Advice in problem solving:
“You must always invert.” — Carl Gustav Jacob Jacobi
“Whenever you can, count.” — Francis Galton
“Each problem that I solved became a rule, which served afterwards to solve other problems.” — Descartes
“By studying the masters, not their pupils.” — Niels Henrik Abel
“Truth is the offspring of silence and meditation. I keep the subject constantly before me and wait ’til the first dawnings open slowly, by little and little, into a full and clear light.” — Isaac Newton
Twice widowed, English artist Mary Delany (1700-1788) took up a remarkable new career in her 70s: She created a series of detailed and botanically accurate portraits of plants, devising them from tissue paper and coloring them by hand:
With the plant specimen set before her she cut minute particles of coloured paper to represent the petals, stamens, calyx, leaves, veins, stalk and other parts of the plant, and, using lighter and darker paper to form the shading, she stuck them on a black background. By placing one piece of paper upon another she sometimes built up several layers and in a complete picture there might be hundreds of pieces to form one plant. It is thought she first dissected each plant so that she might examine it carefully for accurate portrayal …
She kept it up until she lost her eyesight at 88, filling 10 volumes with 985 of these “paper mosaiks.” Eventually they were bequeathed to the British Museum.
(Ruth Hayden, Mrs Delany: Her Life and Her Flowers, 1980.)
Here’s one way to end a composition: Johann Jakob Froberger’s Suite II in C major, Lamento sopra la dolorosa perdita della Real Msta di Ferdinando IV, Re de Romani, ends with a picture of the clouds of heaven welcoming the soul of Ferdinand IV as it climbs up a scale of three octaves.
It was written to lament the death of the King of the Romans in 1654. Elsewhere Froberger had marked the fatal fall of lutenist Charles Fleury down a flight of stairs with a descending scale of two octaves. Perhaps he was just very literal-minded.
(From Wilfrid Hodges, “The Geometry of Music,” in John Fauvel et al., Music and Mathematics, 2006.)
This sounds apocryphal, but it’s interesting. According to Texas lore, John Higginson, owner of the Memphis, El Paso & Pacific Railroad, faced a problem in the 1860s. He had committed to serve the route between Marshall, Texas, and Shreveport, La., but the Civil War had reduced his stock to three boxcars. How could he maintain regular service between two cities 40 miles apart with no engine?
He did it (the story goes) by loading a team of oxen into the first boxcar; freight and passengers into the second car; and the train’s crew and management into the third car. At Marshall, Higginson would start the train coasting down a long descending grade, and at the bottom they’d unload the oxen, hitch them to the front of the train, and drive them until they reached the top of the next hill. Then they’d load the oxen into the first car again and ride down the hill. At Shreveport they’d turn around and use the same method to get home.
In The Humor and Drama of Early Texas (2002), George U. Hubbard writes, “With gravity for the downgrades and oxen for the level areas and the upgrades, the little railroad managed to operate in both directions on a timely and consistent schedule.” With no competition on the Marshall-Shreveport line, Higginson (supposedly) maintained a profitable railroad with no engine at all.
Hubbard cites B.A. Botkin’s A Treasury of Railroad Folklore, from 1953. I find that the Southwestern Historical Quarterly ran an item giving essentially the same details in the early 1950s, citing a Texas newspaper of 1918, and Railway World mentioned it in 1911, quoting the Fort Worth Record (which calls it an “old railroad story”). I can’t find any corroboration beyond that. Good story, though!
A puzzle by V. Dubrovsky, from Quantum, January-February 1992:
In a certain planetary system, no two planets are separated by the same distance. On each planet sits an astronomer who observes the planet closest to hers. Prove that if the total number of planets is odd, there must be a planet that no one is observing.
Here’s one solution to the population problem: In 1967 Buckminster Fuller patented a giant floating geodesic sphere enclosing a city, hoping to reduce the economic and environmental costs of using land for housing. Each sphere would be a mile in diameter, enclosing an enormous volume of air that would be warmed by the sun, enabling it to carry buildings bearing thousands of people. The structural weight of even a half-mile sphere would be a thousandth the weight of the air inside, and heating the air even 1 degree would raise the whole structure like a hot-air balloon. By opening and closing polyethylene “curtains,” the occupants could keep the sphere floating at a chosen altitude. The cities could be tethered to mountaintops or float freely, enabling them (for example) to travel to disaster sites in a matter of days, and permitting humans to “converge and deploy around Earth without its depletion.”
“Cloud Nine is probably possible, but even Bucky didn’t expect to see one soon,” writes J. Baldwin in BuckyWorks: Buckminster Fuller’s Ideas for Today (1996). “He offered it as a jarring exercise, intended to stimulate the imaginative thinking we’re going to need if the billions of new Earth citizens predicted to arrive soon are to have decent housing.”
I don’t know why I find this so striking: It’s a diagram that accompanies the article on deer hunting in Denis Diderot’s Encyclopédie. Rather than presenting a single image with a caption, it combines a vignette of a deer hunt with an illustration of the antlers and piles of dung that are dropped by stags of different ages, together with a musical score showing the notes sounded by the hunting party at different stages of the pursuit.
“Taken as a whole the hunting plates offer few clues for reading their complex hybrid of imagery and notations as an ensemble,” write John Bender and Michael Marrinan in The Culture of the Diagram (2010). Perhaps as a result, the encyclopedia article requires nearly 10 pages.
“The Encyclopedia’s treatment of stag hunting is extraordinary for mobilizing a full range of written language, abstract and arbitrary notations, indexical icons, and pictorial tableaux in an attempt to diagram the highly ritualized, courtly craft of tracking animals under the Ancien Régime.”
Diderot provided similarly remarkable diagrams for “hunting at force,” the kill, boar hunting, and wolf hunting.
In his 1946 essay “Politics and the English Language,” George Orwell “translates” Ecclesiastes 9:11 into “modern English of the worst sort”:
I returned and saw under the sun, that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all.
Objective consideration of contemporary phenomena compels the conclusion that success or failure in competitive activities exhibits no tendency to be commensurate with innate capacity, but that a considerable element of the unpredictable must invariably be taken into account.
Orwell’s English instructor at St. Cyprian’s School, Cicely Vaughan Wilkes, had translated the parable of the Good Samaritan into “oratory and journalese” to illustrate the principles of good writing. Orwell’s companion Walter John Christie wrote that Wilkes had emphasized “simplicity, honesty, and avoidance of verbiage” — and pointed out that these qualities can be seen in Orwell’s writing.
Strategies to memorize π sometimes rely on devising a sentence with words of representative lengths. Isaac Asimov offered this one:
How I want a drink, alcoholic, of course, after the heavy lectures involving quantum mechanics!
Count the letters in each word and you get 3.14159265358979. That’s not the best strategy, though: What shall we do about zero? And it’s hard to reel off the digits impressively for friends when you have to stop and count the letters in each word.
Arthur Benjamin, a mathematician at Harvey Mudd College, suggests using a phonetic code instead:
1 = t or th or d
2 = n
3 = m
4 = r
5 = l
6 = sh, ch, or j
7 = k or hard g
8 = f or v
9 = p or b
0 = s or z
Once we’ve memorized this list, we can turn a whole string of digits into a single word simply by inserting vowels between the consonants. So, for example, 314 could be represented by meter, motor, meteor, matter, mother, etc. The consonants h, w, and y, which don’t appear in the list, can be used like additional vowels.
Using this method, it takes only three sentences to encode the first 60 digits of π:
3. 1415 926 5 3 58 97 9 3 2 384 6264
My turtle Pancho will, my love, pick up my new mover Ginger.
3 38 327 950 2 8841 971
My movie monkey plays in a favorite bucket.
69 3 99 375 1 05820 97494
Ship my puppy Michael to Sullivan’s backrubber!
Now we just need a way to remember the sentences …
(Arthur T. Benjamin, “A Better Way to Memorize Pi: The Phonetic Code,” Math Horizons 7:3 [February 2000], 17.)
Samuel Pepys wrote his famous diary in shorthand, but he took a further precaution when writing about his amorous adventures — he adopted words based on Spanish, French, and Italian:
“I did come to sit avec [with] Betty Michell, and there had her main [hand], which elle [she] did give me very frankly now, and did hazer [make] whatever I voudrais avec l’ [would have with her], which did plaisir [pleasure] me grandement [greatly].”
“The garbled foreign phrases he often used for sexual incidents had something to do with concealment perhaps, much more with his pleasure in marking off sexual experiences through special words and so heightening the excitement of reliving them,” writes Claire Tomalin in Samuel Pepys: The Unequalled Self. “It is the clever schoolboy as lover, showing off to himself in two ways at once.”