Settling Accounts

In 1880 art collector Charles Ephrussi commissioned Manet to paint A Bundle of Asparagus for 800 francs.

When Manet delivered the painting, Ephrussi gave him 1,000 francs.

So later that year Manet delivered the small painting below with a note: “There was one [sprig] missing from your bundle.”

(Thanks, Jon.)


The following story is true. There was a little boy, and his father said, ‘Do try to be like other people. Don’t frown.’ And he tried and tried, but could not. So his father beat him with a strap; and then he was eaten up by lions.

Reader, if young, take warning by his sad life and death. For though it may be an honour to be different from other people, if Carlyle’s dictum about the 30 millions be still true, yet other people do not like it. So, if you are different, you had better hide it, and pretend to be solemn and wooden-headed. Until you make your fortune. For most wooden-headed people worship money; and, really, I do not see what else they can do. In particular, if you are going to write a book, remember the wooden-headed. So be rigorous; that will cover a multitude of sins. And do not frown.

— Oliver Heaviside, “Electromagnetic Theory,” in The Electrician, Feb. 23, 1900

(When asked the population of England, Thomas Carlyle had said, “Thirty million, mostly fools.”)

Black and White

gold chess problem

Henry Dudeney in Strand, June 1911: “It would be difficult to find a prettier little chess problem in three moves, produced from such limited material as a rook and a pawn, than the one given this month, by Dr. S. Gold. The novice will probably find the task of discovering the key move quite perplexing. White plays and checkmates in three moves.”

Click for Answer

Court Order

From Chapter 12 of Ken Follett’s novel The Pillars of the Earth:

‘My stepfather, the builder, taught me how to perform certain operations in geometry: how to divide a line exactly in half, how to draw a right angle, and how to draw one square inside another so that the smaller is half the area of the larger.’

‘What is the purpose of such skills?’ Josef interrupted.

‘Those operations are essential in planning buildings,’ Jack replied pleasantly, pretending not to notice Josef’s tone. ‘Take a look at this courtyard. The area of the covered arcades around the edges is exactly the same as the open area in the middle. Most small courtyards are built like that, including the cloisters of monasteries. It’s because these proportions are most pleasing. If the middle is bigger, it looks like a marketplace, and if it’s smaller, it just looks as if there’s a hole in the roof. But to get it exactly right, the builder has to be able to draw the open part in the middle so that it’s precisely half the area of the whole thing.’

How is this done? Inscribe a diamond within a square and then rotate it 45 degrees:

court order

A Silver Lining

The opening of England’s Liverpool and Manchester Railway in 1830 took a direful turn when William Huskisson, a member of Parliament for Liverpool, approached the Duke of Wellington’s railway carriage. Huskisson became so engrossed in their conversation that he failed to notice an oncoming train, and when he realized his danger and tried to climb into Wellington’s carriage, the door swung outward and deposited him in its path. His leg was badly mangled.

“Immediately after the accident, he was placed on the ‘Northumbrian’ — another of Stephenson’s engines — and raced to Liverpool at the then unprecedented speed of 36 m.p.h., with Stephenson himself as driver,” writes Ernest Frank Carter in Unusual Locomotives. “It was the news of this accident, and the speed of the engine, which was one of the causes of the immediate adoption and rapid spread of railways over the world. Thus was the death of the first person to be involved in a railway accident turned to some good account.”

Langley’s Adventitious Angles
Image: Wikimedia Commons

Edward Mann Langley, founder of the Mathematical Gazette, posed this problem in its pages in 1922:

ABC is an isosceles triangle. B = C = 80 degrees. CF at 30 degrees to AC cuts AB in F. BE at 20 degrees to AB cuts AC in E. Prove angle BEF = 30 degrees.

(Langley’s description makes no mention of D; perhaps this is at the intersection of BE and CF.)

A number of solutions appeared. One, offered by J.W. Mercer in 1923, proposes drawing BG at 20 degrees to BC, cutting CA in G. Now angle GBF is 60 degrees, and angles BGC and BCG are both 80 degrees, so BC = BG. Also, angles BCF and BFC are both 50 degrees, so BF = BG and triangle BFG is equilateral. But angles GBE and BEG are both 40 degrees, so BG = GE = GF. And angle FGE is 40 degrees, so GEF is 70 degrees and BEF is 30 degrees.

Miniatures Image: Wikimedia Commons
Image: Wikimedia Commons

Russian artist Anatoly Konenko works on an absurdly, almost unthinkably tiny scale. Trained as an engineer, he took up microminiature art in 1981, inventing his own instruments and techniques. Soon he was writing on grains of rice, poppy seeds, and even a human hair, and in 1994 he began to publish miniature versions of books by Mikhail Koltsov, Yevgeny Yevtushenko, and Alexander Pushkin. His 1996 edition of Chekhov’s Chameleon assembled 29 pages, three color illustrations, and a portrait of the author into a volume 0.9 millimeters square, at the time the smallest book in the world.

In other media, his feats include a 10-milliliter aquarium, a shod flea, a 1:40,000 scale balalaika, and a 17-millimeter chess set with 2-millimeter pieces. Above: a 3.2-millimeter proboscis midge holds a model of the Eiffel Tower. See his website for more — including a caravan of camels passing through the eye of a needle.


Pains and other such sensory processes may be long or short, continuous or intermittent; but in spite of Longfellow’s ‘long, long thoughts’, I do not think a thought (say, that the pack of cards is on the table, or that Geach’s arguments are fallacious) can significantly be called long or short; nor are we obliged to say that in that case every thought must be strictly instantaneous.

… [W]hat I am suggesting is that thoughts have not got all the kinds of time-relations that physical events, and I think also sensory processes, have. One may say that during half an hour by the clock such-and-such a series of thoughts occurred to a man; but I think it is impossible to find a stretch of physical events that would be just simultaneous, or even simultaneous to a good approximation, with one of the thoughts in the series. I think Norman Malcolm was right when he said at a meeting in Oxford that a mental image could be before one’s mind’s eye for just as long as a beetle took to crawl across a table; but I think it would be nonsense to say that I ‘was thinking’ a given thought for the period of the beetle’s crawl — the continous past of ‘think’ has no such use.

— Peter Geach, God and the Soul, 1969

Extra Credit

Critic Harold C. Schonberg called Leopold Godowsky’s Studies on Chopin’s Études “the most impossibly difficult things ever written for the piano”; Godowsky said they were “aimed at the transcendental heights of pianism.” In the “Badinage,” above, the pianist plays Chopin’s “Black Key” étude with the left hand while simultaneously playing the “Butterfly” étude with the right and somehow preserving the melodies of both. One observer calculated that this requires 1,680 independent finger movements in the space of about 80 seconds, an average of 21 notes per second. “The pair go laughing over the keyboard like two friends long ago separated, now happily united,” marveled James Huneker in the New York World. “After them trails a cloud of iridescent glory.”

The studies’ difficulty means that they’re rarely performed even today; Schonberg said they “push piano technique to heights undreamed of even by Liszt.” Only Italian pianist Francesco Libetta, above, has performed the complete set from memory in concert.

Even Up

rolling die puzzle

Suppose we cover a chessboard with 32 dominoes so that each domino covers two squares. What is the likelihood that there will be an even number of dominoes in each of the two orientations (horizontal and vertical)?

In fact this will always be the case. Consider the 32 squares in the odd-numbered horizontal rows. Each horizontal domino on the board covers either two of these squares or none of them. And each vertical domino covers exactly one of these squares. So the horizontal dominoes cover an even number of these squares (call it n), and the number of squares remaining in this group (32 – n) must also be even. This latter number is also equal to the number of vertical dominoes, so both quantities are even.

(By Vyacheslav Proizvolov.)