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:

video


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 include 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!