, I chose the edition with the most comfortable form factor: the Norton edition pictured above. 

  • In case useful, my proven strategy for finishing long novels is to read at least 50 pages every day. A dosage schedule like that will give the book a chance to gel in your mind. It will also carry you through the dull spots that most long novels have. At very least, it establishes an end date for the experience!

Yasunari Kawabata, Snow Country (1948)

A good description appears on the book jacket:
A powerful tale of wasted love set amid the desolate beauty of western Japan. At an isolated mountain hot spring, with snow blanketing every surface, Shimamura, a wealthy dilettante meets Komako, a lowly geisha. She gives herself to him fully and without remorse, despite knowing that their passion cannot last and that the affair can have only one outcome. In chronicling the course of this doomed romance, Kawabata has created a story for the ages—a stunning novel dense in implication and exalting in its sadness.  … Kawabata brings the brushstroke suggestiveness and astonishing grasp of motive that earned him the Nobel Prize for Literature.  

I failed in my first three attempts at reading this book, even though its breathtaking beauty was evident from the opening pages. The trouble was that the conversations between Shimamura and Komako were so elliptical that I was simply unable to follow the psychological action. That says more about my obtuseness than anything else, and I'm glad I kept at it.

There don't seem to be any convenient reviews of Snow Country, so instead I'll link to this NYT book review of First Snow on Fuji, a collection of Kawabata short stories.

Buy it online: Snow Country.

W.G. Sebald, The Rings of Saturn (1995)

Rings of Saturn recounts a solitary walking tour that failed to produce the desired spiritual rest, leading instead to a hospitalization. (All this is told on the first page.) The first-person narrative is a series of digressions: personal reminiscences, historical accounts, and meditations on artworks and writings ranging from the famous to the obscure. The remarkable prose consists of beautifully formed, archaically long sentences and paragraphs.

The theme of Rings of Saturn is human history as a process of destruction and decay. The mood of this tale is profoundly leaden (melancholy, saturnine). The action is frequently dreamlike and uncanny. This is a powerful book that might have been merely punishing if not for its soaring artistic qualities. I read it twice in succession.

Buy it online: The Rings of Saturn

Lydia Davis, Can't and Won't (Stories) (2014)

This was my first exposure to Davis, a literary titan. I was impressed by her mastery of form, by her revelations of the everyday, and by her beautifully machined prose.

Some pieces in Can't and Won't consist of a single sentence, while others are about as long as typical short stories. Some of the shorter pieces in the book made me think of Borges's parables (and I actually felt that Davis had succeeded in this difficult genre more frequently than Borges did). Despite the work's reserved tone, I laughed out loud several times.

Lydia Davis is often described as a writer's writer. In this collection she struck me foremost as an artist, one who happens to work in the medium of words. Here is the NYT review.

Buy it online: Can't and Won't.

John Williams, Stoner (1965)

Stoner is a quiet book that tells the entire life of a single person. It's the kind of novel that adds to your understanding of what it means to be alive on this planet. And the prose is astonishing. Here are three hundred pages of uninterrupted, unshowy perfection.

After I recovered, I read John McGahern's elegant introduction. Then I did some digging online and discovered that Stoner was all the rage in Europe last year; its rediscovery was a remarkable story in the world of letters in 2013. A review of the book is here.

Buy it online: Stoner.

Also Recommended: 

Rachel Cusk, Outline (2014/2015). Very little happens in it, but I was constantly intrigued by its narrator, a writer traveling for her work. We don't learn many specifics about her, but thanks to her prose we do see how intelligent and observant she is. The writing is intricate, yet the author's craft is such that she seems to achieve her paragraphs effortlessly. I read it in serial in Paris Review, but soon it will be published in the usual way. I'll probably buy a copy and read it again. Buy it online: Outline.

Jenny Offill, Dept. of Speculation (2014). An acidly funny, deliberately strange novel about the midlife of a writer, and the life of a marriage. Dept. of Speculation is short enough to be read in a single sitting, and it probably ought to be read that way. Here is the NYT review, and here is the Guardian review. Buy it online: Dept. of Speculation.

David Mitchell, Black Swan Green (2007). A coming-of-age novel set in early-1980s England and told in the first person by a precocious (but still believable) teenager. This was my third David Mitchell novel. I'd been awed by two of his earlier books, Cloud Atlas and The Thousand Autumns of Jacob de Zoet. Black Swan Green isn't flashy like those two, but for that very reason it would seem to confirm Mitchell's reputation as a writer who can do anything. Buy Mitchell's books online.

Best Books—Nonfiction

Walter Kirn, Blood Will Out (2014). A chilling true story about the homicidal con-artist "Clark Rockefeller." Also, a lacerating self-portrait of Kirn himself, the noted journalist who fell for his fantasies. Blood Will Out offers some of the pleasures of the true crime genre, and there's also some beautiful writing in it. Here's the Boston Globe review. (And here's an interesting review that I think is better read after finishing the book itself.) Buy it online: Blood Will Out.

The Times Comprehensive Atlas of the World, 14th Edition (2014). Computers are indispensible for geography, but if you want to own a world atlas, this is the one to buy. I got a terrific price on it by using a 20%-off coupon at Barnes & Noble, on top of their standard markdown and the additional member discount. The book is is too tall and too wide to fit on any of my bookshelves, but I like having it within reach on the coffee table. Buy it online: The Times Comprehensive Atlas of the World.

Henri Matisse: The Cut-Outs Exhibition Catalogue (2014). The Museum of Modern Art has published this book to accompany its current exhibition of Matisse's late works with paper. The vibrant catalogue of works is reason enough to own it, and there are also fascinating photos of the artist in his studios and living spaces during the 1940s and 50s. We learn from the historical and critical essays that the initial reception to the cut-outs was cool, at least in France, but viewers soon began realizing that with these works Matisse was making profound advances. Buy it online: Henri Matisse: The Cut-Outs Exhibition Catalogue

Isa Chandra Moskowitz, Isa Does It (2013). This is a vegan cookbook. Trust me, the food is delicious. At home, our practice is to eat one vegan meal and one vegetarian meal each week; that healthful policy only survives because of the recipes in Isa Does It (along with our great vegetarian cookbook from the 1990's, Anna Thomas's New Vegetarian Epicure). Buy it online: Isa Does It.

Best Short Stories

Bill Cotter, "The Window Lion." Read the first few paragraphs online here. You can get the whole thing by purchasing the Spring 2014 Paris Review.

Lydia Davis, "Eating Fish Alone," "Master," "Odon von Horvath Out Walking," "The Letter to the Foundation," "Short Conversation (in Airport Departure Lounge)," "Writing," "Wrong Thank-You in Theater," or any number of other pieces in Can't and Won't. The full text of many of these can be found online, but I'm not going to provide links, because I don't think Davis's art can be appreciated on the computer screen. Buy her book. (If you must have a tiny taste right now, then at least let it be typeset properly: example 1example 2.)

Louise Erdrich, "Nero." Read it online here or buy The O. Henry Prize Stories 2014.

Alice Munro, "Royal Beatings," "Chaddeleys and Flemings," "White Dump," "Carried Away," and "Vandals." These are all in the collection Selected Stories, 1968–1994.

Best Poetry

Jeff Dolven, Speculative Music. Dolven is a friend and a Princeton professor whose poems have appeared in The New YorkerParis Review and elsewhere. Buy the book and you'll never look at a cantaloupe the same way again.

Monica Youn, "Exhibition of the Hanged Man" and "March of the Hanged Men." You can get them by purchasing the Winter 2013 Paris Review. The second of the two poems can be read online here, and the Poetry Foundation has more of her work online.

Best Essays

David Searcy, "Still Life Painting" (2014). Read the first few paragraphs online here. You can get the whole thing by purchasing the Fall 2014 Paris Review.

Ta-Nahesi Coates, "The Case for Reparations" (2014). As a writer, how do you essay a topic that your intended readership would prefer not even to think about? (And how do you awaken feelings of responsibility among your readers, when any suggestion of guilt whatsoever would short-circuit the project?) How do you write so that your readers might hold prior opinions in abeyance long enough to read you? (How do you transform a subject that everybody already "knows"?) Coates handles these rhetorical challenges masterfully in "The Case for Reparations." Read it here.

Flannery O'Connor, "Writing Short Stories" (1969). This piece, from the collection Mystery and Manners: Occasional Prose, is apparently a set of remarks delivered by O'Connor before a fiction-writing class. "My calm was shattered," says O'Connor, "when I was sent seven of your manuscripts to read." I enjoyed these profound, imperious, and occasionally hilarious insights into the art of writing. The book's appetizing typography also made me wish that computers had never been invented. Buy it online: Mystery and Manners: Occasional Prose.

Best Music

Philip Glass and Tim Fain, an evening of chamber music

Philip Glass, 78, is a famous composer who creates "music with repetitive structures"—so much so, in fact, that when I listen to some of his pieces, I can just about visualize the Fourier transform of what I'm hearing. At some point last year I reflected that Glass wasn't getting any younger and that I wanted to see him in person, so in 2014 we drove to small-town Ithaca, New York, for an evening of chamber music at the State Theatre, a restored local landmark.

The State Theatre in Ithaca, NY (I took a photo before Philip Glass took the stage).

It felt a little peculiar to listen to a famous composer out in the boondocks. The acoustics at the State weren't great, the microphone on Tim Fain's violin gave trouble at times, and I wasn't always sure that the house piano was up to the task. But none of that could suppress the music on offer. Seeing Mr. Glass in person was everything I'd hoped for. Here is a taste of Metamorphosis #2.

Here is a review of the same program, performed in Dallas just a few days earlier.

Joshua Redman

Fifteen years ago, I saw saxophonist Josh Redman play at the Village Vanguard. He was an intense young musician who put the horn through its paces as he drew on a variety of forms—avant garde, bebop, and swing. This year I received as a gift his latest album, Walking Shadows. With this album, Redman seemed be entering a middle-aged "I want to be backed by an orchestra" phase. Here's a review.

Jazz at Lincoln Center (I took a photo before the show started).

Middle-aged or not, when Redman took the stage this year at Jazz at Lincoln Center, backed by his familiar lineup, I saw no evidence of a man slowing down. Under his nice grey suit he was still teenager-thin, and his opening number, from the 2000 album Beyond, began with a long and technically demanding solo that struck me as playful, profound, and risky all at the same time. Redman also performed songs from Walking Shadows, including the beautiful "Adagio", based on a Bach melody. On this video, you can watch the same trio playing "Adagio" at a Jazz festival:

Best Movies

I rented a lot of movies this year. Here were the best:

Before Sunrise / Sunset / Midnight – a romantic trilogy that exceeds the sum of its parts
Boyhood – a tour de force of realism, and not just because of its central device
Pan's Labyrinth – a dark fairy tale powerfully intertwined with fascist history
Rocky and Get Carter – a good movie and a great one, both co-starring the bleak 1970s
The Searchers – complex, need to see it again, preferably on a big screen
Spring Breakers – sexploitation trash … avant-garde genius … or both
Under the Skin – eerie, erotic, profound … unforgettable

Biggest disappointment: American Hustle. Not that it wasn't good, but I thought it was disjointed and odd, and I'd gone into it with very high expectations.

Word Puzzle I Wish I'd Thought Of

I actually saw this one in 2013, but variants of it kept popping up in early 2014.

Best Hamburger

Shake Shack

The second bite of a Shake Shack burger is always accompanied by a feeling of panic—a panic that there is still some hamburger yet located outside my body. Only by finishing the burger can the error be corrected. Only by continuing to eat the burger can the maw of the abyss be shut.

The only problem with my usual Shake Shack on the Upper West Side is that it's an utterly miserable place. Always, always a line out the door. Never, never a seat upstairs. So down you go to the claustrophobic basement, there to gnash your burger like Kafka's burrowing animal crunching small fry between its jaws. And yet I return, and shall return.

Best Meal in an Airport—Breakfast Division

1. Cafe con leche at ZaZa Cuban Coffee, MCO. Worth a quick side trip even if you aren't flying out of Terminal B.

2. Sausage McMuffin with egg at McDonald's, any airport. Wash it down with a Coke and it's one of the best breakfasts there is. For the upscale version, go for the eggs benedict at Bobby Van's in LGA.

Best Visual Art

Henri Matisse, Henri Matisse: The Cut-Outs, Museum of Modern Art, New York

This historic exhibition rescues Matisse's cut-outs from the greeting-card rack and establishes their importance to Twentieth-Century art. Seeing the pieces in the gallery was a revelation. With the pins still sticking into the bits of paper in some cases, the cut-outs become tangible objects. They can be seen at proper scale, and they can be contemplated in the unusual context of their creation. These galleries, wallpapered, it seems, with bursting fireworks, are an almost overpowering profusion of creative invention, artistic intelligence, and human spirit. The exhibition, which started at the Tate Modern in London, continues in New York until February 8, 2015. If you've been looking for an excuse to spend a weekend in the city, look no further. Online: Henri Matisse: The Cut-Outs.

Honorable mention:

Jeff Koons, Jeff Koons: A Retrospective, the Whitney Museum of American Art, New York. In the 1990s it was easy to dismiss Jeff Koons, and I did. Yet whenever I thought of him afterwards, I also remembered with a prick of the conscience the unreasonable beauty, as it seemed to me, of his Michael Jackson and Bubbles, which I had viewed at SFMOMA. Twenty years later it's easy to celebrate Koons, and here I am again following the crowd. The artist himself still mystifies (ignore whatever he says about his art), but I've come to regard the work itself as significant. Here is the NYT review of the show, and here is the more negative WSJ review.  Online: Jeff Koons at the Whitney

Kara Walker, A Subtlety, Domino Sugar Factory, BrooklynWalker is best known for her silhouette installations. For "A Subtlety" she created a monumental, sugary sphinx in the deserted Domino Sugar factory in Brooklyn. The sculpture and the site worked together powerfully to invoke history. Here's a review in New York magazine.

Best Small-Gallery Show: "Looking East," at The Frist Center for the Visual Arts, Nashville, Tennessee. The Frist Center occupies a landmark Art Deco-plus-Neoclassical building that used to be the city's main post office. There my sister and I saw the traveling "Looking East" exhibit, about Japanese art and its influence on European artists in the 19th Century. Two highlights were Hiroshige's vertiginous view of a rain-lashed bridge (the picture here doesn't do it justice) and a first edition of Hokusai's Hundred Views of Mt. Fuji, with its remarkable postscript. Also shown were memorable works by Toulouse-Lautrec, Cassatt, and others that showed the Japanese influence (sometimes more, sometimes less). The show was organized by the the Museum of Fine Arts in Boston, which publishes a book of the exhibition. P.S., I also found a nice "online exhibition" of Hokusai here.

Best of the Year—Period. 

Twelfth Night, Shakespeare's Globe Company

Twelfth Night was the most stimulating—most amazing—most hilarious play I've ever seen in my life. This was the Shakespeare's Globe production from London, an authentic Elizabethan production with period musical instruments, all male actors, and an open stage design that wasn't quite the Globe experience itself, but that by Broadway standards really strengthened the connection between the actors and the audience. And the actors were phenomenal.

We went to the play because a couple of weeks earlier, I'd been sitting next to a colleague who is an English language arts teacher, and he said it was the best Shakespeare he'd ever seen. So we scrambled to get seats right before the run ended. We were so glad we did. Here is the first paragraph of the review from the New York Review of Books:
The production of Shakespeare’s Twelfth Night by the English theatrical company Shakespeare’s Globe, currently at the Belasco Theatre, brings this play to life in a way I have only very rarely seen equaled in a Shakespearean production. The performances are so uniformly skillful, the interpretation of the play so intelligent and imaginative, and the costumes and stage set so accurate and evocative that the entire experience is exhilarating. Audiences at the performances I’ve attended have been overcome with delight, clearly somewhat surprised by the affecting immediacy of the theatrical experience they have undergone, unaccustomed to a Shakespeare so readily comprehensible and so vividly alive. You may, if you’re lucky, see another Shakespearean production that’s as good as this one, but it’s unlikely you will ever see one that’s better.
A briefer review of the play is here (NYT). And here's a brief scene from the London production in 2013. Check it out! The audience laughter can be heard here because in the context of the story, these jokes really are laugh-out-loud funny—at least, when performed by this incredible cast.

That's it for the list. Let me know your thoughts, as well as your own favorites from the year!

Sunday, November 16, 2014

"Bookend Words" + Trip to the Zoo!

The "First and Last" puzzle in my last post might also have been called a puzzle about "bookend words." For each given letter, what are the words that bookend the dictionary entries for that letter? My results are below, where I gave myself a time limit (5 or 10 minutes, I don't remember which).

aardvark, azure
babble, byway
cabal, cyst
dabbed, dyne
each, eyrie
fable, fuzzy
gabardine, gyre
habanero, hymn
iamb, ivied
jabbed, jutting
kabbala, kumquat
label, lyre
macadam, mythology
nabbed, nuzzled
oafish, oxygen
pablum, python
quack, quoth
rabble, rutted
sabbat, sylph
tabbed, tyrant
ubiquitous, uvula
vacant, vying
wacky, wryer
xanthan, xylophone
yacht, yurt
zapped, zyzzyva

Here are some good answers:

aardvark, azure
baba, byword
cabal, czar
dab (or daal), dystopian
each, eyrie
fable, fuzzy
gabardine, gyroscope
habanero, hysterical
iamb, ivory
jabbed, juxtaposition
kabbala, kuru
label, lyrist
maam, mythos
naan, nymphomania
oaf, ozone
pablum, python
qabalah, quotient
rabbi, rutty
sabbat, syzygy
tabbed, tzar
ubiquitous, uxorious
vacancy, vying
wacky, wurst
xanthan, xylophone
yacht, yurt
zabaglione, zyzzyva


We seem to have developed a custom of visiting zoos in the winter months, when the place is quiet and there aren't any crowds. You do have to bundle up…and you don't get to see all the best animals. But there's still plenty to see, and we also go to zoos in the summer so it evens out. Below, some photos from yesterday's trip to the Bronx Zoo. This was our first visit to North America's largest urban zoological garden, a surprisingly naturalistic setting with 265 acres of exhibits, habitats, and preserved land along the Bronx River.

Giraffes wintering indoors—they looked like a diorama in person, too.

The facility resembles a horse stable.

Bug carousel!

Nice to get out of Manhattan for the tail end of Fall!

Sunday, October 12, 2014

First and Last

For each letter A–Z, what is the alphabetically earliest word you can think of? How about the alphabetically latest? Words must have four or more letters.

No peeking at dictionaries! This is a mental exercise. I had fun with it by giving myself a time limit of 10 minutes. Alternatively one could take a perfectionist approach to each letter.

You can use any variety of words you like, but the words I like to work with are the “standard” word game variety: this means we avoid proper nouns, foreign words, slang, contractions, hyphenated words, or acronyms; personally, I also frown on jargon.

I'll post some answers next week. In the meantime, here's a worksheet you can use to record your answers (the letter A is done as an example).

Feel free to put answers in the comments, or post them online by taking a picture of the completed worksheet with your phone and uploading the image to

(Imgur is nice because you don't need to sign in. Just click the "upload images" button.)

Wednesday, August 13, 2014

Book Review: Jonathan Lethem

Two lesser-known books by Jonathan Lethem, author of Motherless Brooklyn and Fortress of Solitude.

Gun, With Occasional Music
Harvest Books, 2003 (reprint; originally published 1994)
Paperback, 271 pages

Fans of Raymond Chandler and Philip K. Dick will enjoy this hard-boiled science-fiction story. The novel is told in the first person by a Marlowe-type detective, Conrad Metcalf, who gumshoes his way around a futuristic East Bay, California. To solve his case, Metcalf must navigate the usual obstacles (femme fatales, bent cops) as well as some new ones: designer pharmaceuticals, animals who've evolved the capacity for speech, and the comically disturbing "babyheads," about which let me say no more. The prose is occasionally uneven, but this seems intentional, part of the narrator's actual voice (and a spoof on genre dialogue). The plot sometimes meanders, but no more than in Chandler, and again this seemed intentional. As in Chandler, the atmosphere is so rich you don't mind taking the long way round.

The Disappointment Artist: Essays
Vintage, 2006
Paperback, 160 pages

This short book of autobiographical essays is worth reading if you're a student of Lethem, or if you share his enthusiasms (such as comic books or Philip K. Dick). The confessional style hit me over the head a few too many times. I also have to admit that I seldom enjoyed teenaged Jonathan's company. By the end, however, the author's adolescent frailties came more subtly and sympathetically into focus in relation to his mother's death---a tragic death, from brain cancer, when Lethem was just 13.

Up to What Point?

This post is about the abortion debate, an atypical subject for me but I'll be writing about it in relation to my customary concerns about quantitative thinking. All I want to say about it is that whenever I try to follow the debate, I become frustrated by the vague propositions people use to frame the question. Here's an example, a graph from Pew:

The poll asks, "Do you think abortion should be legal or illegal in all or most cases?" The trouble with this is that there's no time axis in the question. I think for a large fraction of people---not everyone, certainly, but for a significant number of people---their actual views about this question depend strongly on how far along the pregnancy in question happens to be.

For a significant number of people the answer is, "Legal up to a point"---which makes it important to ask, "Up to what point?" But none of the polls seem to ask this question.

For each n = 1, 2, 3, ..., what percentage of Americans would outlaw abortion-on-demand in week if they could do so? Is there any research on this?

Sunday, August 3, 2014

Notes from a cross-country drive

From Vermont to Detroit and back, October 2011

The meteor flew from left to right, halfway above the horizon, leaving behind an incandescent green trail that measured the width of my windshield. There was a sense of nearness, as if it weren't more than a couple of thousand feet high. The meteor moved more slowly than any I'd ever seen. In the end, the long green line forked into two dull-orange tracks: the meteor exploding in two. The pieces were extinguished, and the sky went black again.

Sitting on Scott's back deck, looking out over his neighbors' houses toward the flat horizon, I saw in the middle-distance a rank of towering maples, their huge canopies darkening to silhouettes as the evening fell. I'd forgotten about the ancient maple trees that still stand in the Detroit suburbs, like woody pins on a giant map.

The cornfields along my drive were being cut down to stubble. Good time of year to be a bird of prey. I saw plenty of falconiformes swooping down, coming up, or perching on fence-posts. Many held mice, or unidentifiable tasty-bits in their talons.

Passed through Ontario's level farmlands at one or two in the morning. A clear, cold sky had drawn vapor from the ground. Fog bunched itself in the hollows and lay roped across the land in silver skeins. Intermittently, all was white as my car pierced one of the swirling banks.

The beautiful weather in Detroit has followed me, and it's just the time to be tooling around the back-roads. The clouds are whipped cream and the hills are giant heaps of colored popcorn.

Concerning the meteor, I later searched the Web for similar reports and found these.

Friday, July 18, 2014

Book Review: Robert Hass (Ed.),The Essential Haiku

The Essential Haiku: Versions of Basho, Buson, and Issa

Edited and with Verse Translations by Robert Hass

Paperback, HarperCollins 1994

Years ago, a friend who was visiting Japan sent me a beautiful volume of haiku by Kobayashi Issa. I also needed a general collection of haiku for my library, so a few weeks ago I purchased The Essential Haiku, edited by former Poet Laureate Robert Hass. I'm very happy with it.

Hass covers the three great masters, Basho (1644–1694), Buson (1716–1783), and Issa (1763–1827). The language in the translations is gorgeous. In addition to the poems, there are some pages from the poets' journals. The book also includes introductory material for each poet (which looks interesting but I'm saving it for another day).

Here are just a few of the many poems that made a strong impression.

     Even in Kyoto—
hearing the cuckoo's cry—
     I long for Kyoto.

     Year after year
on the monkey's face
     a monkey's face.

     Felling a tree
and seeing the cut end—
     tonight's moon.

     That wren—
looking here, looking there.
     You lose something?

     Lime blossoms!
let's talk about the old days
     making dinner in the kitchen.

     New Year's Day—
everything is in blossom!
     I feel about average.

     These sea slugs,
they just don't seem

     A bee
staggers out
     of the peony.

     A snail
waving its horns

The journal entries include a section from Issa's Journal of My Father's Last Days, translated by Robert N. Huey. In these pages, Issa recorded his father's grueling illness and eventual death. An entry from the early days of the journal: 

Tears were streaming down my face. I just sat there, looking down and unable to say anything. Because of his kindness, my debt to Father is deeper than the matchless snows of Mount Fuji, unmelted by the summer sun, and deeper than a twice-dyed crimson. Yet I hadn't stayed at home and taken care of him. Instead, I had drifted around just like a floating cloud; before I could wonder whether I had gone east, off I was to the west. The days and nights rolled on like a wheel going downhill, and twenty-five years had passed. To have stayed away from my father's side till my hair was white as frost--I wondered whether even the Five Violations could be worse than this.

In my heart, I prostrated myself and thanked Father. But if I were to openly shed tears, it would surely make him feel even worse. So I wiped my eyes and said with a smile, "Now put things like that out of your mind. Just hurry up and get well." I gave him some medicine, and added, "If you get better soon, I'll become the perfect son of a farmer, the Yataro you used to know. I'll cut hay, plow the land, and really set your mind at rest. Please forgive me for what I've been until now." When Father heard this, his joy was boundless.

The most direct reflection of this time, among the poems in the book, is this:

     Last time, I think,
I'll brush the flies
     from my father's face.

Thursday, June 26, 2014

An Interesting Web Magazine: Public Domain Review

Recently I stumbled across a web magazine called The Public Domain Review. Here is what the homepage looks like today:

There are illustrated articles about such diverse things as the Krakatoa sunsets, the first children's picture book, medieval-era animal trials, and a time when Maupassant and Swinburne shared a pretty bizarre lunch.

Oh one more thing: I have made some updates to the previous post, about the brachistochrone problem.

Thursday, June 19, 2014

A Brachistochrone Problem

A dog running on a polished floor spies a bone out of the corner of its eye and begins scrabbling like mad to reach it.

Given the dog's initial speed and initial distance from the bone, and taking the dog's acceleration vector to be constant in magnitude, find the shortest time in which the dog may reach the bone.

Mathematically, the problem I'm posing is this: Given parameters \(d>0\), \(v>0\), and \(a>0\) (defined below), find the smallest time \(T>0\) so that there exists a piecewise-continuous direction function \(\theta(t)\) governing the motion \((x(t), y(t))\) as follows:

\[(x(0),y(0))=(0,d)\] \[(x(T),y(T))=(0,0)\]

There is no constraint on the final velocity.

I haven't solved the problem. I conjecture that the shortest time is

\[T=\sqrt{2}\cdot \frac{d}{v}\cdot\sigma\cdot\sqrt{1+\sqrt{1+\frac{1}{\sigma^2}}}\]
where \(\sigma\) is a "slipperiness factor" defined by \(\sigma = v^2/ad\). This value of \(T\) is achieved with a constant acceleration vector (i.e., \(\theta(t)\) constant). Here is a movie showing this motion:

This is better than a guess in that I've looked at some particular perturbations of the constant-acceleration trajectory and have also computed some specific non-nearby solutions as well. It'd be cool to see a faster, less obvious solution if there is one!

This problem is related to one solved a few years ago by Williams College faculty and students, concerning the optimal path for baserunning. More generally, both problems are about how best to navigate in space subject to a condition on the acceleration vector.

Be sure to let me know if you can improve upon my answer! And in the meantime, here's a funny video of dogs sliding on wood floors. (You can see the physics better if you mute the audio.)

Monday, May 26, 2014

The One Trick All Great Parents Know

Ha - just thought I'd try writing one of those Upworthy-style headlines. Anyway, it's fair to say that riddles are my new parenting trick. When the kids are bickering, I can usually press the reset button by giving them a new puzzle to chew on. Lately we've been doing "two-word rhyming riddles." The kids are starting to get the hang of it now, but most of the time I still give lots of hints. Enjoy!

Answer each riddle with two words that rhyme:

A large hog is a...
A kitten who eats a lot will grow up to be a... 
A rabbit who tells a lot of jokes is a...
A skinny baby horse is a...
A baby bear takes a bath in a...
When Donald stepped in glue, he was a...
When I dropped my lollipop at the beach, I had...
A black bird who can't fly fast is a…
A highchair for a boy chicken:
When I licked a bee, I got a...
A hammerhead playground is a...
If you fill the tub with numbers, you can take a...
An insect who doesn't like to be around others is a...
She rides a broom and has lots of money: 
Thunder comes from a...
Scary flashes in the sky:    
A boy humpback is a...
When the fox on Dora steals Pampers, he is a...
When father is unhappy, he is a...
When father is happy, he is a...
Can you make up your own two-word rhyming riddle?

For grownups: 

An averagely boring task has...
It comes out of a teakettle:
When the festivities began late, it was a...

Saturday, April 19, 2014

How I lost Fifteen Pounds (Not Spam!)

Once upon a time, I was a pretty good athlete. NCAA recruiting letter in my senior year of high school; set a record in track and field when I was in college; played a lot of basketball on playgrounds across the country. Then, as I got older, the demands of work took over and I basically didn't exercise for twenty years.

Eventually it began to sink in that my lifestyle was significantly lowering my odds of watching my kids graduate from high school or college. That lit a fire under me, and I did something about it. So I'm happy to say that after two decades of dissipation, I'm finally on an upswing. I wanted to share my model, in case it might help some other workaholics out there.

Here's the plan:

  • Exercise three times a week: on Friday, Saturday, and Sunday. 

  • Exercise for just 15 minutes--with high intensity.

  • Spend those 15 minutes on the exercise bike (to minimize risk of injury when returning to fitness). 

Some features of this plan:

1) I don't sacrifice any working time.

Although I certainly work on Fridays and on weekends, my schedule on weekends is very flexible, and to some extent that is true of Fridays as well.

The 15-minute time frame is essential. If need be, I can make my entire workout cycle fit into a half-hour--exercise, shower, and transit time. This lowers the competition with work.

Principle (1) is the most important reason for the success of my program. Twenty years of data has proven that work never loses--I finally got wise and stopped trying to beat an invincible opponent.

2) I don't miss any workouts when I travel for work.

It's hard to maintain a workout schedule when you're on the road. You might be on planes all day; the equipment in the fitness room might be dysfunctional; or the business at hand might simply take all of the available time. Often I'll be at dinner with colleagues who are bemoaning a missed workout. Neither of us will have exercised that day, but one of us is right on schedule. Given how the days fall, I rarely pack running shoes.

3) I don't sacrifice any sleep.

Many people with young kids and demanding jobs make room for exercise by sacrificing sleep. They wake up at 5am, or they extend the evening long after the kids have gone to bed. For me that would be a Pyrrhic victory. Why lose sleep?

4) I do have to work out during family time.

On Saturday and Sunday, there comes a point in the day when I slip out of the house to exercise. That is certainly time when I could have been playing with the kids. But again it matters that this is a weekend, when there is plenty of time to go around. And it matters as well that the workout is 15 minutes, so I'm not gone long. Meanwhile it's a positive thing that I don't have to spend any time working out during the week, when family time is scarce.

5) Vacations are a pain.

When visiting family or vacationing, the travel tends to be concentrated around weekends. So I always have to do research ahead of time to find out where I can buy a day pass to a gym. And then I have to fit the gym time into the vacation agenda. However, since the work-to-vacation ratio is enormous, this is a small drawback.

You might wonder about how diet plays into all this. The answer is, I basically eat whatever I want. Food is, and ought to be, a pleasurable thing. I can't be guilting myself all the time over food.

There are however some second-order effects. Like, I actually prefer the way I look I sometimes make eating decisions with that in mind. And unrelated to fitness, I paused my alcohol consumption recently in order to help my sleep and sinus problems. That has probably helped out waist-wise.

Looking ahead

I find that when I talk about my exercise plan, the first thing everyone does is try to optimize it. "Add another workout during the week." "Do some weights, too." "You need to work on your flexibility."

People. What I need to do is not spend another 20 years doing jack-shit.

Of course, I do expect this workout to be different in a year or two. But the simple plan above was like a miracle drug for turning my fitness around after decades of failure. Maybe it will work for you or someone you love!

UPDATE (5/26/14): Clarified the third part of the plan.

Sunday, April 13, 2014

Saturday School

Every Saturday at 10am is "Saturday School" at our house. It's only an hour or so. The important thing is that it's firmly on our calendars, so we schedule around it. Playdates, museum trips, and errands begin later than they used to; lazy mornings end a bit sooner. It's an extra commitment, though it's far short of the commitment homeschooling parents make on any given weekday.

Reading has always been a daily thing in our house, so Saturday School has just been about math. On any given Saturday, my wife and/or I might work with the kids on such things as:
  • A maze, a dot-to-dot, or some other warm-up activity--this gets them comfortable at their desks.
  • A worksheet that I make up (I just create it quickly while they're doing the warm-up activity). Our kids are at different stages mathematically, so I create separate worksheets for them. 
  • Some pages from this series of workbooks, which the kids really enjoy (individual grades here). We also have some word problem books that we draw from.
  • A dice game, using these fabulously glitzy dice that I found on Amazon.
  • Another fun dice game that we got at the Museum of Mathematics in New York. (This is my kids' favorite museum in the city.)
With the younger we're aiming to build up some number bonds, get to recall with some of the single-digit sums, get some practice with counting on, build understanding of two-digit numbers, use some properties of operations, and do some easy word problems (she's practicing reading too). With the elder, the main goals are number bonds, recall of single-digit sums, practice with making ten, and the harder kinds of word problems about the uses/meanings of addition and subtraction. I also include a multi-digit sum for her. (She thought it was really cool when she was able to calculate 3,258,152 + 5,131,646 lickety-split using the standard right-to-left algorithm. Of course, she doesn't really have a sense of those numbers, but that didn't stop her from getting the answer...such is the nature of an algorithm, after all. And computing such a stupendous sum made her feel like a little big shot!)

Here's one of the games we play with the pink dice. Roll the dice. Now mentally compute, or know from memory, the sum of the two values shown; record the sum on paper; and keep a running total. The first player to reach 100 points wins. It sounds boring, but they love the dice, and we've had some exciting contests that came down to the last roll. A friend says she plays a similar game, with the added wrinkle that if you roll a 1, you have to start all over again at zero. 

Clearly it's not a real curriculum by any means. But I think we might be seeing some learning progress even over the short time we've been doing this. We expect to do a bit more during the summer, maybe with some Core Knowledge lessons, and these phonics workbooks that worked so well for our elder when she was in kindergarten.

In any case, my wife and I are enjoying teaching our kids. It's exciting to watch their minds work. It's also a way to share with them our own values about persistence, effort, and the pleasure of academic ideas. And after a long week when it can seem that the family is a convoy of ships passing in the night, it's just nice to sit next to each other for an hour.

Friday, April 4, 2014

Some music puzzles

1. What is the most romantic song you know that doesn't use the word "love"?

No instrumentals, please--the song should have lyrics. And there shouldn't be any inflected forms (loved, loving, lover, etc.).

2. What's the rockingest song you know that doesn't use drums?


I think some good solutions to #1 are here.

My answer to #2 is pretty strong I think.

(For #2, a song with a very different feel also came to mind - not usually thought of as a rock song, though it did make #12 on the "Mainstream Rock" charts.)

These are subjective, so I suppose they're more like discussion questions. But they do have something in common with puzzles, in that they ask for an optimum subject to given constraints.

Friday, March 21, 2014

Soft Landing

A problem for physics-major types.

At the instant of time shown, a particle has the velocity vector shown. Find the constant acceleration vector that brings the particle to the given landing point with the smallest possible speed of impact.

Introduce any variables you like.

(Note, this problem has nothing to do with gravity or any other forces. Alternatively, if you prefer to think of it this way instead, you might say that there was initially no gravity or any other force, and your task is to create a uniform force field of your own.)

A follow-up:

Given any initial position vector, and given any initial velocity vector, does there always exist a constant acceleration vector such that the particle passes through the origin?

Monday, March 17, 2014

Polishing off that pizza

I know that I said I wouldn't be posting the solution to Sicilian Solution III: Pepperoni Protocol, but I think there have been enough developments to warrant one more post....

Reader jeff worked through a few approaches, including finding the solution shown below. The picture of the solution comes to us from Harold Jacobs---it works because the cut goes through the center of the rectangle; goes through the center of one of the circles; and separates the other two circles.

By the way, I should say that Harold was also a colleague of Martin Gardner's, and he reminded me about the nice problems in Gardner's book Penrose Tiles to Trapdoor Ciphers. For example, how do you bisect both halves of a yin-yang symbol with a single cut? This problem dates back to at least 1917.

The "Penrose" of Gardner's book title is of course Sir Roger Penrose. It was both a privilege and a pleasure to be one of Roger's research students during the 1990s. In addition to his researches in physics and mathematics, Roger is widely read in the history of mathematics and knows a lot about mathematical games. He has held the W.W. Rouse Ball Professorship in Mathematics at Oxford, which is named for another mathematician with a historical sensibility and a taste for puzzles.

This morning, my wife found herself facing two hungry little girls, both of them eyeing the last chocolate-chip pancake. She kept the peace with this Solomonic cut:

Google "pancake theorem" for some results along these lines. Some other references you might want to check out:

Hope you enjoyed these forays into bisection! Lately I've been thinking about a physics puzzle...if I get anywhere with it, I'll post. :-)

Thursday, March 13, 2014

Sicilian Solution III: Pepperoni Protocol

This time it's artisanal.

Sonny and Fredo are finishing a Sicilian pie. Both men are hungry, and there's only one slice left.

Nobody wants any trouble here. By making a single straight-line cut, can you produce two pieces with equal areas and equal ratios of cheese to pepperoni?

(I won't be posting the answer to this one - feel free to post in the comments.)

This puzzle and the last one are part of an interesting genre called bisection problems. I looked around online, but I couldn't find a good collection of problems to link to, just various items scattered here and there. But some classic problems in this area include bisecting a triangle; bisecting the area and perimeter of a shape simultaneously; bisecting two shapes simultaneously, and so on. By googling around, I found this book which looks interesting. If others have any gems in this area, feel free to link in the comments.

Hope you enjoyed the Sicilian Trilogy! Thanks as always for all the comments and emails.

Tuesday, March 11, 2014

Sicilian Solution II: The Sicilian Solution Solution

Sonny and Fredo are finishing a Sicilian pie. Both men are hungry, and there's only one slice left.

Nobody wants any trouble here. By making a single straight-line cut, can you produce two pieces with equal areas and equal ratios of crust to cheese?

That was the puzzle in my last post. Reader jeff posted the correct solution in the comments and also sent the following diagram:

Another friend used the same method too.

It was an honor to receive an email about this puzzle from Harold Jacobs, author of Mathematics: A Human Endeavor. Harold mentioned several different ways to solve the problem, including an elegant approach based on the fact that a line bisects the area of a rectangle if and only if the line passes through the center of the rectangle. I reproduce his diagram here:

Both diagrams are reminiscent of the genre of "proofs without words" (a book of which is here).

What if the dimensions of the slice are made general?

We can apply Harold's method: To share the cheese equally, cut through the center of the cheese rectangle. To share the crust equally, share the whole equally, thus, cut through the center of the whole. These two centers determine the cut:

The equation of this line is easy to find. The center of the cheese is point (a + (b/2), a + (c/2)). The center of the whole is point ((a + b)/2, (a + b)/2). The line through these two points is immediately found using the two-point formula as y = x + (1/2)(c - b). This is the desired cut.

Just as in the specific case that we started with, the desired cut has a slope of 1.

So next time you're in a pizzeria and there's trouble brewing, just make a 45-degree cut passing through the center of the cheese rectangle!

Another way to think about it is to go for a 45-degree cut that is equidistant from the two corners.

One can view the y-intercept (1/2)(c - b) as the height of the parallelogram that the bottom player deserves to have, in view of the "excess rectangle" of dimensions (c - b) x (a + b) that sits in the top player's territory atop the (a + b) x (a + b) square:

The parallelogram is half as tall and just as wide as the dashed rectangle, so it has half the area. Similarly, the crusty part of the parallelogram (a smaller parallelogram) has half the area of the crusty part of the dashed rectangle (a smaller rectangle). And likewise the cheesy part of the parallelogram has half the area of the cheesy part of the dashed rectangle.

For the sake of amusement, I also tried to come up with the most abstruse solution possible:

Indeed, Harold told me that he liked this puzzle precisely because there are so many ways to solve it. He gave it to some mathematician friends to see what they come up with--if we hear back, I'll let you know.

Thanks for all the responses! And get ready for Sicilian Solution III: Pepperoni Protocol, coming soon to a blog near you!

UPDATE 1/8/2015 - fixed a typo or error in the solution image, thanks to a reader who pointed out the mistake.

Tuesday, March 4, 2014

The Sicilian Solution

Sonny and Fredo are finishing a Sicilian pie. Both men are hungry, and there's only one slice left.

Nobody wants any trouble here. By making a single straight-line cut, can you produce two pieces with equal areas and equal ratios of crust to cheese?